Proof. The proof of existence of a unique solution and the estimate (19 ) are due to [34] (\(d=2\)) and [35] (\(d=3\)). The proof of existence of a unique solution and the estimate (20 ) are due to [36] (non-convex and \(d=2\)) and [35] (convex). The estimate (21 ) is due to [16]. ◻

Proof. We choose \(v_{h}\in V_{h}\) and \(q\in [1,+\infty]\) arbitrarily. We denote by \(C\) a constant independent of \(v_{h}\) and \(q \in [1,+\infty]\). According to Lemma 3, there is \(\boldsymbol{\psi}_{h}\in [H^{1}(\Omega)\cap P_{r}(\mathcal{T}_{h})]^{d}\) such that \[\begin{align} \label{H195interpolation95ineq1} \Vert \boldsymbol{\psi}_{h} - \nabla v_{h} \Vert_{L^{q}(\Omega)} \leq C h \left( \Sigma_{F \in \mathcal{F}_{h}^{I}} h_{F}^{1-q}\Vert [ \nabla v_{h}] \Vert_{L^{q}(F)}^{q} \right)^{\frac{1}{q}}. \end{align}\tag{25}\]

Since the proof for the case \(d=2\) is similar but simpler, we assume \(d=3\) in the following. Since \(v_{h} = 0\) on \(\partial\Omega\), \(\boldsymbol{n}\times (\nabla v_{h}) = \boldsymbol{0}\) on \(\partial\Omega\). Then by (25 ), we have \[\begin{align} \label{H195interpolation95ineq2} & \Vert \boldsymbol{n} \times \boldsymbol{\psi}_{h} \Vert_{L^{q}(\partial\Omega)} = \Vert \boldsymbol{n} \times (\boldsymbol{\psi}_{h}-\nabla v_{h} )\Vert_{L^{q}(\partial\Omega)} \\ \nonumber \leq & C h^{1-\frac{1}{q}}\left( \Sigma_{F \in \mathcal{F}_{h}^{I}} h_{F}^{1-q}\Vert [ \nabla v_{h}] \Vert_{L^{q}(F)}^{q} \right)^{\frac{1}{q}} \leq C \Vert v_{h}\Vert_{W_{h}^{2,q}(\Omega)}. \end{align}\tag{26}\]

Let \(\{ \psi_{h}^{(i)} \}_{i=1}^{3}\) be the collection of all components of \(\boldsymbol{\psi}_{h}\). Since the domain \(\Omega\) is a polyhedra, we can assume that there is a face \(\tilde{F}\) on \(\partial\Omega\) which is parallel to both \(x_{1}\)-axis and \(x_{2}\)-axis. Then by (26 ), we have \[\begin{align} \label{H195interpolation95ineq3} \Vert \psi_{h}^{(1)} \Vert_{L^{q}(\tilde{F})} + \Vert \psi_{h}^{(2)} \Vert_{L^{q}(\tilde{F})} \leq C \Vert v_{h}\Vert_{W_{h}^{2,q}(\Omega)}. \end{align}\tag{27}\]

In the following, we will prove \[\begin{align} \label{H195interpolation95ineq4} \Vert \psi_{h}^{(i)} \Vert_{L^{q}(\Omega)} \leq C \left( \Vert \psi_{h}^{(i)} \Vert_{L^{q}(\tilde{F})} + \Vert \nabla \psi_{h}^{(i)} \Vert_{L^{q}(\Omega)} \right), \qquad \forall i\in \{1,2\}. \end{align}\tag{28}\] In fact, if (28 ) holds, then (25 ,27 ) imply \(\Vert \dfrac{\partial v_{h}}{\partial x_{1}} \Vert_{L^{q}(\Omega)} + \Vert \dfrac{\partial v_{h}}{\partial x_{2}} \Vert_{L^{q}(\Omega)} \leq C \Vert v_{h}\Vert_{W_{h}^{2,q}(\Omega)}\). Then by applying the same argument to another face on \(\partial\Omega\) which is not parallel to \(\tilde{F}\), we will have \(\Vert \nabla v_{h} \Vert_{L^{q}(\Omega)} \leq C \Vert v_{h}\Vert_{W_{h}^{2,q}(\Omega)}\).

Therefore, we only need to prove (28 ). For the sake of simplicity, we will only prove (28 ) for \(i=1\). We decompose the domain \(\Omega\) into finitely many non-overlapping simplexes \(\{ T_{i} \}_{i=1}^{N}\), such that \(\overline{\Omega} = \cup_{i=1}^{N} \overline{T}_{i}\). We assume this decomposition of \(\Omega\) is conforming such that it is like a conforming mesh of \(\Omega\). We would like to point out that this decomposition of \(\Omega\) is not relevant to the mesh \(\mathcal{T}_{h}\), such that \(N\) is independent of \(h\). If we take an element \(K\in\mathcal{T}_{h}\) arbitrarily, \(K\) may have non-trivial overlapping with two or more simplexes among \(\{ T_{i} \}_{i=1}^{N}\).

Without losing of generality, we assume that a triangular face \(F_{1}\) of \(T_{1}\) is contained in \(\tilde{F}\). Let \(T\) be an arbitrary simplex among \(\{ T_{i} \}_{i=2}^{N}\). By relabelling \(\{ T_{i} \}_{i=2}^{N}\), there is a positive integer \(1< \tilde{N} \leq N\) such that \(\partial T_{1} \cap \partial T_{2} = F_{2}\), \(\partial T_{2} \cap \partial T_{3} = F_{3}\),\(\cdots\),\(\partial T_{\tilde{N}-1}\cap \partial T_{\tilde{N}} = F_{\tilde{N}}\) where \(F_{2},\cdots,F_{\tilde{N}}\) are triangular faces. We claim that for any \(1\leq i \leq N\), \[\begin{align} \label{H195interpolation95ineq5} \Vert \psi_{h}^{(1)} \Vert_{L^{q}(T_{i})} + \Vert \psi_{h}^{(1)} \Vert_{L^{q}(F_{i}^{\prime\prime})} \leq C \left( \Vert \psi_{h}^{(1)} \Vert_{L^{q}(F_{i}^{\prime})} + \Vert \nabla \psi_{h}^{(1)} \Vert_{L^{q}(T_{i})} \right), \end{align}\tag{29}\] where \(F_{i}^{\prime}\) and \(F_{i}^{\prime\prime}\) can be any two triangular faces of \(T_{i}\). If (29 ) holds, then (28 ) is true for \(i=1\) because (27 ) holds and \(N\) is independent of \(h\).

Therefore, we only need to prove (29 ). Since \(\psi_{h}^{(1)} \in H^{1}(\Omega)\cap P_{r}(\mathcal{T}_{h})\), \(\psi_{h}^{(1)}\) is globally Lipschitz on \(\overline{\Omega}\). So \(\psi_{h}^{(1)}\) is globally Lipschitz on \(\overline{T_{i}}\) for any \(1\leq i \leq N\). We notice that the Lipschitz constant of \(\psi_{h}^{(i)}\) on \(\overline{T}_{i}\) is bounded by (independent of \(h\)) \(\Vert \nabla \psi_{h}^{(i)} \Vert_{L^{\infty}(T_{i})}\) for any \(1\leq i \leq N\). So (29 ) holds for \(q = + \infty\).

Now we are going to prove (29 ) for \(q \in [1,+\infty)\). Via affine transformation, we can assume \(T_{i}\) is the simplex determined by the four vertexes \((0,0,0)\), \((1,0,0)\), \((0,1,0)\) and \((0,0,1)\). We further assume \(F_{i}^{\prime}\) is the triangular face through \((0,0,0)\), \((1,0,0)\) and \((0,1,0)\), and \(F_{i}^{\prime\prime}\) is the triangular face through \((1,0,0)\), \((0,1,0)\) and \((0,0,1)\). Since \(\psi_{h}^{(i)}\) is globally Lipschitz on \(\overline{T}_{i}\), we have that for any \((x_{1},x_{2},x_{3}) \in \overline{T}_{i}\), \[\begin{align} \label{H195interpolation95ineq6} \psi_{h}^{(1)}(x_{1},x_{2},x_{3}) = \psi_{h}^{(1)}(x_{1},x_{2},0) + \int_{0}^{x_{3}} \dfrac{\partial \psi_{h}^{(1)}}{\partial x_{3}}(x_{1},x_{2},s) ds. \end{align}\tag{30}\] Since \(\psi_{h}^{(1)} \in H^{1}(\Omega)\cap P_{r}(\mathcal{T}_{h})\), we have that for any \((x_{1},x_{2},0) \in \overline{F}_{i}^{\prime}\), \(\dfrac{\partial \psi_{h}^{(1)}}{\partial x_{3}}(x_{1},x_{2},s)\) on the right hand side of (30 ) is the same as the weak derivative of \(\psi_{h}^{(i)}\) with respect to \(x_{3}\) on the line segment \(l(x_{1},x_{2}):=\{ (x_{1},x_{2},s): 0< s < 1 - x_{1} - x_{2} \}\) except at most finitely many points. In fact, these points are intersections between the line segment \(l(x_{1},x_{2})\) and interior mesh interfaces in \(\mathcal{F}_{h}^{I}\) which are not parallel to \(l(x_{1},x_{2})\). Therefore, (30 ) implies that for any \(q \in [1,+\infty)\), \[\begin{align} & \Vert \psi_{h}^{(1)} \Vert_{L^{q}(F_{i}^{\prime\prime})} \leq \left(\sqrt{3} \int_{F_{i}^{\prime}} \vert \psi_{h}^{(1)}(x_{1},x_{2},1-x_{1}-x_{2}) \vert^{q}dx_{1}dx_{2} \right)^{\frac{1}{q}} \\ \leq & \sqrt{3} \left(\int_{F_{i}^{\prime}} \vert \psi_{h}^{(1)}(x_{1},x_{2},1-x_{1}-x_{2}) \vert^{q}dx_{1}dx_{2} \right)^{\frac{1}{q}} \leq C \left( \Vert \psi_{h}^{(1)} \Vert_{L^{q}(F_{i}^{\prime})} + \Vert \nabla \psi_{h}^{(1)}\Vert_{L^{q}(T_{i})}\right), \\ & \Vert \psi_{h}^{(1)} \Vert_{L^{q}(T_{i})} \leq C \left( \Vert \psi_{h}^{(1)} \Vert_{L^{q}(F_{i}^{\prime})} + \Vert \nabla \psi_{h}^{(1)}\Vert_{L^{q}(T_{i})}\right). \end{align}\]

Therefore, we have that (29 ) for \(q \in [1,+\infty]\). We can conclude that the proof is complete. ◻

Proof. We choose \(\boldsymbol{x}_{0} \in \overline{\Omega}\) arbitrarily. For any \(R>0\), we denote by \(\eta\) the cut-off function to be \(1\) inside of \(\mathcal{B}_{R}(\boldsymbol{x}_{0}):= \{ \boldsymbol{x}\in \mathbb{R}^{3}: \vert \boldsymbol{x}- \boldsymbol{x}_{0}\vert < R \}\) and to be \(0\) outside of \(\mathcal{B}_{2R}(\boldsymbol{x}_{0}):= \{ \boldsymbol{x}\in \mathbb{R}^{3}: \vert \boldsymbol{x}- \boldsymbol{x}_{0}\vert < 2R \}\).

We define \(v = \eta \phi\). We notice that in \(\Omega\), \[\begin{align} & A:D^{2}v = A:D^{2}(\eta \phi) \\ = & \eta A:D^{2}\phi + 2 A:(\nabla \eta \otimes \nabla \phi) + \phi A:D^{2} \eta \\ = & - \eta \left( \boldsymbol{b}\cdot \nabla \phi + c \phi \right) + 2 A:(\nabla \eta \otimes \nabla \phi) + \phi A:D^{2} \eta, \end{align}\] since \(A:D^{2}\phi + \boldsymbol{b}\cdot \nabla \phi + c \phi = 0\) in \(\Omega\). Therefore, by Sobolev’s embedding inequality, we have that \[\begin{align} \label{improved95regularity1} A:D^{2}v \in L^{q}(\Omega) \text{ with } q := \min (\dfrac{3p}{3-p}, 2) > \frac{3}{2}. \end{align}\tag{31}\] Obviously, \(1 < p < q\).

We define \(A_{0} = A(\boldsymbol{x}_{0}) \in \mathbb{R}^{3 \times 3}\). We define the linear mapping \(M: L^{\tilde{q}}(\Omega) \rightarrow W^{2,\tilde{q}}(\Omega) \cap W_{0}^{1,\tilde{q}}(\Omega)\) by \[\begin{align} \nabla \cdot \left( A_{0} \nabla M(f) \right) = f,\qquad \forall f \in L^{\tilde{q}}(\Omega), \forall 1 < \tilde{q} \leq q. \end{align}\] By Lemma 1, the mapping \(M\) is not only well-defined but also continuous for any \(1 < \tilde{q} \leq q\).

It is easy to see that in \(\Omega\) \[\begin{align} A_{0}:D^{2} v = (A_{0} - A):D^{2}v + A:D^{2}v. \end{align}\] Since \(1 < p < q\), we can apply the mapping \(M\) on both sides of the above equation, to obtain that \[\begin{align} v = M \left( (A_{0} - A):D^{2}v \right) + M\left( A:D^{2} v \right) \text{ in } \Omega. \end{align}\]

We define \(B_{2R}(\boldsymbol{x}_{0}):=\{ \boldsymbol{x}\in \Omega: \vert \boldsymbol{x}- \boldsymbol{x}_{0} \vert < 2R\} = \mathcal{B}_{2R}(\boldsymbol{x}_{0}) \cap \Omega\). We define a linear mapping \(T : W^{2,\tilde{q}}(B_{2R}(\boldsymbol{x}_{0})) \rightarrow W^{2,\tilde{q}}(B_{2R}(\boldsymbol{x}_{0}))\) by \[\begin{align} \label{map95T} Tw = \left( M \tilde{w}\right)|_{B_{2R}(\boldsymbol{x}_{0})} \end{align}\tag{32}\] where \(w \in W^{2,\tilde{q}}(B_{2R}(\boldsymbol{x}_{0}))\) and \[\begin{align} \label{tilde95w95T} \tilde{w} = (A_{0} - A):D^{2} w \text{ in } B_{2R}(\boldsymbol{x}_{0}),\qquad \tilde{w} = 0 \text{ in } \Omega \setminus \overline{B_{2R}(\boldsymbol{x}_{0})}. \end{align}\tag{33}\] Here \(1 < \tilde{q} \leq q\). Therefore, for any \(1 < \tilde{q} \leq q\), we have that \[\begin{align} & \Vert Tw \Vert_{W^{2,\tilde{q}}(B_{2R}(\boldsymbol{x}_{0}))} \leq C_{\tilde{q}} \Vert \tilde{w}\Vert_{L^{\tilde{q}}(\Omega)} \\ = & C_{\tilde{q}} \Vert (A_{0} - A):D^{2} w \Vert_{L^{\tilde{q}}(B_{2R}(\boldsymbol{x}_{0}))} \\ \leq & C_{\tilde{q}} \Vert A_{0} - A \Vert_{L^{\infty}(B_{2R}(\boldsymbol{x}_{0}))} \Vert w\Vert_{W^{2,\tilde{q}}(B_{2R}(\boldsymbol{x}_{0}))}, \qquad \forall w \in W^{2,\tilde{q}}(B_{2R}(\boldsymbol{x}_{0})). \end{align}\] The constant \(C_{\tilde{q}}\) is due to Lemma 1.

Since \(A \in [C^{0}(\overline{\Omega})]^{3 \times 3}\), there is a \(R_{0} > 0\) such that \(\Vert T\Vert \leq \frac{1}{2}\) from \(W^{2,\tilde{q}}(B_{2R_{0}}(\boldsymbol{x}_{0}))\) to \(W^{2,\tilde{q}}(B_{2R_{0}}(\boldsymbol{x}_{0}))\) for all \(p \leq \tilde{q} \leq q\). By (31 ) and Lemma 1, \(M\left( A:D^{2} v \right) \in L^{\tilde{q}}(\Omega)\) for any \(1 < \tilde{q} \leq q\). Therefore for any \(p \leq \tilde{q} \leq q\), there is a unique \(w \in W^{2,\tilde{q}}(B_{2R_{0}}(\boldsymbol{x}_{0}))\) such that \[\begin{align} \label{w95T95unique} w = T w + M\left( A:D^{2} v \right)|_{B_{2R_{0}(\boldsymbol{x}_{0})}} \quad \text{ in }B_{2R_{0}(\boldsymbol{x}_{0})}. \end{align}\tag{34}\]

On the other hand, since \(v = 0\) in \(\Omega \setminus \overline{B_{2R}(\boldsymbol{x}_{0})}\) and \(v \in W^{2,p}(\Omega)\) (\(1< p < q\)), (32 ) and (33 ) imply that \[\begin{align} T (v|_{B_{2R_{0}}(\boldsymbol{x}_{0})}) = M\left( (A_{0} - A):D^{2} v \right)|_{B_{2R_{0}}(\boldsymbol{x}_{0})}. \end{align}\] Then we have \[\begin{align} \label{map95T95contraction} & v|_{B_{2R_{0}}(\boldsymbol{x}_{0})} = M(A_{0}:D^{2}v)|_{B_{2R_{0}}(\boldsymbol{x}_{0})} \\ \nonumber = & T (v|_{B_{2R_{0}}(\boldsymbol{x}_{0})}) + M\left( A:D^{2} v \right)|_{B_{2R_{0}}(\boldsymbol{x}_{0})} \quad \text{ in } B_{2R_{0}}(\boldsymbol{x}_{0}). \end{align}\tag{35}\] (34 ) and (35 ) imply that \(\phi |_{B_{R_{0}}(\boldsymbol{x}_{0})} \in W^{2,q}(B_{R_{0}}(\boldsymbol{x}_{0}))\).

Since \(\boldsymbol{x}_{0} \in \overline{\Omega}\) is chosen arbitrarily, we can conclude that \(\phi \in W^{2,q}(\Omega)\) where \(q = \min (\dfrac{3p}{3 - p} ,2) > \frac{3}{2}\). So \(\phi \in C^{0}(\overline{\Omega})\). Furthermore, by the interior estimate in [23], we have \(\phi \in W_{\text{loc}}^{2,3}(\Omega)\) since \(A:D^{2} \phi + \boldsymbol{b}\cdot \nabla \phi + c \phi = 0\) in \(\Omega\). Therefore, by the ABP estimate [23], we can conclude that \(\phi = 0\) \(\Omega\). ◻

Proof. When \(d=2\), \(W^{2,p}(\Omega) \Subset C^{0}(\overline{\Omega})\) due to Sobolev’s embedding theorem. Furthermore, by the interior estimate in [23], we have \(u \in W_{\text{loc}}^{2,2}(\Omega)\). Therefore, Theorem 1 can be proven by mimicking the proofs of [30] and [30] with the global \(W^{2,p}\) estimate of Poisson equation (see [34]) and the Alexandroff–Bakelman–Pucci estimate estimate [23].

Now we focus on the case \(d=3\). Due to [30], Theorem 1 holds if we further assume \(\Omega\) is a convex polyhedra in \(\mathbb{R}^{3}\) and \(\frac{3}{2} < p \leq 2\). In fact, if we replace [40] by Lemma 1 in the proof of [30], it is easy to have that for any \(1 < p \leq 2\) and for any \(w \in W^{2,p}(\Omega) \cap W_{0}^{1,p}(\Omega)\), \[\begin{align} \label{global95estimate195A95continuous} \Vert w\Vert_{W^{2,p}(\Omega)} \leq C_{p}^{\prime} \left( \Vert A:D^{2}w + \boldsymbol{b}\cdot\nabla w + cw\Vert_{L^{p}(\Omega)} + \Vert w \Vert_{L^{p}(\Omega)} \right). \end{align}\tag{37}\] Here \(C_{p}^{\prime}\) depends on \(1 < p \leq 2\) and \(\Omega\) is an open bounded convex domain in \(\mathbb{R}^{3}\).

As the proof of [30], Theorem 1 will be proven via (37 ) and the proof of contraction with the uniqueness of solution in \(W^{2,p}(\Omega) \cap W_{0}^{1,p}(\Omega)\): \[\begin{align} A:D^{2}w + \boldsymbol{b}\cdot \nabla w + cw = 0 \text{ in } \Omega \text{ with } w \in W^{2,p}(\Omega) \cap W_{0}^{1,p}(\Omega) \Longrightarrow u = 0 \text{ in } \Omega. \end{align}\]

When \(\frac{3}{2} < p \leq 2\), it is straightforward by the Alexandroff–Bakelman–Pucci estimate estimate with the fact \(W^{2,p}(\Omega) \Subset C^{0}(\overline{\Omega})\) (see the proof of [30]). Therefore, in order to complete the proof of Theorem 1, we only need the uniqueness for \(1<p \leq \frac{3}{2}\), which is provided by Lemma 5. Therefore, we can conclude that the proof of Theorem 1 is complete. ◻

Proof. Obviously, we have \[\begin{align} \label{H195estimate} \Vert \nabla (w_{h} - w)\Vert_{L^{2}(\Omega)} \leq C \Vert \nabla(\chi_{h} - w) \Vert_{L^{2}(\Omega)}, \qquad \forall \chi_{h} \in V_{h}. \end{align}\tag{41}\] By [41] (\(d=3\)), [42] (\(d=2\)), \[\begin{align} \label{W1infinity95estimate} \Vert \nabla (w_{h} - w)\Vert_{L^{\infty}(\Omega)} \leq C \Vert \nabla(\chi_{h} - w) \Vert_{L^{\infty}(\Omega)}, \qquad \forall \chi_{h} \in V_{h}, \end{align}\tag{42}\] if \(h <h_{0}\). Here the constant \(C\) in (41 ,42 ) may depend on \(\tilde{A}_{0}\). Therefore, according to (41 ,42 ), there is a constant \(C\) such that for any \(2 \leq p \leq +\infty\), if \(h<h_{0}\), \[\begin{align} \label{W1p95estimate951} \Vert w_{h} - w \Vert_{W^{1,p}(\Omega)} \leq C \Vert \chi_{h} - w \Vert_{W^{1,p}(\Omega)}, \qquad \forall \chi_{h} \in V_{h}. \end{align}\tag{43}\]

Now, for any \(p \in (1,2)\), we denote by \(p^{\prime} \geq 2\) such that \(\frac{1}{p} + \frac{1}{p^{\prime}}\). We choose \(\chi_{h} \in V_{h}\) arbitrarily. Then by Lemma 1, we have for any \(p \in (1,2)\) if \(\Omega\) is a convex polyhedra in \(\mathbb{R}^{d}\) (\(d=2,3\)) and for any \(p \in (\frac{4}{3}-\epsilon_{1},2]\) if \(\Omega\) is a Lipschitz polygon in \(\mathbb{R}^{2}\), \[\begin{align} & \Vert w_{h} - \chi_{h}\Vert_{W^{1,p}(\Omega)} \leq C \Vert \nabla \cdot ( \tilde{A}_{0}\nabla (w_{h} - \chi_{h}) ) \Vert_{W^{-1,p}(\Omega)} \\ = & C \sup_{\tilde{v} \in W_{0}^{1,p^{\prime}}(\Omega)} \dfrac{-( \nabla \cdot ( \tilde{A}_{0}\nabla (w_{h} - \chi_{h}), \tilde{v}))_{\Omega}} {\Vert \tilde{v}\Vert_{W^{1,p^{\prime}}(\Omega)}} = C \sup_{\tilde{v} \in W_{0}^{1,p^{\prime}}(\Omega)} \dfrac{( \tilde{A}_{0}\nabla (w_{h} - \chi_{h}), \nabla \tilde{v})_{\Omega}} {\Vert \tilde{v}\Vert_{W^{1,p^{\prime}}(\Omega)}} \\ = & C \sup_{\tilde{v} \in W_{0}^{1,p^{\prime}}(\Omega)} \dfrac{( \tilde{A}_{0}\nabla (w_{h} - \chi_{h}), \nabla \tilde{v}_{h})_{\Omega}} {\Vert \tilde{v}\Vert_{W^{1,p^{\prime}}(\Omega)}} = C \sup_{\tilde{v} \in W_{0}^{1,p^{\prime}}(\Omega)} \dfrac{( \tilde{A}_{0}\nabla (w - \chi_{h}), \nabla \tilde{v}_{h})_{\Omega}} {\Vert \tilde{v}\Vert_{W^{1,p^{\prime}}(\Omega)}}, \end{align}\] where \(\tilde{v}_{h} \in V_{h}\) satisfying \[\begin{align} (\tilde{A}_{0}\nabla \tilde{v}_{h}, \nabla \tilde{w}_{h})_{\Omega} = (\tilde{A}_{0}\nabla \tilde{v}, \nabla \tilde{w}_{h}), \qquad \forall \tilde{w}_{h} \in V_{h}. \end{align}\] By (43 ), \[\begin{align} \Vert \tilde{v}_{h}\Vert_{W^{1,p^{\prime}}(\Omega)} \leq C \Vert \tilde{v}\Vert_{W^{1,p^{\prime}}(\Omega)}. \end{align}\] So we have \[\begin{align} \Vert w_{h} - \chi_{h}\Vert_{W^{1,p}(\Omega)} \leq C \Vert w - \chi_{h}\Vert_{W^{1,p}(\Omega)}. \end{align}\] Therefore, for any \(p \in (1,2)\), we also have that if \(h<h_{0}\), \[\begin{align} \label{W1p95estimate952} \Vert w_{h} - w \Vert_{W^{1,p}(\Omega)} \leq C \Vert \chi_{h} - w \Vert_{W^{1,p}(\Omega)}, \qquad \forall \chi_{h} \in V_{h}. \end{align}\tag{44}\]

(43 , 44 ) imply that for any \(p\in (1,+\infty]\), \[\begin{align} \Vert w_{h} - w \Vert_{W^{1,p}(\Omega)} \leq C \inf_{\chi_{h}\in V_{h}} \Vert \chi_{h} - w \Vert_{W^{1,p}(\Omega)}. \end{align}\] Therefore, the proof of the estimates (40 ) is complete. ◻

Proof. We choose \(w_{h} \in V_{h}\) arbitrarily. We notice that \(\mathcal{L}_{\tilde{A}_{0},h}w_{h} \in V_{h}\). We denote by \(w\in H_{0}^{1}(\Omega)\) satisfying \[\begin{align} \tilde{A}_{0}: D^{2} w = \nabla \cdot \left( \tilde{A}_{0} \nabla w\right) = \mathcal{L}_{\tilde{A}_{0},h}w_{h} \quad \text{ in } \Omega. \end{align}\]

According to (38 ), \[\begin{align} \label{fem95projection} (\tilde{A}_{0}\nabla w_{h}, \nabla v_{h})_{\Omega} = (\tilde{A}_{0}\nabla w, \nabla v_{h})_{\Omega}, \qquad \forall v_{h} \in V_{h}. \end{align}\tag{46}\]

Then by (46 ), the construction of \(w\) and Lemma 6, we have that for any \(p\in (1,+\infty]\) and \(0< h < h_{0}\), \[\begin{align} \label{W1p95estimate95constant95A95complete} \Vert w_{h} - w \Vert_{W^{1,p}(\Omega)} \leq C \inf_{\chi_{h}\in V_{h}} \Vert \chi_{h} - w \Vert_{W^{1,p}(\Omega)}\leq C h \Vert w\Vert_{W^{2,p}(\Omega)}. \end{align}\tag{47}\] (47 ) and Lemma 1 imply that for any \(p \in (1, 2]\) if \(\Omega\) is a convex polyhedra in \(\mathbb{R}^{d}\) (\(d=2,3\)) and for any \(p \in (\frac{4}{3}-\epsilon_{1}, \frac{4}{3}) + \epsilon_{2}\), if \(h < h_{0}\), \[\begin{align} \label{W1p95estimate95constant95A} \Vert w_{h} - w \Vert_{W^{1,p}(\Omega)} \leq C h \Vert w\Vert_{W^{2,p}(\Omega)} \leq CC^{\prime} h \Vert \mathcal{L}_{\tilde{A}_{0},h}w_{h}\Vert_{L^{p}(\Omega)}. \end{align}\tag{48}\] Here the constant \(C^{\prime}\) may also depend on \(\tilde{A}_{0}\).

We denote by \(\boldsymbol{\phi}_{h}\) the standard \(L^{2}\)-orthogonal projection of \(\nabla w\) into \([P_{0}(\mathcal{T}_{h})]^{d}\). Then by the discrete trace inequality, we have \[\begin{align} \label{w1p95estiamte95constant95A95projection} & \Vert \boldsymbol{\phi}_{h} - \nabla w \Vert_{L^{p}(\Omega)} \leq C h \Vert w\Vert_{W^{2,p}(\Omega)} \leq C C^{\prime} h \Vert \mathcal{L}_{\tilde{A}_{0},h}w_{h}\Vert_{L^{p}(\Omega)},\\ \nonumber & \left( \Sigma_{F \in \mathcal{F}_{h}^{I}} h_{F}^{1-p}\Vert [\boldsymbol{\phi}_{h}] \Vert_{L^{p}(F\cap \overline{D})}^{p} \right)^{\frac{1}{p}} \leq C \Vert w\Vert_{W^{2,p}(\Omega)} \leq C C^{\prime} \Vert \mathcal{L}_{\tilde{A}_{0},h}w_{h}\Vert_{L^{p}(\Omega)}. \end{align}\tag{49}\]

Then by (48 ,49 ) and the discrete inverse inequality, we can conclude that the proof is complete. ◻

Proof. By (38 ), \(\mathcal{L}_{\tilde{A}_{0},h}w_{h}\) is the same as \(\mathcal{L}_{0,h}\) in [22]. Therefore, by replacing [22] by Lemma 7, the proof in [22] is also valid for this Lemma. It is easy to verify that the proof of [22] is independent of the choice of \(\boldsymbol{x}_{0}\), \(R\) and \(R^{\prime}\). ◻

Proof. We denote by \(B_{t}:= \{ \boldsymbol{x}\in \Omega: \vert \boldsymbol{x}- \boldsymbol{x}_{0} \vert < t \} \subset \Omega\) for any \(t>0\). For any \(v_{h} \in V_{h}(B_{t})\), we have \[\begin{align} ((\mathcal{L}_{\tilde{A},h} - \mathcal{L}_{\tilde{A}_{0},h})w, v_{h})_{\Omega} = \int_{\Omega}\left((\tilde{A}_{0} - \tilde{A}):\overline{\nabla}_{h}(\nabla w)\right)v_{h} d\boldsymbol{x}. \end{align}\] We recall that \(V_{h}(B_{t})=\{ v \in V_{h}: v|_{\Omega \setminus B_{t}} = 0\}\). We define \(\mathcal{T}_{t,h} = \{ K \in \mathcal{T}_{h}: \mathcal{H}^{d}(K \setminus B_{t}) = 0 \}\), where \(\mathcal{H}^{d}(\cdot)\) is the \(d\)-dimensional Hausdorff measure.

We define \(p^{\prime}\in (1, + \infty)\) such that \(\frac{1}{p} + \frac{1}{p^{\prime}} = 1\). So we have \[\begin{align} & ((\mathcal{L}_{\tilde{A},h} - \mathcal{L}_{\tilde{A}_{0},h})w, v_{h})_{\Omega} = \Sigma_{K \in \mathcal{T}_{t,h}}\int_{K}\left((\tilde{A}_{0} - \tilde{A}):\overline{\nabla}_{h}(\nabla w)\right) v_{h} d\boldsymbol{x}\\ \leq & \Vert \tilde{A}_{0} - \tilde{A} \Vert_{L^{\infty}(B_{t})} \left( \Sigma_{K \in \mathcal{T}_{t,h}} \Vert \overline{\nabla}_{h}(\nabla w)\Vert_{L^{p}(K)}^{p} \right)^{\frac{1}{p}} \Vert v_{h}\Vert_{L^{p^{\prime}}(\Omega)}. \end{align}\]

We denote by \(\mathcal{F}_{t,h}\) the collection of all mesh interface of \(\mathcal{T}_{t,h}\). By the construction of \(\mathcal{T}_{t,h}\), we have \[\begin{align} \mathcal{H}^{d-1}(F \cap \overline{B_{t}}) = \mathcal{H}^{d-1}(F), \qquad \forall F \in \mathcal{F}_{t,h}. \end{align}\] Then with Definition 1 and (9 ), we have \[\begin{align} & \Sigma_{K \in \mathcal{T}_{t,h}} \Vert \overline{\nabla}_{h}(\nabla w)\Vert_{L^{p}(K)}^{p} \\ \leq & C_{*} \left(\Sigma_{K \in \mathcal{T}_{t,h}} \Vert D_{h}^{2} w\Vert_{L^{p}(K)}^{p} + \Sigma_{F \in (\mathcal{F}_{t,h} \cap \mathcal{F}_{h}^{I}) }h_{F}^{1-p}\Vert [\nabla w]\Vert_{L^{p}(F)}^{p} \right) \leq C_{*} \Vert w\Vert_{W_{h}^{2,p}(B_{t})}^{p}, \end{align}\] where \(\mathcal{F}_{h}^{I}\) is the collection of all interior mesh faces of \(\mathcal{T}_{h}\). The constant \(C_{*}\) is independent of \(\boldsymbol{x}_{0}\) and \(t\).

We choose \(t_{0}>0\) small enough, which is independent of \(h\), such that \[\begin{align} \Vert \tilde{A}_{0} - \tilde{A} \Vert_{L^{\infty}(B_{t_{0}})} \leq \dfrac{\delta}{C_{*} + 1}. \end{align}\] Since \(\tilde{A}\in [C(\overline{\Omega})]^{d \times d}\), \(t_{0}\) can be chosen to be independent of \(\boldsymbol{x}_{0} \in \overline{\Omega}\).

Then we have \[\begin{align} \vert ((\mathcal{L}_{\tilde{A},h} - \mathcal{L}_{\tilde{A}_{0},h})w, v_{h})_{\Omega} \vert \leq \delta \Vert w\Vert_{W_{h}^{2,p}(B_{t_{0}})}^{p} \Vert v_{h}\Vert_{L^{p^{\prime}}(\Omega)}, \qquad \forall v_{h} \in V_{h}(B_{t_{0}}). \end{align}\] We choose \(h_{\delta} = \frac{t_{0}}{4}\) such that for any \(0<h\leq h_{\delta}\), \(V_{h}(B_{t_{0}})\) is not empty. Then we can conclude that the proof is complete. ◻

Proof. The proof is an immediate consequence of Lemma 8 and Lemma 9. ◻

Proof. We divided the proof into two steps.

Step \(1\). For any \(\boldsymbol{x}_{0} \in \overline{\Omega}\), we denote by \(B_{R}:=\{ \boldsymbol{x}\in \Omega: \vert \boldsymbol{x}- \boldsymbol{x}_{0} \vert < R \}\) for any \(R > 0\). Let \(\eta \in C^{3}(\mathbb{R}^{d})\) be a cut-off function satisfying \[\begin{align} \label{cut95off95func} 0\leq \eta \leq 1, \quad \eta|_{B_{\frac{4}{3}R}} = 1, \quad \eta|_{\mathbb{R}^{d} \setminus B_{\frac{5}{3}R}} = 0, \quad \Vert \eta \Vert_{W^{m,\infty}(\mathbb{R}^{d})} \leq C R^{-m} (m=0,1,2). \end{align}\tag{54}\]

From now on, we assume \(h < \frac{R}{24}\). We define \(\tilde{A}_{0} = \tilde{A}(\boldsymbol{x}_{0})\in \mathbb{R}^{d \times d}\).

We choose \(w_{h} \in V_{h}\) arbitrarily. We notice that \(\eta w_{h} \in W_{h}^{(p)}(B_{\frac{5}{3}R})\) and \(I_{h}(\eta w_{h}) \in V_{h}(B_{\frac{11}{6}R})=\{ v_{h} \in V_{h} : v_{h}|_{\Omega \setminus B_{\frac{11}{6} R}} = 0 \}\). Here \(I_{h}: C^{0}(\overline{\Omega})\rightarrow V_{h}\) is the standard nodal interpolant onto \(V_{h}\).

We notice that \(I_{h}(\eta v_{h}) = \eta v_{h} = 0\) in \(\Omega \setminus B_{\frac{11}{6}R}\) for any \(v_{h} \in V_{h}\). Thus [22] implies that for any \(q \in (1, +\infty)\), there is a constant \(C_{q}\) such that \[\begin{align} \label{nodal95interpolation95error} & \Vert I_{h}(\eta v_{h}) - \eta v_{h} \Vert_{L^{q}(\Omega)} \leq C_{q} h R^{-1} \Vert v_{h}\Vert_{L^{q}(B_{2R})},\\ \nonumber & \Vert \nabla (I_{h}(\eta v_{h}) -\eta v_{h} ) \Vert_{L^{q}(\Omega)} \leq C_{q}R^{-1} \Vert v_{h} \Vert_{L^{q}( B_{2R} )}, \\ \nonumber & \Vert I_{h}(\eta v_{h}) - \eta v_{h}\Vert_{W_{h}^{2,q}(\Omega)} \leq C_{q} R^{-2}\Vert v_{h} \Vert_{W^{1,q}(B_{2R})}, \quad \forall v_{h} \in V_{h}. \end{align}\tag{55}\]

By Lemma 10, there are two constants \(\lambda_{0}>0\) and \(R > 0\) (we choose \(R\) to be \(R_{1}\) in Lemma 10) such that \[\begin{align} \label{gloabl95estimate95with95reaction95ineq1} \lambda_{0}\Vert w_{h} \Vert_{W_{h}^{p}(B_{R})} = \lambda_{0}\Vert I_{h}(\eta w_{h}) \Vert_{W_{h}^{p}(B_{R})} \leq \Vert \mathcal{L}_{\tilde{A},h}(I_{h}(\eta w_{h})) \Vert_{L_{h}^{p}(B_{2R})}. \end{align}\tag{56}\] Here the constant \(\lambda_{0}\) may depend on \(\tilde{A}(\boldsymbol{x}_{0})\).

By (56 ,10 ) and Definition 1, we have \[\begin{align} & \lambda_{0}\Vert w_{h} \Vert_{W_{h}^{p}(B_{R})} = \lambda_{0}\Vert I_{h}(\eta w_{h}) \Vert_{W_{h}^{p}(B_{R})} \\ \leq & \Vert \mathcal{L}_{\tilde{A},h}(\eta w_{h}) \Vert_{L_{h}^{p}(B_{2R})} + \Vert \mathcal{L}_{\tilde{A},h}(I_{h}(\eta w_{h})-\eta w_{h}) \Vert_{L_{h}^{p}(B_{2R})} \\ \leq & \Vert \mathcal{L}_{\tilde{A},h}(\eta w_{h}) \Vert_{L_{h}^{p}(B_{2R})} + \Vert \mathcal{L}_{\tilde{A},h}(I_{h}(\eta w_{h})-\eta w_{h}) \Vert_{L^{p}(\Omega)} \\ \leq & \Vert \mathcal{L}_{\tilde{A},h}(\eta w_{h}) \Vert_{L_{h}^{p}(B_{2R})} + \Vert \tilde{A} \Vert_{L^{\infty}(\Omega)}\Vert \overline{\nabla}_{h}\nabla (I_{h}(\eta w_{h})-\eta w_{h})\Vert_{L^{p}(\Omega)} \\ \leq & \Vert \mathcal{L}_{\tilde{A},h}(\eta w_{h}) \Vert_{L_{h}^{p}(B_{2R})} + C\Vert \tilde{A} \Vert_{L^{\infty}(\Omega)}\Vert I_{h}(\eta w_{h})-\eta w_{h}\Vert_{W_{h}^{2,p}(\Omega)} \end{align}\] Then by (55 ), we have \[\begin{align} \label{gloabl95estimate95with95reaction95ineq2} \lambda_{0}\Vert w_{h} \Vert_{W_{h}^{p}(B_{R})} \leq \Vert \mathcal{L}_{\tilde{A},h}(\eta w_{h}) \Vert_{L_{h}^{p}(B_{2R})} + C R^{-2}\Vert w_{h}\Vert_{W^{1,p}(B_{2R})}. \end{align}\tag{57}\]

We define \(p^{\prime} \in (1, + \infty)\) such that \(\frac{1}{p} + \frac{1}{p^{\prime}}\). By (10 ), \[\begin{align} \Vert \mathcal{L}_{\tilde{A},h}(\eta w_{h}) \Vert_{L_{h}^{p}(B_{2R})} = \sup_{v_{h} \in V_{h}(B_{2R})} \dfrac{(\mathcal{L}_{\tilde{A},h}(\eta w_{h}), v_{h})_{\Omega}} {\Vert v_{h}\Vert_{L^{p^{\prime}}(B_{2R})}}, \end{align}\] where \(V_{h}(B_{2R}) = \{ v_{h} \in V_{h} : v_{h}|_{\Omega \setminus B_{2R}} = 0 \}\). We choose \(v_{h} \in V_{h}(B_{2R})\) such that \[\begin{align} \label{local95norm95variable95A} \Vert \mathcal{L}_{\tilde{A},h}(\eta w_{h}) \Vert_{L_{h}^{p}(B_{2R})} = (\mathcal{L}_{\tilde{A},h}(\eta w_{h}), v_{h})_{\Omega}, \quad \Vert v_{h}\Vert_{L^{p^{\prime}}(B_{2R})} = \Vert v_{h}\Vert_{L^{p^{\prime}}(\Omega)} = 1. \end{align}\tag{58}\]

By (58 ,51 ) and Definition 1, we have \[\begin{align} \label{gloabl95estimate95with95reaction95ineq3} & \Vert \mathcal{L}_{\tilde{A},h}(\eta w_{h}) \Vert_{L_{h}^{p}(B_{2R})} = (\tilde{A}:\overline{\nabla}_{h}(\nabla (\eta w_{h})), v_{h})_{\Omega} \\ \nonumber = & (\tilde{A}:\overline{\nabla}_{h}(\eta \nabla w_{h}), v_{h})_{\Omega} + (\tilde{A}:\overline{\nabla}_{h}(w_{h}\nabla \eta ), v_{h})_{\Omega}\\ \nonumber \leq & (\tilde{A}:\overline{\nabla}_{h}(\eta \nabla w_{h}), v_{h})_{\Omega} + \Vert \tilde{A}\Vert_{L^{\infty}(\Omega)} \Vert \overline{\nabla}_{h}(w_{h}\nabla \eta )\Vert_{L^{p}(\Omega)} \\ \nonumber \leq & (\tilde{A}:\overline{\nabla}_{h}(\eta \nabla w_{h}), v_{h})_{\Omega} + C \Vert w_{h}\nabla \eta\Vert_{W_{h}^{2,p}(\Omega)} \\ \nonumber \leq & (\tilde{A}:\overline{\nabla}_{h}(\eta \nabla w_{h}), v_{h})_{B_{2R}} + C R^{-2}\Vert w_{h}\Vert_{W^{1,p}(\Omega)}. \end{align}\tag{59}\]

Therefore, (57 ,59 ) imply that \[\begin{align} \label{gloabl95estimate95with95reaction95ineq4} \lambda_{0}\Vert w_{h} \Vert_{W_{h}^{p}(B_{R})} \leq & (\tilde{A}:\overline{\nabla}_{h}(\eta \nabla w_{h}), v_{h})_{\Omega} + C R^{-2}\Vert w_{h}\Vert_{W^{1,p}(B_{2R})}. \end{align}\tag{60}\]

We denote by \(\overline{\eta}_{h} \in P_{0}(\mathcal{T}_{h})\) the element-wise average of \(\eta\). Then we have \[\begin{align} \label{gloabl95estimate95with95reaction95ineq5} & (\tilde{A}:\overline{\nabla}_{h}(\eta \nabla w_{h}), v_{h})_{\Omega} \\ \nonumber = & (\tilde{A}:\overline{\nabla}_{h}( (\eta - \overline{\eta}_{h} ) \nabla w_{h}), v_{h})_{\Omega} + (\tilde{A}:\left(\overline{\nabla}_{h}( \overline{\eta}_{h} \nabla w_{h}) - \overline{\eta}_{h}\overline{\nabla}_{h}( \nabla w_{h})\right), v_{h})_{\Omega} \\ \nonumber & \qquad + (\tilde{A}:\overline{\nabla}_{h}( \nabla w_{h}), \overline{\eta}_{h} v_{h} - I_{h}(\eta v_{h}))_{\Omega} + (\tilde{A}:\overline{\nabla}_{h}( \nabla w_{h}), I_{h}(\eta v_{h}))_{\Omega} \end{align}\tag{61}\]

We give estimates of all four terms on the right hand side of (61 ) in the following.

By (58 ), we have \[\begin{align} (\tilde{A}:\overline{\nabla}_{h}( (\eta - \overline{\eta}_{h} ) \nabla w_{h}), v_{h})_{\Omega} \leq \Vert \tilde{A} \Vert_{L^{\infty}(\Omega)} \Vert \overline{\nabla}_{h}( (\eta - \overline{\eta}_{h} ) \nabla w_{h}) \Vert_{L^{p}(\Omega)} \end{align}\] Since \(\overline{\nabla}_{h}( (\eta - \overline{\eta}_{h} ) \nabla w_{h}) \in [\overline{V}_{h}]^{d \times d}\), we denote by \(\underline{\varphi}_{h} \in [\overline{V}_{h}]^{d \times d}\) such that \[\begin{align} \Vert \overline{\nabla}_{h}( (\eta - \overline{\eta}_{h} ) \nabla w_{h}) \Vert_{L^{p}(\Omega)} = \left( \overline{\nabla}_{h}( (\eta - \overline{\eta}_{h} ) \nabla w_{h}), \underline{\varphi}_{h}\right)_{\Omega}, \qquad \Vert \underline{\varphi}_{h} \Vert_{L^{p^{\prime}}(\Omega)} = 1. \end{align}\] By Definition 1 and the fact \(\eta = \overline{\eta}_{h} = 0\) in \(\Omega \setminus B_{\frac{11}{6}R}\), we have \[\begin{align} &\left( \overline{\nabla}_{h}( (\eta - \overline{\eta}_{h} ) \nabla w_{h}), \underline{\varphi}_{h}\right)_{\Omega} = - \langle [(\eta - \overline{\eta}_{h} ) \nabla w_{h}]\otimes \boldsymbol{n}, \{ \underline{\varphi}_{h} \} \rangle_{\mathcal{F}_{h}^{I}} + (\nabla ((\eta - \overline{\eta}_{h} ) \nabla w_{h}), \underline{\varphi}_{h})_{\mathcal{T}_{h}} \\ = & - \langle [\eta - \overline{\eta}_{h}]\{ \nabla w_{h}\} \otimes \boldsymbol{n}, \{ \underline{\varphi}_{h} \} \rangle_{\mathcal{F}_{h}^{I}} - \langle \{\eta - \overline{\eta}_{h}\} [\nabla w_{h}]\otimes \boldsymbol{n}, \{ \underline{\varphi}_{h} \} \rangle_{\mathcal{F}_{h}^{I}} + (\nabla ((\eta - \overline{\eta}_{h} ) \nabla w_{h}), \underline{\varphi}_{h})_{\mathcal{T}_{h}} \\ \leq & C \left( R^{-1} \Vert w_{h} \Vert_{W^{1,p}(B_{\frac{11}{6}R})} + hR^{-1}\Vert w_{h}\Vert_{W_{h}^{2,p}(B_{\frac{11}{6} R})}\right) \Vert \underline{\varphi}_{h} \Vert_{L^{p^{\prime}}(\Omega)}. \end{align}\] By [22], we have \[\begin{align} \Vert w_{h}\Vert_{W_{h}^{2,p}(B_{\frac{11}{6} R})} \leq Ch^{-1} \Vert w_{h}\Vert_{W^{1,p}(B_{2R})}. \end{align}\] Therefore, we have \[\begin{align} \label{gloabl95estimate95with95reaction95ineq6} (\tilde{A}:\overline{\nabla}_{h}( (\eta - \overline{\eta}_{h} ) \nabla w_{h}), v_{h})_{\Omega} \leq C R^{-1} \Vert w_{h} \Vert_{W^{1,p}(B_{2R})}. \end{align}\tag{62}\]

By (58 ), we have \[\begin{align} (\tilde{A}:\left(\overline{\nabla}_{h}( \overline{\eta}_{h} \nabla w_{h}) - \overline{\eta}_{h}\overline{\nabla}_{h}( \nabla w_{h})\right), v_{h})_{\Omega} \leq \Vert \tilde{A}\Vert_{L^{\infty}(\Omega)} \Vert \overline{\nabla}_{h}( \overline{\eta}_{h} \nabla w_{h}) - \overline{\eta}_{h}\overline{\nabla}_{h}( \nabla w_{h})\Vert_{L^{p}(\Omega)}. \end{align}\] We notice \(\overline{\nabla}_{h}( \overline{\eta}_{h} \nabla w_{h}) - \overline{\eta}_{h}\overline{\nabla}_{h}( \nabla w_{h}) \in [\overline{V}_{h}]^{d\times d}\). we denote by \(\underline{\varphi}_{h} \in [\overline{V}_{h}]^{d \times d}\) such that \[\begin{align} \Vert \overline{\nabla}_{h}( \overline{\eta}_{h} \nabla w_{h}) - \overline{\eta}_{h}\overline{\nabla}_{h}( \nabla w_{h})\Vert_{L^{p}(\Omega)} = \left(\overline{\nabla}_{h}( \overline{\eta}_{h} \nabla w_{h}) - \overline{\eta}_{h}\overline{\nabla}_{h}( \nabla w_{h}), \underline{\varphi}_{h} \right)_{\Omega}, \qquad \Vert \underline{\varphi}_{h} \Vert_{L^{p^{\prime}}(\Omega)} = 1. \end{align}\] Since \(\overline{\eta}_{h}\underline{\varphi}_{h} \in [\overline{V}_{h}]^{d \times d}\), Definition 1 implies that \[\begin{align} & \left(\overline{\nabla}_{h}( \overline{\eta}_{h} \nabla w_{h}) - \overline{\eta}_{h}\overline{\nabla}_{h}( \nabla w_{h}), \underline{\varphi}_{h} \right)_{\Omega} \\ = & - \langle [\overline{\eta}_{h} \nabla w_{h}] \otimes \boldsymbol{n}, \{ \underline{\varphi}_{h} \} \rangle_{\mathcal{F}_{h}^{I}} + \langle [\nabla w_{h}] \otimes \boldsymbol{n}, \{ \overline{\eta}_{h} \underline{\varphi}_{h} \} \rangle_{\mathcal{F}_{h}^{I}} \\ = & - \langle [\overline{\eta}_{h}] \{ \nabla w_{h}\} \otimes \boldsymbol{n}, \{ \underline{\varphi}_{h} \} \rangle_{\mathcal{F}_{h}^{I}} + \frac{1}{4}\langle [\nabla w_{h}] \otimes \boldsymbol{n}, [ \overline{\eta}_{h}] [\underline{\varphi}_{h} ] \rangle_{\mathcal{F}_{h}^{I}}. \end{align}\] Then by discrete trace inequality and the fact \(\eta = \overline{\eta}_{h} = 0\) in \(\Omega \setminus B_{\frac{11}{6}R}\), we have \[\begin{align} \label{gloabl95estimate95with95reaction95ineq7} (\tilde{A}:\left(\overline{\nabla}_{h}( \overline{\eta}_{h} \nabla w_{h}) - \overline{\eta}_{h}\overline{\nabla}_{h}( \nabla w_{h})\right), v_{h})_{\Omega} \leq C R^{-1} \Vert w_{h}\Vert_{W^{1,p}(B_{2R})}. \end{align}\tag{63}\]

By the fact \(\eta = \overline{\eta}_{h} = 0\) in \(\Omega \setminus B_{\frac{11}{6}R}\) and Definition 1, \[\begin{align} & (\tilde{A}:\overline{\nabla}_{h}( \nabla w_{h}), \overline{\eta}_{h} v_{h} - I_{h}(\eta v_{h}))_{\Omega} \\ \leq & \Vert \tilde{A} \Vert_{L^{\infty}(\Omega)} \Vert \overline{\nabla}_{h}( \nabla w_{h}) \Vert_{L^{p}(B_{\frac{11}{6}R})} \Vert \overline{\eta}_{h} v_{h} - I_{h}(\eta v_{h}) \Vert_{L^{p^{\prime}}(\Omega)} \\ \leq & C \Vert w_{h}\Vert_{W_{h}^{2,p}(B_{\frac{23}{12}}R)} \Vert \overline{\eta}_{h} v_{h} - I_{h}(\eta v_{h}) \Vert_{L^{p^{\prime}}(\Omega)}. \end{align}\] By [22], we have \[\begin{align} \Vert w_{h} \Vert_{W_{h}^{2,p}(B_{\frac{23}{12} R})} \leq Ch^{-1} \Vert w_{h}\Vert_{W^{1,p}(B_{2R})}. \end{align}\] So we have \[\begin{align} (\tilde{A}:\overline{\nabla}_{h}( \nabla w_{h}), \overline{\eta}_{h} v_{h} - I_{h}(\eta v_{h}))_{\Omega} \leq C h^{-1}\Vert w_{h}\Vert_{W^{1,p}(B_{2R})} \Vert \overline{\eta}_{h} v_{h} - I_{h}(\eta v_{h}) \Vert_{L^{p^{\prime}}(\Omega)}. \end{align}\] By (55 ,58 ), we have \[\begin{align} \label{gloabl95estimate95with95reaction95ineq8} & (\tilde{A}:\overline{\nabla}_{h}( \nabla w_{h}), \overline{\eta}_{h} v_{h} - I_{h}(\eta v_{h}))_{\Omega} \\ \nonumber \leq & C R^{-1} \Vert w_{h}\Vert_{W^{1,p}(B_{2R})} \Vert v_{h} \Vert_{L^{p^{\prime}}(\Omega)} = C R^{-1} \Vert w_{h}\Vert_{W^{1,p}(B_{2R})}. \end{align}\tag{64}\]

We notice \(I_{h}(\eta v_{h}) \in V_{h}(B_{\frac{11}{6}R})\). Then by (51 ,10 ,55 ,58 ), we have \[\begin{align} \label{gloabl95estimate95with95reaction95ineq9} & (\tilde{A}:\overline{\nabla}_{h}( \nabla w_{h}), I_{h}(\eta v_{h}))_{\Omega} = (\mathcal{L}_{\tilde{A},h}w_{h}, I_{h}(\eta v_{h}))_{\Omega} \\ \nonumber \leq & \Vert \mathcal{L}_{\tilde{A},h}w_{h}\Vert_{L_{h}^{p}(B_{2R})} \Vert I_{h}(\eta v_{h})\Vert_{L^{p^{\prime}}(\Omega)} \leq C \Vert \mathcal{L}_{\tilde{A},h}w_{h}\Vert_{L_{h}^{p}(B_{2R})} \Vert v_{h} \Vert_{L^{p^{\prime}}(B_{2R})} = C \Vert \mathcal{L}_{\tilde{A},h}w_{h}\Vert_{L_{h}^{p}(B_{2R})}. \end{align}\tag{65}\]

According to (60 ,61 , 62 ,63 ,64 , 65 ), we have \[\begin{align} \label{gloabl95estimate95with95reaction95ineq10} \lambda_{0}\Vert w_{h}\Vert_{W_{h}^{2,p}(B_{R})} \leq C \left( \Vert \mathcal{L}_{\tilde{A},h}w_{h}\Vert_{L_{h}^{p}(B_{2R})} + R^{-2} \Vert w_{h} \Vert_{W^{1,p}(B_{2R})} \right), \qquad \forall w_{h} \in V_{h}. \end{align}\tag{66}\] We recall that \(R\) is chosen to be \(R_{1}\) in Lemma 10. The constant \(\lambda_{0}\) may depend on \(\tilde{A}(\boldsymbol{x}_{0})\).

Step \(2\). Let \(\{\boldsymbol{x}_{j}\}_{j=1}^{N} \subset \overline{\Omega}\) such that \(\overline{\Omega} = \cup_{j=1}^{N}\overline{B}_{R_{j}^{\prime}}(\boldsymbol{x}_{j})\) where \(R_{j}^{\prime}\) is the constant \(R\) for \(\boldsymbol{x}_{j}\) in (66 ), and \(B_{R_{j}^{\prime}}:= \{ \boldsymbol{x}\in \Omega : \vert \boldsymbol{x}- \boldsymbol{x}_{j} \vert < R_{j}^{\prime}\}\). Since \(\overline{\Omega}\) is compact, \(N\) is finite and independent of \(h\).

We denote by \(S_{j} = B_{R_{j}^{\prime}}\) and \(\tilde{S}_{j} = B_{2R_{j}^{\prime}} := \{ \boldsymbol{x}\in \Omega : \vert \boldsymbol{x}- \boldsymbol{x}_{j} \vert < 2R_{j}^{\prime}\}\) for any \(1\leq j \leq N\).

Since \(R_{j}\) is independent of \(h\) for any \(1 \leq j \leq N\), (66 ) implies that \[\begin{align} \Vert w_{h}\Vert_{W_{h}^{2,p}(\Omega)}^{p} \leq \Sigma_{j=1}^{N} \Vert w_{h} \Vert_{W_{h}^{2,p}(S_{j})}^{p} \leq C \left( \Sigma_{j=1}^{N} \Vert \mathcal{L}_{\tilde{A},h} w_{h} \Vert_{L_{h}^{p}(\tilde{S}_{j})}^{p} + \Vert w_{h}\Vert_{W^{1,p}(\Omega)}^{p} \right). \end{align}\]

By (10 ), \[\begin{align} & \Sigma_{j=1}^{N} \Vert \mathcal{L}_{\tilde{A},h} w_{h} \Vert_{L_{h}^{p}(\tilde{S}_{j})}^{p} = \Sigma_{j=1}^{N} \left| \sup_{v_{h} \in V_{h}(\tilde{S}_{j})}\dfrac{(\mathcal{L}_{\tilde{A},h} w_{h}, v_{h})_{\Omega}}{\Vert v_{h}\Vert_{L^{p^{\prime}}(\tilde{S}_{j})}} \right|^{p}\\ \leq & N \left| \sup_{v_{h} \in V_{h}}\dfrac{(\mathcal{L}_{\tilde{A},h} w_{h}, v_{h})_{\Omega}}{\Vert v_{h}\Vert_{L^{p^{\prime}}(\Omega)}} \right|^{p} = N \Vert \mathcal{L}_{\tilde{A}, h}w_{h} \Vert_{L_{h}^{p}(\Omega)}. \end{align}\]

Therefore we can conclude that the proof is complete. ◻

Proof. Since \(\Vert w_{h} \Vert_{W_{h}^{2,p}(\Omega)} \leq M_{1}\) for any \(h>0\), Lemma 4 implies \[\begin{align} \Vert w_{h}\Vert_{W^{1,p}(\Omega)} \leq C M_{1}, \qquad \forall h>0. \end{align}\] Thus there is \(w \in W_{0}^{1,p}(\Omega)\) such that \[\begin{align} \label{conv95w} \Vert w_{h} - w\Vert_{L^{p}(\Omega)} \rightarrow 0, \qquad w_{h} \rightharpoonup w \text{ in } W_{0}^{1,p}(\Omega), \end{align}\tag{67}\] for a subsequence of \(h \rightarrow 0\). We denote by \(\boldsymbol{\phi}_{h} \in [H^{1}(\Omega)\cap \overline{V}_{h}]^{d}\) the standard \(H^{1}\)-conforming averaging of \(\nabla w_{h}\). By Lemma 3, we have \[\begin{align} \label{interpolation95gradient95w} \Vert \boldsymbol{\phi}_{h} - \nabla w_{h} \Vert_{L^{p}(\Omega)} \leq C h \left( \Sigma_{F \in \mathcal{F}_{h}^{I}} h_{F}^{1-p}\Vert [\nabla w_{h}] \Vert_{L^{p}(F)}^{p} \right)^{\frac{1}{p}}. \end{align}\tag{68}\] (68 ) and the fact \(\Vert w_{h}\Vert_{W^{1,p}(\Omega)} + \Vert w_{h} \Vert_{W_{h}^{2,p}(\Omega)} \leq (1+C)M_{1}\) imply that there is \(\boldsymbol{\phi} \in [W^{1,p}(\Omega)]^{d}\) such that \[\begin{align} \label{conv95phi} \Vert \boldsymbol{\phi}_{h} - \boldsymbol{\phi}\Vert_{L^{p}(\Omega)} + \Vert \nabla w_{h} - \boldsymbol{\phi} \Vert_{L^{p}(\Omega)} \rightarrow 0, \quad \text{ and } \quad \boldsymbol{\phi}_{h} \rightharpoonup \boldsymbol{\phi} \text{ in } [W^{1,p}(\Omega)]^{d}, \end{align}\tag{69}\] for a subsequence of \(h \rightarrow 0\). (67 ,69 ) imply that \(\nabla w = \boldsymbol{\phi}\) almost everywhere in \(\Omega\). Therefore, we have that \(w \in W^{2,p}(\Omega) \cap W_{0}^{1,p}(\Omega)\) and \[\begin{align} \label{conv95w95second95order} \Vert w_{h} - w \Vert_{W^{1,p}(\Omega)} + \Vert \boldsymbol{\phi}_{h} - \nabla w \Vert_{L^{p}(\Omega)} \rightarrow 0, \quad \text{ and } \boldsymbol{\phi}_{h} \rightharpoonup \nabla w \text{ in } [W^{1,p}(\Omega)]^{d}, \end{align}\tag{70}\] for a subsequence of \(h \rightarrow 0\).

According to Definition 1 with the fact \(\Vert w_{h} \Vert_{W_{h}^{2,p}(\Omega)} \leq M_{1} < + \infty\) and \(w\in W^{2,p}(\Omega)\), there is a constant \(M_{2}\) such that \[\begin{align} \label{discrete95second95deri95bound} \Vert \overline{\nabla}_{h}(\nabla w) \Vert_{L^{p}(\Omega)} + \Vert \overline{\nabla}_{h} (\nabla w_{h})\Vert_{L^{p}(\Omega)} \leq M_{2} < + \infty. \end{align}\tag{71}\] We denote by \(P_{h}\) the standard \(L^{2}\)-orthogonal projection onto \(V_{h}\). By Lemma 2, there is a constant \(\tilde{C}_{1}\) independent of \(h\) such that for any \(v \in W_{0}^{1,p^{\prime}}(\Omega)\), \[\begin{align} \label{L295projection95bound1} \Vert P_{h} v\Vert_{L^{p^{\prime}}(\Omega)} \leq \tilde{C}_{1} \Vert v\Vert_{L^{p^{\prime}}(\Omega)}, \qquad \Vert P_{h} v\Vert_{W^{1,p^{\prime}}(\Omega)} \leq \tilde{C}_{1} \Vert v\Vert_{W^{1,p^{\prime}}(\Omega)}. \end{align}\tag{72}\]

By (71 ), there is \(\underline{\psi} \in [L^{p}(\Omega)]^{d \times d}\) such that \[\begin{align} \overline{\nabla}_{h}(\nabla w_{h}) \rightharpoonup \underline{\psi} \in [L^{p}(\Omega)]^{d \times d}, \end{align}\] for a subsequence of \(h \rightarrow 0\). On the other hand, we have that for any \(\underline{\zeta} \in [C_{0}^{\infty}(\Omega)]^{d\times d}\), \[\begin{align} (\overline{\nabla}_{h}(\nabla w_{h}), \underline{\zeta})_{\Omega} = (\overline{\nabla}_{h}(\nabla w_{h}), P_{h}(\underline{\zeta}))_{\Omega} + (\overline{\nabla}_{h}(\nabla w_{h}), P_{h}(\underline{\zeta}) - \underline{\zeta})_{\Omega}. \end{align}\] Since \(P_{h}(\underline{\zeta}) \in [V_{h}]^{d \times d}\), then by Definition 1 \[\begin{align} & (\overline{\nabla}_{h}(\nabla w_{h}), \underline{\zeta})_{\Omega} = -(\nabla w_{h}, \nabla \cdot P_{h}(\underline{\zeta}))_{\Omega} + (\overline{\nabla}_{h}(\nabla w_{h}), P_{h}(\underline{\zeta}) - \underline{\zeta})_{\Omega} \\ = & -(\nabla w_{h}, \nabla \cdot \underline{\zeta})_{\Omega} - (\nabla w_{h}, \nabla \cdot (P_{h}(\underline{\zeta}) - \underline{\zeta}))_{\Omega} + (\overline{\nabla}_{h}(\nabla w_{h}), P_{h}(\underline{\zeta}) - \underline{\zeta})_{\Omega} \end{align}\] In the following of the proof, \(\lim_{h\rightarrow 0}\) means the limit when a subsequence mentioned above of \(h \rightarrow 0\). By (70 ), we have \[\begin{align} \lim_{h \rightarrow 0} -(\nabla w_{h}, \nabla \cdot \underline{\zeta})_{\Omega} = -(\nabla w, \nabla\cdot \underline{\zeta})_{\Omega} = (D^{2}w, \underline{\zeta})_{\Omega}. \end{align}\] By (71 ) and the fact \(\Vert w_{h} \Vert_{W_{h}^{2,p}(\Omega)} \leq M_{1} < + \infty\), (72 ) implies that \[\begin{align} \lim_{h\rightarrow 0} (\nabla w_{h}, \nabla \cdot (P_{h}(\underline{\zeta}) - \underline{\zeta}))_{\Omega} = \lim_{h \rightarrow 0} (\overline{\nabla}_{h}(\nabla w_{h}), P_{h}(\underline{\zeta}) - \underline{\zeta})_{\Omega} = 0. \end{align}\] Therefore, we have \[\begin{align} \lim_{h \rightarrow 0} (\overline{\nabla}_{h}(\nabla w_{h}), \underline{\zeta})_{\Omega} = (D^{2}w, \underline{\zeta})_{\Omega}, \qquad \forall \underline{\zeta} \in [C_{0}^{\infty}(\Omega)]^{d \times d}. \end{align}\] Therefore, we have \(\underline{\psi} = D^{2}w\) almost everywhere in \(\Omega\). So we have \[\begin{align} \label{conv95w95second95order95weak} \overline{\nabla}_{h}(\nabla w_{h}) \rightharpoonup D^{2}w \in [L^{p}(\Omega)]^{d \times d}, \end{align}\tag{73}\] for a subsequence of \(h \rightarrow 0\).

According to (70 , 71 , 73 ), We can conclude the proof is complete. ◻

Proof. We notice that by (14 ), \(A \in [C(\overline{\Omega})]^{d\times d}\) is equivalent to \(\tilde{A} \in [C(\overline{\Omega})]^{d\times d}\), and \(c \leq 0\) is equivalent to \(\tilde{c} \leq 0\).

For the sake of simplicity, we only give detailed proof of the case \(d=3\). The proof of the case \(d=2\) (\(\Omega\) can be non-convex) can be proven in the same manner.

According to Lemma 11, it is sufficient to prove that for any \(p \in (1, 2]\), \[\begin{align} \Vert w_{h}\Vert_{W^{1,p}(\Omega)} \leq C \Vert \tilde{A}: \overline{\nabla}_{h}(\nabla w_{h}) +\tilde{\boldsymbol{b}}\cdot\nabla w_{h} + \tilde{c} w_{h} \Vert_{L_{h}^{p}(\Omega)}, \qquad \forall w_{h} \in V_{h}. \end{align}\] Here the constant \(C\) may depend on \(p\).

We prove by contradiction. Without losing generality, we assume that there is one \(p \in (1, 2]\) and a sequence of \(\{w_{h} \in V_{h}\}_{h>0}\), such that \[\begin{align} \label{assumption95contradition1} \Vert w_{h}\Vert_{W^{1,p}(\Omega)} = 1, \quad \text{and } \quad \lim_{h\rightarrow 0}\Vert \tilde{A}: \overline{\nabla}_{h}(\nabla w_{h}) +\tilde{\boldsymbol{b}}\cdot\nabla w_{h} + \tilde{c}w_{h}\Vert_{L_{h}^{p}(\Omega)} = 0. \end{align}\tag{75}\] So there is a positive constant \(C\) such that \[\begin{align} & \Vert \mathcal{L}_{\tilde{A},h}w_{h} \Vert_{L_{h}^{p}(\Omega)} = \Vert \tilde{A}: \overline{\nabla}_{h}(\nabla w_{h})\Vert_{L_{h}^{p}(\Omega)} \\ \leq & C \left( \Vert \tilde{A}: \overline{\nabla}_{h}(\nabla w_{h}) +\tilde{\boldsymbol{b}}\cdot\nabla w_{h} + \tilde{c}w_{h}\Vert_{L_{h}^{p}(\Omega)} + \Vert w_{h} \Vert_{W^{1,p}(\Omega)} \right). \end{align}\] The equality above is due to (51 ). According to Lemma 11, there is \(M_{1} < +\infty\) such that \[\begin{align} \Vert w_{h} \Vert_{W_{h}^{2,p}(\Omega)} \leq M_{1}, \qquad \forall h>0. \end{align}\] Therefore, Lemma 12 implies that there is \(w \in W^{2} \cap W_{0}^{1,p}(\Omega)\) and \(0<M_{2}< + \infty\) such that \[\begin{align} \tag{76} & \Vert w_{h} - w\Vert_{W^{1,p}(\Omega)} \rightarrow 0, \\ \tag{77} & \overline{\nabla}_{h}(\nabla w_{h}) \rightharpoonup D^{2}w \text{ in } L^{p}(\Omega), \end{align}\] for a subsequence of \(h \rightarrow 0\); and \[\begin{align} \label{discrete95second95deri95bound95proof} \quad \Vert \overline{\nabla}_{h}(\nabla w) \Vert_{L^{p}(\Omega)} + \Vert \overline{\nabla}_{h} (\nabla w_{h})\Vert_{L^{p}(\Omega)} \leq M_{2}, \qquad \forall h> 0. \end{align}\tag{78}\]

In the following of the proof, we don’t distinguish \(h \rightarrow 0\) and a subsequence of \(h \rightarrow 0\). Then (75 ) and (76 ) imply that \(\Vert w\Vert_{W^{1,p}(\Omega)} \geq \frac{1}{2}\). By Theorem 1, in order to achieve contradiction with \(\Vert w\Vert_{W^{1,p}(\Omega)} \geq \frac{1}{2}\), we only need to prove that \[\begin{align} \tilde{A}: D^{2} w + \tilde{\boldsymbol{b}}\cdot \nabla w + \tilde{c} w = 0 \text{ almost everywhere in } \Omega. \end{align}\] Since \(w \in W^{2,p}(\Omega)\), it is sufficient to prove that \[\begin{align} \label{statement95contradiction1} (\tilde{A}:D^{2}w + \tilde{\boldsymbol{b}}\cdot \nabla w + \tilde{c} w, v)_{\Omega} = 0, \qquad \forall v \in C_{0}^{\infty}(\Omega). \end{align}\tag{79}\]

We choose \(v \in C_{0}^{\infty}(\Omega)\) arbitrarily. It is easy to see \[\begin{align} & (\tilde{A}:D^{2}w + \tilde{\boldsymbol{b}}\cdot \nabla w + \tilde{c} w, v)_{\Omega} \\ = & (\tilde{A}:(D^{2}w -\overline{\nabla}_{h}(\nabla w_{h})), v )_{\Omega} + (\tilde{A}:\overline{\nabla}_{h}(\nabla w_{h}), v )_{\Omega} + (\tilde{\boldsymbol{b}}\cdot \nabla w + \tilde{c} w, v)_{\Omega} \\ = & (\tilde{A}:(D^{2}w -\overline{\nabla}_{h}(\nabla w_{h})), v )_{\Omega} + (\tilde{A}:\overline{\nabla}_{h}(\nabla w_{h}) + \tilde{\boldsymbol{b}}\cdot \nabla w_{h} + \tilde{c} w_{h}, P_{h} v )_{\Omega} \\ & \qquad + (\tilde{A}:\overline{\nabla}_{h}(\nabla w_{h}) + \tilde{\boldsymbol{b}}\cdot \nabla w_{h} + \tilde{c} w_{h}, v - P_{h}v )_{\Omega} + (\tilde{\boldsymbol{b}}\cdot \nabla (w-w_{h}) + \tilde{c}(w-w_{h}), v)_{\Omega}. \end{align}\] Here we denote by \(P_{h}\) the standard \(L^{2}\)-orthogonal projection onto \(V_{h}\). We define \(p^{\prime} \in (1, + \infty)\) by \(\frac{1}{p} + \frac{1}{p^{\prime}} = 1\). By (77 ) and the fact that \(\tilde{A}v\in [L^{p^{\prime}}(\Omega)]^{d \times d}\), \[\begin{align} \lim_{h \rightarrow 0} (\tilde{A}:(D^{2}w -\overline{\nabla}_{h}(\nabla w_{h})), v )_{\Omega} = 0. \end{align}\] By Lemma 2, there is a constant \(\tilde{C}_{1}\) independent of \(h\), \[\begin{align} \label{L295projection95bound2} \Vert P_{h} v\Vert_{L^{p^{\prime}}(\Omega)} \leq \tilde{C}_{1} \Vert v\Vert_{L^{p^{\prime}}(\Omega)}, \qquad \Vert P_{h} v\Vert_{W^{1,p^{\prime}}(\Omega)} \leq \tilde{C}_{1} \Vert v\Vert_{W^{1,p^{\prime}}(\Omega)}. \end{align}\tag{80}\] Therefore, by (75 ) and (80 ), we have \[\begin{align} & \lim_{h\rightarrow 0} \vert (\tilde{A}:\overline{\nabla}_{h}(\nabla w_{h} + \tilde{\boldsymbol{b}}\cdot\nabla w_{h}+\tilde{c}w_{h}), P_{h}v)_{\Omega}\vert\\ \leq & \lim_{h \rightarrow 0} \Vert \tilde{A}: \overline{\nabla}_{h}(\nabla w_{h}) + \tilde{\boldsymbol{b}}\cdot \nabla w_{h} + \tilde{c} w_{h}\Vert_{L_{h}^{p}(\Omega)} \Vert P_{h} v\Vert_{L^{p^{\prime}}(\Omega)} \\ \leq & \tilde{C}_{1} \lim_{h \rightarrow 0} \Vert \tilde{A}: \overline{\nabla}_{h}(\nabla w_{h}) + \tilde{\boldsymbol{b}}\cdot \nabla w_{h} + \tilde{c} w_{h}\Vert_{L_{h}^{p}(\Omega)} \Vert v\Vert_{L^{p^{\prime}}(\Omega)} = 0. \end{align}\] By (75 ,78 ,80 ), we have \[\begin{align} \lim_{h\rightarrow 0} (\tilde{A}:\overline{\nabla}_{h}(\nabla w_{h}) +\tilde{\boldsymbol{b}}\cdot\nabla w_{h}+\tilde{c}w_{h},v-P_{h}v)_{\Omega} = 0. \end{align}\] By (76 ), we have \[\begin{align} \lim_{h\rightarrow 0} (\tilde{\boldsymbol{b}}\cdot \nabla (w-w_{h}) + \tilde{c}(w-w_{h}), v)_{\Omega} = 0. \end{align}\] Therefore, we have \[\begin{align} \label{verification95contradiction1} (\tilde{A}:D^{2}w + \tilde{\boldsymbol{b}}\cdot \nabla w + \tilde{c}w, v)_{\Omega} = 0, \qquad \forall v \in C_{0}^{\infty}(\Omega). \end{align}\tag{81}\]

(81 ) implies that the proof is complete. ◻

Proof. We divide the proof into two steps. In step \(1\), We will first prove the estimates (82 ) and (85 ,86 ) for \(1< p \leq 2\) if \(\Omega\) is convex polyhedra in \(\mathbb{R}^{d}\) (\(d=2,3\)), and \(\frac{4}{3} - \epsilon_{1} <p < \frac{4}{3} + \epsilon_{2}\) if \(\Omega\) is a polygon (maybe non-convex) in \(\mathbb{R}^{2}\). In step \(2\), we will prove the estimate (82 ) for \(1< p \leq \frac{4}{3} - \epsilon_{1}\) and \(\Omega\) is a polygon (maybe non-convex) in \(\mathbb{R}^{2}\).

Step \(1\). We consider \(1< p \leq 2\) if \(\Omega\) is convex polyhedra in \(\mathbb{R}^{d}\) (\(d=2,3\)), and \(\frac{4}{3} - \epsilon_{1} <p < \frac{4}{3} + \epsilon_{2}\) if \(\Omega\) is a polygon (maybe non-convex) in \(\mathbb{R}^{2}\).

By the design of the finite element method (16 ) and Theorem 2, there is a unique \(u_{h} \in V_{h}\) to be the numerical solution of (16 ) and \[\begin{align} \Vert u_{h} \Vert_{W_{h}^{2,p}(\Omega)} \leq C \Vert f\Vert_{L^{p}(\Omega)}, \end{align}\] if \(h\) is small enough. Then by Lemma 12, there is \(u \in W^{2,p}(\Omega) \cap W_{0}^{1,p}(\Omega)\) such that \[\begin{align} \Vert u_{h} - u\Vert_{W^{1,p}(\Omega)} \rightarrow 0, \qquad \overline{\nabla}_{h}(\nabla u_{h}) \rightharpoonup D^{2}u \text{ in } [L^{p}(\Omega)]^{d \times d}, \end{align}\] for a sequence of \(h \rightarrow 0\); and there is \(0 < M_{2} < +\infty\) such that \(\Vert \overline{\nabla}(\nabla u_{h})\Vert_{L^{p}(\Omega)} \leq M_{2}\) for any \(h > 0\).

Now we need to prove (83 ). We define \(p^{\prime} \in (1, + \infty)\) by \(\frac{1}{p} + \frac{1}{p^{\prime}} = 1\). We denote by \(P_{h}\) the standard \(L^{2}\)-orthogonal projection onto \(V_{h}\). It is well-known that there is a constant \(\tilde{C}_{1}\) independent of \(h\), \[\begin{align} \label{L295projection95bound3} \Vert P_{h} v\Vert_{L^{p^{\prime}}(\Omega)} \leq \tilde{C}_{1} \Vert v\Vert_{L^{p^{\prime}}(\Omega)}, \qquad \Vert P_{h} v\Vert_{W^{1,p^{\prime}}(\Omega)} \leq \tilde{C}_{1} \Vert v\Vert_{W^{1,p^{\prime}}(\Omega)}. \end{align}\tag{87}\]

We choose \(v \in C_{0}^{\infty}(\Omega)\) arbitrarily. By the design of the finite element method (16 ), We have \[\begin{align} & (\tilde{A}:D^{2}u + \tilde{\boldsymbol{b}}\cdot\nabla u + \tilde{c}u, v)_{\Omega} \\ = & (\tilde{A}: (D^{2}u - \overline{\nabla}(\nabla u_{h})), v)_{\Omega} + (\tilde{A}:\overline{\nabla}(\nabla u_{h}) + \tilde{\boldsymbol{b}}\cdot\nabla u + \tilde{c}u, v)_{\Omega} \\ = & (\tilde{A}: (D^{2}u - \overline{\nabla}(\nabla u_{h})), v)_{\Omega} + (\tilde{A}:\overline{\nabla}(\nabla u_{h}) + \tilde{\boldsymbol{b}}\cdot\nabla u + \tilde{c}u, v - P_{h}v)_{\Omega} \\ & \qquad + (\tilde{A}:\overline{\nabla}(\nabla u_{h}) + \tilde{\boldsymbol{b}}\cdot\nabla u + \tilde{c}u, P_{h}v)_{\Omega} \\ = & (\tilde{A}: (D^{2}u - \overline{\nabla}(\nabla u_{h})), v)_{\Omega} + (\tilde{A}:\overline{\nabla}(\nabla u_{h}) + \tilde{\boldsymbol{b}}\cdot\nabla u + \tilde{c}u, v - P_{h}v)_{\Omega} \\ & \qquad + (\tilde{A}:\overline{\nabla}(\nabla u_{h}) + \tilde{\boldsymbol{b}}\cdot\nabla u_{h} + \tilde{c}u_{h}, P_{h}v)_{\Omega} + (\tilde{\boldsymbol{b}}\cdot \nabla(u - u_{h}) + \tilde{c}(u - u_{h}), P_{h}v)_{\Omega} \\ = & (\tilde{A}: (D^{2}u - \overline{\nabla}(\nabla u_{h})), v)_{\Omega} + (\tilde{A}:\overline{\nabla}(\nabla u_{h}) + \tilde{\boldsymbol{b}}\cdot\nabla u + \tilde{c}u, v - P_{h}v)_{\Omega} \\ & \qquad + (\tilde{f}, v)_{\Omega} + (\tilde{f}, P_{h}v - v)_{\Omega} + (\tilde{\boldsymbol{b}}\cdot \nabla(u - u_{h}) + \tilde{c}(u - u_{h}), P_{h}v)_{\Omega}. \end{align}\]

Since \(\overline{\nabla}_{h}(\nabla u_{h}) \rightharpoonup D^{2}u \text{ in } [L^{p}(\Omega)]^{d \times d}\), we have \[\begin{align} (\tilde{A}:(D^{2}u - \overline{\nabla}_{h}(\nabla u_{h})), v)_{\Omega} + (\tilde{\boldsymbol{b}}\cdot \nabla(u - u_{h}) + \tilde{c}(u - u_{h}), P_{h}v)_{\Omega}\rightarrow 0, \end{align}\] for a sequence of \(h \rightarrow 0\).

By (87 ) and the fact that \(\Vert \overline{\nabla}(\nabla u_{h})\Vert_{L^{p}(\Omega)} \leq M_{2}\), we have \[\begin{align} & \vert (\tilde{A}:\overline{\nabla}(\nabla u_{h}) + \tilde{\boldsymbol{b}}\cdot \nabla u + \tilde{c}u, v - P_{h}v)_{\Omega} \vert \\ \leq & C \left(\Vert \overline{\nabla}(\nabla u_{h})\Vert_{L^{p}(\Omega)} + \Vert u\Vert_{W^{1,p}(\Omega)}\right)\Vert v - P_{h} v\Vert_{L^{p^{\prime}}(\Omega)} \rightarrow 0, \end{align}\] as \(h \rightarrow 0\).

Similarly, we have \[\begin{align} \vert (\tilde{f}, P_{h}v - v)_{\Omega} \vert \rightarrow 0, \end{align}\] as \(h \rightarrow 0\).

So we have \[\begin{align} (\tilde{A}:D^{2}u + \tilde{\boldsymbol{b}}\cdot\nabla u + \tilde{c}u, v)_{\Omega} = (\tilde{f}, v)_{\Omega}. \end{align}\]

Since \(v \in C_{0}^{\infty}(\Omega)\) is chosen arbitrarily, \(\tilde{A}:D^{2}+\tilde{\boldsymbol{b}}\cdot\nabla u + \tilde{c}u = \tilde{f}\) in \(\Omega\) almost everywhere. By the definition of the function \(\gamma\) (see (14 )) and the fact that \(A\) is uniformly elliptic and uniformly bounded in \(\Omega\), \(\gamma\) is uniformly bounded from below and above by positive constants. So we have \(A:D^{2}u +\boldsymbol{b}\cdot\nabla u + cu = f\) in \(\Omega\) almost everywhere. So (83 ) is proven. Theorem 1 implies the uniqueness of \(u \in W^{2,p}(\Omega) \cap W_{0}^{1,p}(\Omega)\), which results in \[\begin{align} \Vert u_{h} - u\Vert_{W^{1,p}(\Omega)} \rightarrow 0, \qquad \overline{\nabla}_{h}(\nabla u_{h}) \rightharpoonup D^{2}u \text{ in } [L^{p}(\Omega)]^{d \times d}, \end{align}\] as \(h \rightarrow 0\).

The proof of (86 ) is straightforward due to Theorem 2 and the construction of the finite element method (16 ). Thus the proof in Step \(1\) is complete.

Step \(2\). We consider \(1< p \leq \frac{4}{3} - \epsilon_{1}\) if \(\Omega\) is a polygon (maybe non-convex) in \(\mathbb{R}^{2}\).

Let \(\{{f_{i}}\}_{i \geq 1} \subset L^{\frac{4}{3}}(\Omega)\) such that \(\Vert f_{i} -f \Vert_{L^{p}(\Omega)} \rightarrow 0\) as \(i \rightarrow +\infty\). For any \(i \geq 1\), we denote by \(u_{i} \in W^{2,\frac{4}{3}}(\Omega) \cap W_{0}^{1, \frac{4}{3}}(\Omega)\) satisfying \[\begin{align} A:D^{2} u_{i} + \boldsymbol{b}\cdot \nabla u_{i} + c u_{i} = f_{i} \text{ in } \Omega \text{ almost everywhere}. \end{align}\]

Since \(\Vert f_{i} -f \Vert_{L^{p}(\Omega)} \rightarrow 0\) as \(i \rightarrow +\infty\), \(\{ \Vert f_{i} \Vert_{L^{p}(\Omega)} \}_{i \geq 1}\) have a uniform upper bound. So \(\{ \Vert u_{i}\Vert_{W^{2,p}(\Omega)} \}_{i \geq 1}\) have a uniform upper bound as well. So there is \(u \in W^{2,p}(\Omega) \cap W_{0}^{1,p}(\Omega)\) such that \[\begin{align} u_{i} \rightharpoonup u \text{ in } W^{2,p}(\Omega), \end{align}\] for a subsequence of \(i \rightarrow + \infty\).

We choose \(v \in C_{0}^{\infty}(\Omega)\) arbitrarily. Then \[\begin{align} & (A:D^{2} u + \boldsymbol{b}\cdot \nabla u + cu, v)_{\Omega} \\ = & (A:D^{2}(u - u_{i}) + \boldsymbol{b}\cdot \nabla (u - u_{i}) + c(u - u_{i}), v)_{\Omega} + (A:D^{2}u_{i} + \boldsymbol{b}\cdot\nabla u_{i} + c u_{i}, v)_{\Omega} \\ = & (A:D^{2}(u - u_{i}) + \boldsymbol{b}\cdot \nabla (u - u_{i}) + c(u - u_{i}), v)_{\Omega} + (f_{i} - f, v)_{\Omega} + (f, v)_{\Omega}. \end{align}\]

Since \(u_{i} \rightharpoonup u \text{ in } W^{2,p}(\Omega)\) for a subsequence of \(i \rightarrow + \infty\) and \(\Vert f_{i} - f \Vert_{L^{p}(\Omega)} \rightarrow 0\) as \(i \rightarrow + \infty\), we have \[\begin{align} (A:D^{2} u + \boldsymbol{b}\cdot \nabla u + c u, v)_{\Omega} = (f, v)_{\Omega}. \end{align}\]

Since \(v \in C_{0}^{\infty}(\Omega)\) is chosen arbitrarily, \(A:D^{2}u = f\) in \(\Omega\) almost everywhere. By Theorem 1, \(u \in W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)\) is the unique solution of (1 ). Thus the proof of Step \(2\) is complete. ◻

Proof. According to the assumption (8 ), it is easy to see that for any \(\boldsymbol{x}\in \Omega \setminus E_{0}\) and for any \(\alpha\in\Lambda\), \(a_{ij}^{\alpha}(\boldsymbol{x}),b_{i}^{\alpha}(\boldsymbol{x}),c^{\alpha}(\boldsymbol{x}), f^{\alpha}(\boldsymbol{x})\) are all finite numbers for any \(1\leq i,j \leq d\). Since \(B \in [L^{1}(\Omega)]^{d\times d}, \boldsymbol{c}\in [L^{1}(\Omega)]^{d}, w \in L^{1}(\Omega)\), there is a set \(E_{1} \subset \Omega\) with zero \(d\)-dimensional Lebesgue measure, such that \(b_{ij}, c_{i}, w\) are all finite numbers in \(\Omega\setminus E_{1}\) for any \(1\leq i,j \leq d\).

We choose \(t \in \mathbb{R}\) arbitrarily. It is easy to see \[\begin{align} & \{\boldsymbol{x}\in \Omega: \sup_{\alpha \in \Lambda} [A^{\alpha}:B + \boldsymbol{b}^{\alpha}\cdot \boldsymbol{c} + c^{\alpha}w - f^{\alpha}] (\boldsymbol{x}) > t \} \\ = & \cup_{\alpha \in \Lambda} \{ \boldsymbol{x}\in \Omega : [A^{\alpha}:B + \boldsymbol{b}^{\alpha}\cdot \boldsymbol{c} + c^{\alpha}w - f^{\alpha}](\boldsymbol{x}) > t \}. \end{align}\] We denote by \(S_{t} := \{\boldsymbol{x}\in \Omega: \sup_{\alpha \in \Lambda} [A^{\alpha}:B + \boldsymbol{b}^{\alpha}\cdot \boldsymbol{c} + c^{\alpha}w - f^{\alpha}](\boldsymbol{x}) > t \}\).

Therefore, for any \(\boldsymbol{x}_{0} \in S_{t} \setminus (E_{0}\cup E_{1})\), there is \(\alpha_{0} \in \Lambda\) such that \[\begin{align} [A^{\alpha_{0}}:B + \boldsymbol{b}^{\alpha_{0}}\cdot \boldsymbol{c} + c^{\alpha_{0}}w - f^{\alpha_{0}}](\boldsymbol{x}_{0}) > t \end{align}\]

We define \(\epsilon_{0} = [A^{\alpha_{0}}:B + \boldsymbol{b}^{\alpha_{0}}\cdot \boldsymbol{c} + c^{\alpha_{0}}w - f^{\alpha_{0}}](\boldsymbol{x}_{0}) - t > 0\). According to the assumption (8 ), there is \(\tilde{\alpha}_{0} \in \tilde{\Lambda}\) such that \[\begin{align} & \vert (A^{\tilde{\alpha}_{0}} - A^{\alpha_{0}}):B(\boldsymbol{x}_{0})\vert + \vert (\boldsymbol{b}^{\tilde{\alpha}_{0}} - \boldsymbol{b}^{\alpha_{0}}) \cdot \boldsymbol{c} (\boldsymbol{x}_{0}) \vert \\ & \qquad + \vert (c^{\tilde{\alpha}_{0}}- c^{\alpha_{0}}) w(\boldsymbol{x}_{0}) \vert + \vert (f^{\tilde{\alpha}_{0}} - f^{\alpha_{0}})(\boldsymbol{x}_{0}) \vert < \frac{1}{2}\epsilon_{0}. \end{align}\] So we have \[\begin{align} [A^{\tilde{\alpha}_{0}}:B + \boldsymbol{b}^{\tilde{\alpha}_{0}}\cdot \boldsymbol{c} + c^{\tilde{\alpha}_{0}}w - f^{\tilde{\alpha}_{0}}](\boldsymbol{x}_{0}) - t > \epsilon_{0} - \frac{1}{2} \epsilon_{0} > 0. \end{align}\] Therefore, we have \[\begin{align} \boldsymbol{x}_{0} \in \{ \boldsymbol{x}\in\Omega \setminus(E_{0}\cup E_{1}):\sup_{\tilde{\alpha} \in \tilde{\Lambda}} [A^{\tilde{\alpha}}:B + \boldsymbol{b}^{\tilde{\alpha}}\cdot\boldsymbol{c} + c^{\tilde{\alpha}}w - f^{\tilde{\alpha}}](\boldsymbol{x}) > t \}. \end{align}\]

Since \(\boldsymbol{x}_{0} \in S_{t} \setminus (E_{0}\cup E_{1})\) is chosen arbitrarily, we have \[\begin{align} & S_{t}\setminus (E_{0} \cup E_{1}) = \{ \boldsymbol{x}\in \Omega \setminus (E_{0}\cup E_{1}): \sup_{\alpha \in \Lambda} [A^{\alpha}:B + \boldsymbol{b}^{\alpha}\cdot \boldsymbol{c} + c^{\alpha}w - f^{\alpha}](\boldsymbol{x}) > t \} \\ = & \{ \boldsymbol{x}\in \Omega \setminus (E_{0}\cup E_{1}): \sup_{\tilde{\alpha} \in \tilde{\Lambda}} [A^{\tilde{\alpha}}:B + \boldsymbol{b}^{\tilde{\alpha}}\cdot \boldsymbol{c} + c^{\tilde{\alpha}}w - f^{\tilde{\alpha}}] (\boldsymbol{x}) > t \} \\ = & \cup_{\tilde{\alpha}\in \tilde{\Lambda}} \{\boldsymbol{x}\in \Omega \setminus (E_{0}\cup E_{1}): [A^{\tilde{\alpha}}:B + \boldsymbol{b}^{\tilde{\alpha}}\cdot \boldsymbol{c} + c^{\tilde{\alpha}}w - f^{\tilde{\alpha}}] (\boldsymbol{x}) > t \}. \end{align}\]

Since \(\tilde{\Lambda}\) has at most countably many elements and \(E_{0},E_{1}\) have zero \(d\)-dimensional Lebesgue measure, we have that \(S_{t}\) is a measurable subset of \(\Omega\). Since \(t\in \mathbb{R}\) is chosen arbitrarily, we can conclude \[\begin{align} & \sup_{\alpha \in \Lambda} [A^{\alpha}:B + \boldsymbol{b}^{\alpha}\cdot \boldsymbol{c} + c^{\alpha}w - f^{\alpha}] \\ \end{align}\] is a measurable function on \(\Omega\) and (91 ) holds.

According to (91 ), (3 ,4 ) and the fact \(\sup_{\tilde{\alpha} \in \tilde{\Lambda}}\vert f^{\tilde{\alpha}}\vert \in L^{p}(\Omega)\) imply that \(\sup_{\alpha \in \Lambda} [A^{\alpha}:B + \boldsymbol{b}^{\alpha}\cdot \boldsymbol{c} + c^{\alpha}w - f^{\alpha}]\in L^{p}(\Omega)\) and (92 ) holds. ◻

Proof. According to the assumption (8 ), it is easy to see that for any \(\boldsymbol{x}\in \Omega \setminus E_{0}\) and for any \(\alpha\in\Lambda\), \(a_{ij}^{\alpha}(\boldsymbol{x}),b_{i}^{\alpha}(\boldsymbol{x}),c^{\alpha}(\boldsymbol{x}), f^{\alpha}(\boldsymbol{x})\) are all finite numbers for any \(1\leq i,j \leq d\). Then the assumption (8 ) is satisfied by \(\left(\varrho^{\alpha}A^{\alpha}, \varrho^{\alpha} \boldsymbol{b}^{\alpha}, \varrho^{\alpha}c^{\alpha}, \varrho^{\alpha}f^{\alpha}\right)\) with \(\tilde{\Lambda}\subset \Lambda\) and the set \(E_{0}\cup E_{1}\). So Lemma 13 implies that \(\sup_{\alpha\in \Lambda}\varrho^{\alpha}[A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}u - f^{\alpha}]\) is a measurable function on \(\Omega\).

Since \(w \in W^{2,1}(\Omega)\), there is a set \(E_{1}^{\prime}\subset \Omega\) with zero \(d\)-dimensional Lebesgue measure such that \(\frac{\partial^{2}w}{\partial x_{i}\partial x_{j}}(\boldsymbol{x}), \frac{\partial w}{\partial x_{i}}(\boldsymbol{x}), w(\boldsymbol{x})\) are all finite numbers for any \(\boldsymbol{x}\in \Omega \setminus E_{1}^{\prime}\) and for any \(1\leq i,j \leq d\). We choose \(\boldsymbol{x}_{0} \in \Omega \setminus (E_{0}\cup E_{1} \cup E_{1}^{\prime})\) arbitrarily. We claim that if \(\sup_{\alpha\in \Lambda}[A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}w - f^{\alpha}](\boldsymbol{x}_{0}) = 0\), then \[\begin{align} \sup_{\alpha\in \Lambda} \varrho^{\alpha}[A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}w - f^{\alpha}](\boldsymbol{x}_{0}) = 0. \end{align}\] The proof will be complete if this claim is true.

We prove this claim in the following. The fact \(\sup_{\alpha\in \Lambda}[A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}w - f^{\alpha}](\boldsymbol{x}_{0}) = 0\) is equivalent to \[\tag{96} \begin{align} \tag{97} & [A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}w - f^{\alpha}](\boldsymbol{x}_{0}) \leq 0, \qquad \forall \alpha \in \Lambda; \\ \tag{98} & \text{there is a sequence } \{ \alpha_{i} \}_{i=1}^{+\infty} \subset \Lambda \text{ such that } \\ \nonumber & \qquad \lim_{i\rightarrow +\infty}[A^{\alpha_{i}}:D^{2}w + \boldsymbol{b}^{\alpha_{i}} \cdot \nabla w + c^{\alpha_{i}}w - f^{\alpha_{i}}](\boldsymbol{x}_{0}) = 0. \end{align}\] By (94 ,97 ), we have \[\begin{align} \varrho^{\alpha}[A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}w - f^{\alpha}](\boldsymbol{x}_{0}) \leq 0, \qquad \forall \alpha \in \Lambda. \end{align}\] On the other hand, (94 ,97 , 98 ) implies \[\begin{align} \lim_{i\rightarrow +\infty} \varrho^{\alpha}[A^{\alpha_{i}}:D^{2}w + \boldsymbol{b}^{\alpha_{i}} \cdot \nabla w + c^{\alpha_{i}}w - f^{\alpha_{i}}](\boldsymbol{x}_{0}) = 0. \end{align}\] So the claim has been proven. Therefore, the proof is complete. ◻

Proof. Without losing of generality, we assume \(A^{\alpha}, \boldsymbol{b}\) and \(c^{\alpha}\) are zero matrices, zero vectors and zero for any \(\alpha \in \Lambda\), respectively.

Let \(n\) be an arbitrary positive integer. We define \(\Omega_{n} = \Omega \cap \{(l_{1}2^{-n},\cdots, l_{d}2^{-n}): (l_{1},\cdots,l_{d})\in \mathbb{Z}^{d}\}\). We define a countable subset \(\Lambda_{n}\) of \(\Lambda\). For any \(\boldsymbol{y}\in \Omega_{n}\) and any \(l\;in \mathbb{Z}\), if there is \(\alpha \in \Lambda\) such that \(l2^{-n}\leq f^{\alpha} < (l+1)2^{-n}\), then we put this \(\alpha\) into \(\Lambda_{n}\). Notice that though there may be more than one \(\alpha \in \Lambda\) qualified, we only choose at most one \(\alpha\) in the above procedure. Therefore, \(\Lambda_{n}\) has at most countably many elements. We define \(\tilde{\Lambda} = \cup_{n=1}^{\infty}\Lambda_{n} \subset \tilde{\Lambda}\).

We choose \(\boldsymbol{x}\in \Omega\) and \(\alpha \in \Lambda\) and \(\epsilon >0\) arbitrarily. Then there is a sequence \(\{\boldsymbol{y}_{n}\}_{n=1}^{\infty} \subset \tilde{\Lambda}\) such that \(\lim_{n \rightarrow +\infty}\vert \boldsymbol{y}_{n} - \boldsymbol{x}\vert = 0\) and \(\boldsymbol{y}_{n}\in \Lambda_{n}\) for any positive integer \(n\). Then by (99 ), there is a positive integer \(N_{0}\) such that \[\begin{align} \vert f^{\beta}(\boldsymbol{y}_{n}) - f^{\beta}(\boldsymbol{x}) \vert < \frac{\epsilon}{3}, \qquad 2^{-n} < \frac{\epsilon}{3}, \end{align}\] for any \(n \geq N_{0}\) and any \(\beta \in \Lambda\).

There is an integer \(l_{0}\) such that \(l2^{-N_{0}} \leq f^{\alpha}(\boldsymbol{y}_{N_{0}})< (l+1)2^{-N_{0}}\). By the construction of \(\Lambda_{n} \subset \tilde{\alpha}\), there is \(\tilde{\alpha} \in \Lambda_{n}\) such that \[\begin{align} l_{0}2^{-N_{0}} \leq f^{\tilde{\alpha}}(\boldsymbol{y}_{N_{0}}) < (l_{0} +1)2^{-N_{0}}. \end{align}\] Then we have \[\begin{align} \vert f^{\tilde{\alpha}}(\boldsymbol{y}_{N_{0}}) - f^{\alpha}(\boldsymbol{y}_{N_{0}})\vert < 2^{-N_{0}}< \frac{\epsilon}{3}. \end{align}\] We notice that \[\begin{align} \vert f^{\alpha}(\boldsymbol{y}_{n}) - f^{\alpha}(\boldsymbol{x}) \vert < \frac{\epsilon}{3}, \qquad \vert f^{\tilde{\alpha}}(\boldsymbol{y}_{n}) - f^{\tilde{\alpha}}(\boldsymbol{x}) \vert < \frac{\epsilon}{3}. \end{align}\] Then we have that \[\begin{align} \vert f^{\alpha}(\boldsymbol{x}) - f^{\tilde{\alpha}}(\boldsymbol{x}) \vert < \epsilon. \end{align}\] We can conclude that the proof is complete. ◻

Proof. Let \(p\) be any number in the range described in this Theorem.

We denote by \[\begin{align} \Vert v\Vert_{W_{\lambda}^{2,p}(\Omega)}= \big( \Vert D^{2}v\Vert_{L^{p}(\Omega)}^{p} + 2\lambda\Vert \nabla v\Vert_{L^{p}(\Omega)}^{p} + \lambda^{2}\Vert v\Vert_{L^{p}(\Omega)}^{p} \big)^{1/p}, \qquad \forall v \in W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega). \end{align}\]

According to the assumption (8 ), it is easy to see that for any \(\boldsymbol{x}\in \Omega \setminus E_{0}\) and for any \(\alpha\in\Lambda\), \(a_{ij}^{\alpha}(\boldsymbol{x}),b_{i}^{\alpha}(\boldsymbol{x}),c^{\alpha}(\boldsymbol{x}), f^{\alpha}(\boldsymbol{x})\) are all finite numbers for any \(1\leq i,j \leq d\).

For any \(\alpha \in \Lambda\), we define a function \(\gamma^{\lambda, \alpha}\) on \(\Omega\) such that for any \(\boldsymbol{x}\in\Omega\) almost everywhere, \[\begin{align} \label{def95gamma95lambda95alpha} \gamma^{\lambda,\alpha}(\boldsymbol{x}) = \dfrac{\Sigma_{1\leq i \leq d}a_{ii}^{\alpha} (\boldsymbol{x})+ \lambda^{-1}\vert c^{\alpha}(\boldsymbol{x})\vert}{\vert A^{\alpha}(\boldsymbol{x}) \vert^{2} + (2\lambda)^{-1}\vert \boldsymbol{b}^{\alpha}(\boldsymbol{x})\vert^{2} + (\lambda^{-1}c^{\alpha}(\boldsymbol{x}))^{2}}. \end{align}\tag{102}\] Here \(\vert B\vert = \big(\Sigma_{1\leq i,j \leq d}\vert b_{ij} \vert^{2} \big)^{1/2}\) for any \(B \in \mathbb{R}^{d\times d}\) and \(\vert \boldsymbol{\beta} \vert = \big(\Sigma_{1\leq i \leq d} \vert \beta_{i}\vert^{2} \big)^{1/2}\) for any \(\boldsymbol{\beta} \in \mathbb{R}^{d}\).

By (3 ,4 ) and the fact \(\tilde{\Lambda}\) is a countable set, there is a set \(E_{1} \supset E_{0}\) with zero \(d\)-dimensional Lebesgue measure such that

\[\begin{align} & \gamma^{\lambda,\tilde{\alpha}}(\boldsymbol{x}) = \dfrac{\Sigma_{1\leq i \leq d}a_{ii}^{\tilde{\alpha}} (\boldsymbol{x})+ \lambda^{-1}\vert c^{\tilde{\alpha}}(\boldsymbol{x}) \vert}{\vert A^{\tilde{\alpha}}(\boldsymbol{x})\vert^{2}+(2\lambda)^{-1} \vert \boldsymbol{b}^{\tilde{\alpha}}(\boldsymbol{x})\vert^{2} + (\lambda^{-1}c^{\tilde{\alpha}}(\boldsymbol{x}))^{2}}, \\ & C_{1} \leq \gamma^{\lambda,\tilde{\alpha}}(\boldsymbol{x}) \leq C_{2}, \end{align}\] for any \(\boldsymbol{x}\in \Omega \setminus E_{1}\) and any \(\tilde{\alpha} \in \tilde{\Lambda}\). Here \(0< C_{1} \leq C_{2}<+\infty\) are two constants independent of \(\boldsymbol{x}\in \Omega \setminus E_{1}\) and \(\tilde{\alpha} \in \tilde{\Lambda}\). Then by the assumption (8 ), we have \[\begin{align} & \gamma^{\lambda,\alpha}(\boldsymbol{x}) = \dfrac{\Sigma_{1\leq i \leq d}a_{ii}^{\alpha} (\boldsymbol{x})+ \lambda^{-1}\vert c^{\alpha}(\boldsymbol{x})\vert}{ \vert A^{\alpha}(\boldsymbol{x})\vert^{2}+(2\lambda)^{-1} \vert \boldsymbol{b}^{\alpha}(\boldsymbol{x})\vert^{2} +(\lambda^{-1}c^{\alpha}(\boldsymbol{x}))^{2}},\\ & C_{1} \leq \gamma^{\lambda,\alpha}(\boldsymbol{x}) \leq C_{2}, \end{align}\] for any \(\boldsymbol{x}\in\Omega\setminus E_{1}\) and any \(\alpha \in \Lambda\).

Obviously, the assumption (8 ) is satisfied by \((\gamma^{\lambda,\alpha}A^{\alpha},\gamma^{\lambda,\alpha}\boldsymbol{b}^{\alpha}, \gamma^{\lambda,\alpha}c^{\alpha},\gamma^{\lambda,\alpha}f^{\alpha})\) with \(\tilde{\Lambda}\subset \Lambda\) and the set \(E_{1}\). Then by Lemma 13 and (3 ,4 ), we have that for any \(w\in W^{2,p}(\Omega) \cap W_{0}^{1,p}(\Omega)\), \[\begin{align} \label{hjb95sol95bound} & \sup_{\alpha \in \Lambda} \gamma^{\lambda,\alpha}[A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}w - f^{\alpha}] \in L^{p}(\Omega),\\ \nonumber & \Vert \sup_{\alpha \in \Lambda} \gamma^{\lambda,\alpha}[A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}w - f^{\alpha}]\Vert_{L^{p}(\Omega)} \\ \nonumber \leq & C \big( \Vert w\Vert_{W^{2,p}(\Omega)} + \Vert \sup_{\tilde{\alpha} \in \tilde{\Lambda}} \vert f^{\tilde{\alpha}} \vert \Vert_{L^{p}(\Omega)}\big). \end{align}\tag{103}\] Then Lemma 14 implies that for any \(w \in W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)\), \(\sup_{\alpha \in \Lambda} [A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}w - f^{\alpha}] = 0\) in \(\Omega\) is equivalent to \[\begin{align} \sup_{\alpha \in \Lambda} \gamma^{\lambda,\alpha}[A^{\alpha}:D^{2}w + \boldsymbol{b}^{\alpha}\cdot \nabla w + c^{\alpha}w - f^{\alpha}] = 0 \quad \text{ in } \Omega, \end{align}\] which is equivalent to \[\begin{align} \label{hjb95pde95equivalent} \sup_{\tilde{\alpha} \in \tilde{\Lambda}} \gamma^{\lambda,\tilde{\alpha}}[A^{\tilde{\alpha}}:D^{2}w + \boldsymbol{b}^{\tilde{\alpha}}\cdot \nabla w + c^{\tilde{\alpha}}w - f^{\tilde{\alpha}}] = 0 \quad \text{ in } \Omega, \end{align}\tag{104}\] due to (91 ).

We define a mapping \(M_{\lambda}: L^{p}(\Omega)\rightarrow W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)\) by \[\begin{align} (\Delta - \lambda )(M_{\lambda} g) = g \text{ in } \Omega, \end{align}\] for any \(g \in L^{p}(\Omega)\). According to Theorem 3, we have that for any \(g\in L^{p}(\Omega)\), \[\begin{align} \label{coercivity95poisson95lambda95ineq} \Vert M_{\lambda}g\Vert_{W_{\lambda}^{2,p}(\Omega)} \leq \underline{C}_{p,\lambda} \Vert g\Vert_{L^{p}(\Omega)}, \end{align}\tag{105}\] where the constant \(\underline{C}_{p,\lambda}\) may depend on \(p\) and \(\lambda\). In fact, \(\underline{C}_{2,\lambda} = 1\) for any \(\lambda \geq 1\) with convex domain due to [20].

By (104 ) and the definition of \(M_{\lambda}\), \(u \in W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)\) satisfies the Hamilton-Jacobi-Bellman equation (2 ) if and only if \[\begin{align} - u = M_{\lambda}\big( \sup_{\tilde{\alpha} \in \tilde{\Lambda}} \gamma^{\lambda,\tilde{\alpha}}[A^{\tilde{\alpha}}:D^{2}u + \boldsymbol{b}^{\tilde{\alpha}}\cdot \nabla u + c^{\tilde{\alpha}}u - f^{\tilde{\alpha}}] - (\Delta u - \lambda u)\big). \end{align}\] On the other hand, (103 ) and (105 ) imply that \[\begin{align} \label{hjb95pde95contraction} w \rightarrow T_{\lambda}(w):= M_{\lambda}\big( \sup_{\tilde{\alpha} \in \Lambda} \gamma^{\lambda,\tilde{\alpha}} [A^{\tilde{\alpha}}:D^{2}w + \boldsymbol{b}^{\tilde{\alpha}}\cdot \nabla w + c^{\tilde{\alpha}}w - f^{\tilde{\alpha}}] - (\Delta - \lambda)w \big) \end{align}\tag{106}\] is a well defined mapping from \(W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)\) into itself. Therefore, in order to to the existence and uniqueness of the solution of (2 ), it is sufficient to prove that the mapping (106 ) is a contraction with respect to the norm \(\Vert \cdot\Vert_{W_{\lambda}^{2,p}(\Omega)}\).

By (100 ) and the proof of [20] (\(\underline{\kappa}\) will be determined later), we have that for any \(w_{1},w_{2}\in W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)\), \[\begin{align} \label{hjb95operator95diff95ineq} & \Vert \sup_{\tilde{\alpha} \in \Lambda} \gamma^{\lambda,\tilde{\alpha}} [A^{\tilde{\alpha}}:D^{2}w_{1} + \boldsymbol{b}^{\tilde{\alpha}}\cdot \nabla w_{1} + c^{\tilde{\alpha}}w_{1} - f^{\tilde{\alpha}}] \\ \nonumber & \qquad - \sup_{\tilde{\alpha} \in \Lambda} \gamma^{\lambda,\tilde{\alpha}} [A^{\tilde{\alpha}}:D^{2}w_{2} + \boldsymbol{b}^{\tilde{\alpha}}\cdot \nabla w_{2} + c^{\tilde{\alpha}}w_{2} - f^{\tilde{\alpha}}] - (\Delta - \lambda)(w_{1} - w_{2})\Vert_{L^{p}(\Omega)} \\ \nonumber \leq & \underline{C}_{p,\lambda}^{\prime}\sqrt{1 - \epsilon}\Vert (w_{1}-w_{2})\Vert_{W_{\lambda}^{2,p}(\Omega)}. \end{align}\tag{107}\] In fact, \(\underline{C}_{2,\lambda}^{\prime} = 1\) on convex domain.

By (105 ,107 ), we have \[\begin{align} \label{hjb95pde95contraction95bound1} & \Vert M_{\lambda}\big(\sup_{\tilde{\alpha} \in \Lambda} \gamma^{\lambda,\tilde{\alpha}} [A^{\tilde{\alpha}}:D^{2}w_{1} + \boldsymbol{b}^{\tilde{\alpha}}\cdot \nabla w_{1} + c^{\tilde{\alpha}}w_{1} - f^{\tilde{\alpha}}]\big) \\ \nonumber & \qquad - M_{\lambda}\big(\sup_{\tilde{\alpha} \in \Lambda} \gamma^{\lambda,\tilde{\alpha}} [A^{\tilde{\alpha}}:D^{2}w_{2} + \boldsymbol{b}^{\tilde{\alpha}}\cdot \nabla w_{2} + c^{\tilde{\alpha}}w_{2} - f^{\tilde{\alpha}}]\big) - (w_{1} - w_{2})\Vert_{L^{p}(\Omega)} \\ \nonumber \leq & \underline{C}_{p,\lambda}\underline{C}_{p,\lambda}^{\prime} \sqrt{1 - \epsilon}\Vert (w_{1}-w_{2})\Vert_{W_{\lambda}^{2,p}(\Omega)}. \end{align}\tag{108}\]

We can choose \[\begin{align} \label{def95underline95kappa} \underline{\kappa}= 1 - \frac{1}{(\underline{C}_{2,\lambda} \underline{C}_{2,\lambda}^{\prime})^{2}}. \end{align}\tag{109}\] By (105 ) and (107 ), if (100 ) is satisfied with \(\epsilon \in (\underline{\kappa},1)\), \(T_{\lambda}\) is a contraction. Then there is a unique solution \(u\in W^{2,p}(\Omega)\cap W_{0}^{1,p} (\Omega)\) satisfying the Hamilton-Jacobi-Bellman equation (2 ) which is equivalent to (104 ). Since \(\underline{C}_{2,\lambda} = \underline{C}_{2,\lambda}^{\prime} = 1\), \(\underline{\kappa} = 0\) if \(p = 2\) with convex domain.

By (108 ) with \(w_{1}=u\) and \(w_{2}=0\), we have \[\begin{align} \Vert u - M_{\lambda}(\sup_{\tilde{\alpha}\in \tilde{\Lambda}} \gamma^{\tilde{\alpha}}f^{\tilde{\alpha}}) \Vert_{W_{\lambda}^{2,p}(\Omega)} \leq \underline{C}_{p,\lambda}\underline{C}_{p,\lambda}^{\prime} \sqrt{1 - \epsilon}\Vert u\Vert_{W_{\lambda}^{2,p}(\Omega)}. \end{align}\] Since \(\underline{C}_{p,\lambda} \underline{C}_{p,\lambda}^{\prime}\sqrt{1-\epsilon}<1\), we have \[\begin{align} \Vert u\Vert_{W^{2,p}(\Omega)} \leq C \Vert \sup_{\tilde{\alpha} \in \tilde{\Lambda}}\vert f^{\tilde{\alpha}}\vert\Vert_{L^{p}(\Omega)}. \end{align}\] Therefore, the proof is complete. ◻

Proof. Let \(p\) be any number in the range described in this Theorem.

Since (3 ,4 ) and the assumption (8 ) hold, it is obvious to mimic the proof of Theorem 6 (the argument to obtain (104 )) to have that the finite element method (18 ) is equivalent to find \(u_{h}\in V_{h}\) satisfying \[\begin{align} \label{hjb95fem95equivalent} (\sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla u_{h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla u_{h} + c^{\tilde{\alpha}}u_{h} - f^{\tilde{\alpha}}], v_{h})_{\Omega} = 0, \qquad \forall v_{h} \in V_{h}. \end{align}\tag{114}\] We denote by \[\begin{align} \Vert v_{h}\Vert_{W_{h,\lambda}^{2,p}(\Omega)}= \big( \Vert v_{h}\Vert_{W_{h}^{2,p}(\Omega)}^{p} + 2\lambda\Vert \nabla v_{h}\Vert_{L^{p}(\Omega)}^{p} + \lambda^{2}\Vert v_{h}\Vert_{L^{p}(\Omega)}^{p} \big)^{1/p}, \qquad \forall v_{h} \in V_{h}. \end{align}\]

For any \(w_{h} \in V_{h}\), we define \(F_{h}(w_{h}) \in V_{h}\) by \[\begin{align} &(F_{h}(w_{h}), v_{h})_{\Omega} \\ = &(\sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla u_{h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla u_{h} + c^{\tilde{\alpha}}u_{h} - f^{\tilde{\alpha}}], v_{h})_{\Omega}, \quad \forall v_{h} \in V_{h}. \end{align}\] We define a mapping \(M_{\lambda,h}:V_{h} \rightarrow V_{h}\) such that for any \(g_{h} \in V_{h}\), \[\begin{align} -(\nabla (M_{\lambda,h}(g_{h})), \nabla v_{h})_{\Omega} - \lambda (M_{\lambda,h}(g_{h}),v_{h})_{\Omega} = (g_{h}, v_{h})_{\Omega},\qquad \forall v_{h} \in V_{h}. \end{align}\] It is easy to see that for any \(w_{h}\in V_{h}\), \[\begin{align} M_{\lambda,h}(\mathcal{L}_{I_{d},h}w_{h}-\lambda w_{h}) = w_{h}, \end{align}\] where \(\mathcal{L}_{I_{d},h}w_{h} \in V_{h}\) satisfying \((\mathcal{L}_{I_{d},h}w_{h}, v_{h})_{\Omega} =(I_{d}:\overline{\nabla}_{h}(\nabla w_{h}),v_{h})_{\Omega}\) for any \(v_{h} \in V_{h}\). Then by Theorem 2, we have that \[\begin{align} \label{coercivity95poisson95fem95lambda95ineq} \Vert M_{\lambda,h}(g_{h}) \Vert_{W_{h,\lambda}^{2,p}(\Omega)} \leq C_{p,\lambda} \Vert g_{h}\Vert_{L^{p}(\Omega)}, \qquad \forall g_{h}\in V_{h}, \forall 0 < h < h_{3}, \end{align}\tag{115}\] where the constant \(C_{p,\lambda}\) may depend on \(p\) and \(\lambda\).

We notice that \(u_{h}\in V_{h}\) is a numerical solution of the finite element method (114 ) if and only if \(F_{h}(u_{h}) = 0\) in \(\Omega\), which is equivalent to \[\begin{align} -u_{h} = M_{\lambda,h}\big(F_{h}(u_{h}) - (\mathcal{L}_{I_{d},h}u_{h} -\lambda u_{h}) \big). \end{align}\] Therefore, to prove the existence and uniqueness of (114 ), it is sufficient to prove the mapping \[\begin{align} \label{hjb95fem95contraction} w_{h} \rightarrow M_{\lambda,h}\big( F_{h}(w_{h}) - (\mathcal{L}_{I_{d},h}w_{h} -\lambda w_{h}) \big) \end{align}\tag{116}\] is a contraction from \(V_{h}\) into itself with respect to the \(\Vert \cdot \Vert_{W_{h,\lambda}^{2,p}(\Omega)}\) norm.

We choose \(w_{1,h},w_{2,h} \in V_{h}\) arbitrarily. We notice that for any \(v_{h}\in V_{h}\) \[\begin{align} & F_{h}(v_{h}) - (\mathcal{L}_{I_{d},h}v_{h}-\lambda v_{h}) \\ = & P_{h}\big( \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla v_{h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla v_{h} + c^{\tilde{\alpha}}v_{h} - f^{\tilde{\alpha}}] - (\overline{\nabla}_{h}(\nabla v_{h})-\lambda v_{h})\big). \end{align}\] Here \(P_{h}\) is the standard \(L^{2}\)-orthogonal projection onto \(V_{h}\). Therefore, we have \[\begin{align} \label{hjb95fem95opreator95projection} & \Vert \big(F_{h}(w_{1,h})-(\mathcal{L}_{I_{d},h}w_{1,h}-\lambda w_{1,h})\big) \\ \nonumber &\qquad - \big(F_{h}(w_{2,h})-(\mathcal{L}_{I_{d},h}w_{2,h}-\lambda w_{2,h})\big) \Vert_{L^{p}(\Omega)} \\ \nonumber \leq & C_{p}^{\prime} \Vert \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla w_{1,h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla w_{1,h} + c^{\tilde{\alpha}}w_{1,h} - f^{\tilde{\alpha}}] \\ \nonumber & \qquad - \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla w_{2,h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla w_{2,h} + c^{\tilde{\alpha}}w_{2,h} - f^{\tilde{\alpha}}] \\ \nonumber & \qquad -(\overline{\nabla}_{h}(\nabla (w_{1,h}-w_{2,h}))-\lambda(w_{1,h}-w_{2,h})) \Vert_{L^{p}(\Omega)} \end{align}\tag{117}\] According to the proof of [20], we have that for any \(\boldsymbol{x}\in \Omega\) almost everywhere, \[\begin{align} \label{hjb95fem95operator95diff} & \vert \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla w_{1,h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla w_{1,h} + c^{\tilde{\alpha}}w_{1,h} - f^{\tilde{\alpha}}](\boldsymbol{x}) \\ \nonumber & \qquad - \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla w_{2,h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla w_{2,h} + c^{\tilde{\alpha}}w_{2,h} - f^{\tilde{\alpha}}](\boldsymbol{x}) \\ \nonumber & \qquad -(\overline{\nabla}_{h}(\nabla (w_{1,h}-w_{2,h}))-\lambda(w_{1,h}-w_{2,h})) (\boldsymbol{x})\vert \\ \nonumber \leq & \big(\sup_{\tilde{\alpha}\in \tilde{\Lambda}}\sqrt{E^{\tilde{\alpha}}(\boldsymbol{x})} \big)\sqrt{\vert \overline{\nabla}_{h}(\nabla w_{h}(\boldsymbol{x}))\vert^{2} +2\lambda \vert \nabla w_{h}(\boldsymbol{x})\vert^{2} + \lambda^{2}\vert w_{h}(\boldsymbol{x})\vert^{2}}, \end{align}\tag{118}\] where \(w_{h}=w_{1,h} - w_{2,h}\) and \[\begin{align} E^{\tilde{\alpha}}(\boldsymbol{x}) = & \vert \gamma^{\tilde{\alpha}}(\boldsymbol{x}) A^{\tilde{\alpha}}(\boldsymbol{x}) - I_{d} \vert^{2} + \vert \gamma^{\tilde{\alpha}}(\boldsymbol{x}) \vert^{2} \frac{\vert \boldsymbol{b}^{\tilde{\alpha}}(\boldsymbol{x}) \vert^{2}}{2\lambda} + \dfrac{\vert \lambda + c^{\tilde{\alpha}}(\boldsymbol{x}) \vert^{2}}{\lambda^{2}}\\ = & \frac{\vert \gamma^{\tilde{\alpha}}(\boldsymbol{x}) \vert^{2} \vert c^{\tilde{\alpha}}(\boldsymbol{x})\vert^{2}}{\lambda^{2}} +\gamma^{\tilde{\alpha}}(\boldsymbol{x})\big( \frac{ \gamma^{\tilde{\alpha}}(\boldsymbol{x})\vert \boldsymbol{b}^{\tilde{\alpha}}(\boldsymbol{x}) \vert}{2\lambda} - \frac{2\vert c^{\tilde{\alpha}}(\boldsymbol{x})\vert}{\lambda}\big)\\ \nonumber & \qquad + \big(d+1 +\vert \gamma^{\tilde{\alpha}}(\boldsymbol{x}) \vert^{2} \vert A^{\tilde{\alpha}}(\boldsymbol{x})\vert^{2} - 2 \text{Tr} A^{\tilde{\alpha}}(\boldsymbol{x})\big). \end{align}\] The last equation above is due to the fact \(c^{\tilde{\alpha}}(\boldsymbol{x})\leq 0\) for any \(\tilde{\alpha}\in \tilde{\Lambda}\). By the definition of \(\gamma^{\tilde{\alpha}}\) (see (17 )), \[\begin{align} E^{\tilde{\alpha}}(\boldsymbol{x}) = & \frac{(\text{Tr}A^{\tilde{\alpha}}(\boldsymbol{x}))^{2} \vert c^{\tilde{\alpha}}(\boldsymbol{x})\vert^{2}}{\vert A^{\tilde{\alpha}} (\boldsymbol{x}) \vert^{4}}\lambda^{-2} + \frac{\text{Tr}A^{\tilde{\alpha}}(\boldsymbol{x})}{\vert A^{\tilde{\alpha}} (\boldsymbol{x}) \vert^{2}}\big( \frac{\text{Tr}A^{\tilde{\alpha}}(\boldsymbol{x})}{\vert A^{\tilde{\alpha}}(\boldsymbol{x}) \vert^{2}} \frac{\vert \boldsymbol{b}^{\tilde{\alpha}} (\boldsymbol{x})\vert^{2}}{2} - 2 \vert c^{\tilde{\alpha}}(\boldsymbol{x})\vert\big)\lambda^{-1}\\ & \qquad + (d+1 - \frac{(\text{Tr}A^{\tilde{\alpha}} (\boldsymbol{x}))^{2}}{\vert A^{\tilde{\alpha}}(\boldsymbol{x}) \vert^{2}}). \end{align}\] Then for any \(\epsilon \in (0,1]\), \(E^{\tilde{\alpha}}(\boldsymbol{x})\leq 1 - \epsilon\) is equivalent to \[\begin{align} & -\frac{(\text{Tr}A^{\tilde{\alpha}}(\boldsymbol{x}))^{2} \vert c^{\tilde{\alpha}}(\boldsymbol{x})\vert^{2}}{\vert A^{\tilde{\alpha}} (\boldsymbol{x}) \vert^{4}}\lambda^{-2} - \frac{\text{Tr}A^{\tilde{\alpha}}(\boldsymbol{x})}{\vert A^{\tilde{\alpha}} (\boldsymbol{x}) \vert^{2}}\big( \frac{\text{Tr}A^{\tilde{\alpha}}(\boldsymbol{x})}{\vert A^{\tilde{\alpha}}(\boldsymbol{x}) \vert^{2}} \frac{\vert \boldsymbol{b}^{\tilde{\alpha}} (\boldsymbol{x})\vert^{2}}{2} - 2 \vert c^{\tilde{\alpha}}(\boldsymbol{x})\vert\big)\lambda^{-1}\\ & \qquad + \frac{(\text{Tr}A^{\tilde{\alpha}} (\boldsymbol{x}))^{2}}{\vert A^{\tilde{\alpha}}(\boldsymbol{x}) \vert^{2}} \geq d+\epsilon, \end{align}\] which is equivalent to \[\begin{align} \dfrac{\vert A^{\alpha}(\boldsymbol{x})\vert^{2}(\text{Tr} A^{\alpha} (\boldsymbol{x}))^{-2}}{1 + 2\lambda^{-1}\vert c^{\alpha}(\boldsymbol{x})\vert (\text{Tr}A^{\alpha}(\boldsymbol{x}))^{-1} - \vert A^{\alpha}(\boldsymbol{x})\vert^{-2} \big( \lambda^{-2} \vert c^{\alpha}(\boldsymbol{x})\vert^{2} + (2\lambda)^{-1}\vert \boldsymbol{b}^{\alpha}(\boldsymbol{x})\vert^{2}\big)} \leq \dfrac{1}{d + \epsilon}. \end{align}\] By (110 ), we have that for any \(\epsilon \in (\kappa,1)\), \[\begin{align} \label{hjb95fem95operator95constant95bound} E^{\tilde{\alpha}}(\boldsymbol{x}) \leq & 1 - \epsilon, \end{align}\tag{119}\] for any \(\boldsymbol{x}\in \Omega\) almost everywhere and for any \(\tilde{\alpha} \in \tilde{\Lambda}\). Here the constant \(\kappa \in [0,1)\) will be determined later.

By (117 ,118 ,119 ), we have \[\begin{align} \label{hjb95fem95operator95diff95ineq} & \Vert \big(F_{h}(w_{1,h})-(\mathcal{L}_{I_{d},h}w_{1,h}-\lambda w_{1,h})\big) \\ \nonumber &\qquad - \big(F_{h}(w_{2,h})-(\mathcal{L}_{I_{d},h}w_{2,h}-\lambda w_{2,h})\big)\Vert_{L^{p}(\Omega)} \\ \nonumber \leq & C_{p,\lambda}^{\prime}C_{p}^{\prime}\sqrt{1 - \epsilon} \big( \Vert \overline{\nabla}_{h}(\nabla (w_{1,h}-w_{2,h})) \Vert_{L^{p}(\Omega)}^{p} \\ \nonumber & \qquad + 2\lambda \Vert \nabla (w_{1,h}-w_{2,h}) \Vert_{L^{p}(\Omega)}^{p} + \lambda^{2}\Vert w_{1,h}-w_{2,h} \Vert_{L^{p}(\Omega)}^{p}\big)^{1/p}. \end{align}\tag{120}\]

By (1), it is easy to see that \[\begin{align} \label{discrete95deri95bound1} \Vert \overline{\nabla}_{h}(\nabla (w_{1,h}-w_{2,h})) \Vert_{L^{p}(\Omega)}^{p} \leq C_{p}^{\prime\prime} \Vert w_{1,h}-w_{2,h}\Vert_{W_{h}^{2,p}(\Omega)}. \end{align}\tag{121}\] By (115 , 120 , 121 ), we have \[\begin{align} \label{hjb95fem95contraction95ineq} & \Vert M_{\lambda,h}\big(F_{h}(w_{1,h})-(\mathcal{L}_{I_{d},h}w_{1,h}-\lambda w_{1,h})\big)\\ \nonumber &\qquad - M_{\lambda,h}\big(F_{h}(w_{2,h})-(\mathcal{L}_{I_{d},h}w_{2,h}-\lambda w_{2,h})\big)\Vert_{W_{h,\lambda}^{2.p}(\Omega)} \\ \nonumber \leq & \big( C_{p,\lambda}C_{p,\lambda}^{\prime}C_{p}^{\prime} C_{p}^{\prime\prime}\big) \sqrt{1-\epsilon}\Vert w_{1,h}-w_{2,h}\Vert_{W_{h,\lambda}^{2,p}(\Omega)}. \end{align}\tag{122}\] In order to make the mapping (116 ) a contraction, it is sufficient to choose \[\begin{align} \label{def95kappa} \kappa = 1 - \dfrac{1}{\big( C_{p,\lambda} C_{p,\lambda}^{\prime}C_{p}^{\prime}C_{p}^{\prime\prime}\big)^{2}}, \end{align}\tag{123}\] such that \(\big( C_{p,\lambda}C_{p,\lambda}^{\prime}C_{p}^{\prime} C_{p}^{\prime\prime}\big) \sqrt{1-\epsilon}<1\) for any \(\epsilon \in (\kappa,1]\). Therefore, the existence and uniqueness of numerical solution of the finite element method (18 ) is proven.

Let \(u_{h}\in V_{h}\) be the numerical solution of (18 ). Then we have \[\begin{align} F_{h}(u_{h}) = 0 \text{ and } u_{h} = - M_{\lambda,h}\big(F_{h}(u_{h})-(\mathcal{L}_{I_{d},h}u_{h}-\lambda u_{h})\big) \text{ in }\Omega. \end{align}\] By (122 ) with \(w_{1,h}=u_{h}\) and \(w_{2,h} = 0\), we have \[\begin{align} \Vert u_{h} - M_{\lambda,h}\big( \sup_{\tilde{\alpha}\in \tilde{\Lambda}} \gamma^{\tilde{\alpha}} f^{\tilde{\alpha}}\big)\Vert_{W_{h,\lambda}^{2,p}(\Omega)} \leq \big( C_{p,\lambda}C_{p,\lambda}^{\prime}C_{p}^{\prime\prime}\big) \sqrt{1-\epsilon}\Vert u_{h}\Vert_{W_{h,\lambda}^{2,p}(\Omega)}. \end{align}\] Since \(\big( C_{p,\lambda}C_{p,\lambda}^{\prime}C_{p}^{\prime} C_{p}^{\prime\prime}\big) \sqrt{1-\epsilon}<1\) for any \(\epsilon \in (\kappa,1]\), we have that \[\begin{align} \label{hjb95numerical95sol95bound1} \Vert u_{h}\Vert_{W^{2,p}_{h}(\Omega)} \leq C \Vert \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\vert f^{\tilde{\alpha}}\vert \Vert_{L^{p}(\Omega)}. \end{align}\tag{124}\]

It is straightforward to verify that if there are \(\epsilon \in (0,1]\) and \(\lambda>0\) such that (110 ) holds, then (100 ) in Theorem 6 holds as well for the same \(\epsilon\) and \(\lambda\). Furthermore, it is easy to check that \(\kappa \geq \underline{\kappa}\) which is introduced in (109 ). Then by Theorem 6, there is a unique solution \(u\in W_{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)\) of the Hamilton-Jacobi-Bellman equation (2 ).

We choose \(\chi_{h} \in V_{h}\) arbitrarily. By (122 ) with \(w_{1,h} = u_{h}\) and \(w_{2,h}=\chi_{h}\), we have \[\begin{align} \Vert M_{\lambda,h}(F_{h}(\chi_{h})) + (u_{h} - \chi_{h}) \Vert_{W_{h,\lambda}^{2,p} (\Omega)}\leq \big( C_{p,\lambda}C_{p,\lambda}^{\prime}C_{p}^{\prime} C_{p}^{\prime\prime}\big)\sqrt{1-\epsilon} \Vert u_{h} - \chi_{h}\Vert_{W_{h,\lambda}^{2,p}(\Omega)}. \end{align}\] By (115 ) and the fact \(\big( C_{p,\lambda}C_{p,\lambda}^{\prime}C_{p}^{\prime} C_{p}^{\prime\prime}\big)\sqrt{1-\epsilon}<1\), we have \[\begin{align} \Vert u_{h} - \chi_{h}\Vert_{W_{h}^{2,p}(\Omega)} \leq C \Vert F_{h}(\chi_{h})\Vert_{L^{p}(\Omega)}. \end{align}\]

By the same argument in the proof of Theorem 6, it is easy to see that \[\begin{align} \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\alpha} [A^{\tilde{\alpha}}:D^{2}u+\boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla u + c^{\tilde{\alpha}}u - f^{\tilde{\alpha}}] = 0 \text{ in } \Omega. \end{align}\]

We notice that \[\begin{align} &\Vert F_{h}(\chi_{h})\Vert_{L^{p}(\Omega)} \leq C \Vert \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla \chi_{h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla \chi_{h} + c^{\tilde{\alpha}}\chi_{h} - f^{\tilde{\alpha}}]\Vert_{L^{p}(\Omega)}\\ = & C \Vert \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla \chi_{h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla \chi_{h} + c^{\tilde{\alpha}}\chi_{h} - f^{\tilde{\alpha}}] \\ & \qquad - \sup_{\tilde{\alpha}\in \tilde{\Lambda}}\gamma^{\alpha} [A^{\tilde{\alpha}}:D^{2}u+\boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla u + c^{\tilde{\alpha}}u - f^{\tilde{\alpha}}] \Vert_{L^{p}(\Omega)} \\ \leq & C \Vert \sup_{\tilde{\alpha}\in \tilde{\Lambda}} \vert \gamma^{\tilde{\alpha}} [A^{\tilde{\alpha}}:\overline{\nabla}_{h} (\nabla \chi_{h}) + \boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla \chi_{h} + c^{\tilde{\alpha}}\chi_{h} - f^{\tilde{\alpha}}] \\ & \qquad - \gamma^{\alpha} [A^{\tilde{\alpha}}:D^{2}u+\boldsymbol{b}^{\tilde{\alpha}}\cdot\nabla u + c^{\tilde{\alpha}}u - f^{\tilde{\alpha}}]\vert \Vert_{L^{p}(\Omega)}. \end{align}\] Then by (3 ,4 ), we have \[\begin{align} &\Vert F_{h}(\chi_{h})\Vert_{L^{p}(\Omega)} \leq C \big( \Vert \overline{\nabla}_{h}(\nabla \chi_{h}) - D^{2}u\Vert_{L^{p} (\Omega)} + \Vert \chi_{h} - u\Vert_{W^{1,p}(\Omega)}\big) \\ \leq & C \big( \Vert \overline{\nabla}_{h}(\nabla (\chi_{h}-u)) \Vert_{L^{p}(\Omega)} + \Vert \overline{\nabla}_{h}(\nabla u) - \nabla (\nabla u) \Vert_{L^{p}(\Omega)} + \Vert \chi_{h} - u\Vert_{W^{1,p}(\Omega)}\big). \end{align}\] By Definition 1 of the discrete gradient operator \(\overline{\nabla}_{h}\), we have \[\begin{align} & \Vert \overline{\nabla}_{h}(\nabla \chi_{h}) - D^{2}u\Vert_{L^{p} (\Omega)} \leq C \Vert u - \chi_{h}\Vert_{W_{h}^{2,p}(\Omega)}, \\ & \Vert \overline{\nabla}_{h}(\nabla u) - \nabla (\nabla u) \Vert_{L^{p}(\Omega)} = \Vert \overline{P}_{h}D^{2}u - D^{2}u\Vert_{L^{p}(\Omega)}. \end{align}\] Therefore, (111 ) holds.

We can conclude that proof is complete. ◻