May 25, 2026
We study the Laplace equation posed in the unbounded rectangular domain \(\Pi = I \times (0,\infty)\) with \(I= (0,2\pi)\), and subject to nonlocal boundary conditions on \(\partial \Pi\) in the trace sense. The analysis is carried out in the Bochner-Sobolev space \(W^2_{p,1}(\Pi;{\mathrm X})\), associated with the Bochner space \(L^{p,1}(\Pi;{\mathrm X})\), with \(p \in (1,\infty)\) and \({\mathrm X}\) is a suitable Banach space. To solve the problem, we employ a generalized spectral method. In particular, we introduce the notion of \(\otimes\)-basis generated by tensor products and extend the classical scheme known from the scalar case to the present setting.
Moreover, we prove that the system of root functions of the corresponding nonlocal spectral problem forms a \(\otimes\)-basis in \(L^p(I;{\mathrm X})\).
Problems arising in mechanics, mathematical physics, and pure mathematics naturally lead to the study of differential equations in more general functional frameworks, such as Morrey, Grand Lebesgue, and Orlicz spaces (see, e.g., [1]–[6] and the references therein). These spaces provide a more flexible and robust setting, enabling a finer characterization of local regularity, growth conditions, and integrability properties of solutions. In particular, they are well suited for capturing nonstandard phenomena, including variable smoothness, non-uniform integrability, and anisotropic features that frequently arise in applications.
At the same time, many contemporary applications give rise to classes of equations that fall outside the scope of the classical theory (see, e.g., [7]–[12]). These include equations with discontinuous or merely measurable coefficients, nonlinear structures, and operators exhibiting nonstandard growth or degeneracy. Such challenges necessitate the development of new analytical approaches, as well as the extension and refinement of existing solvability results to broader and more realistic settings.
The present work is devoted to the generalization and further development of the aforementioned problems. In particular, we establish unique solvability results for a boundary value problem associated with the Laplace equation in an unbounded strip \(\Pi \subset \mathbb{R}^2\). By imposing minimal regularity assumptions on the boundary data and adopting a non-harmonic analysis approach, we obtain solvability in Sobolev–Bochner spaces. Recall that Bochner integrability is defined with respect to a Banach space and extends the classical notion of the Lebesgue integral to Banach-valued measurable functions through the norm of \(f\).
To this end, we introduce the basic notions of Bochner spaces and Unconditional Martingale Difference (UMD) spaces (see [13]–[16]).
In order to apply the generalized spectral method, we introduce the algebraic tensor product (\(\otimes\)-product) of two Banach spaces. Moreover, the \(\otimes\)-product structure on the Banach space \({\mathrm X}\) allows us to extend the classical notions of basis and completeness to the notions of \(\otimes\)-basis and \(\otimes\)-completeness, respectively.
Since \({\mathrm X}\) is not a Hilbert space, we replace the notion of orthogonality with that of biorthogonality, more precisely, \(\otimes\)-biorthogonality (see [3], [11]–[13], [17]–[21]).
Furthermore, we adapt the classical Fourier method of separation of variables to our framework, which enables us to prove the unique solvability of the modal boundary value problem 13 in the general class of Bochner–Sobolev spaces. This problem serves as a starting point for further investigations of the singular equation \(y^m u_{xx}+u_{yy}=0,\) \(m>-2,\) first studied by Moiseev in [12].
In this paper, we use the following notation:
Let \(I := (0, 2\pi)\), \(J_0 := \{(0,y) \;: \;y \in (0,\infty)\}, \; J_{2\pi}:= \{ (2\pi,y) \;: \;y \in (0,\infty)\}.\)
\(\Pi = (0,2\pi) \times (0,\infty)\), \(\Pi_\xi = I \times (0,\xi)\) for all \(\xi > 0\),
\(J_0^\xi:=(0,y),\) \(J_{2\pi}^\xi:=(2\pi,y)\) for \(0<y<\xi\).
For each Banach space \({\mathrm X}\), \(\|\cdot\|_{{\mathrm X}}\) denotes its norm, and \({\mathrm X}^*\) its dual space.
\([{\mathrm X};{\mathrm Y}]\) denotes the Banach space of bounded linear operators from \({\mathrm X}\) to \({\mathrm Y}\), \([{\mathrm X};{\mathrm X}]=[{\mathrm X}]\).
\(\alpha = (\alpha_1,\ldots,\alpha_n)\) is a multi-index with \(|\alpha| = \sum_{k=1}^n\alpha_k\) and \[\partial^\alpha u(x) = \partial_{x_1}^{\alpha_1} \cdots\partial_{x_n}^{\alpha_n} u(x),\quad x\in \mathbb{R}^n.\]
For any measurable set \(\mathcal{S}\), \(|\mathcal{S}|\) denotes its Lebesgue measure, and \(\overline{\mathcal{S}}\) its closure.
\(f|_\mathcal{S}\) denotes the restriction of \(f\) to \(\mathcal{S}\).
\(p'\) denotes the conjugate exponent of \(p\), i.e., \(\frac{1}{p} + \frac{1}{p'} = 1\).
For any domain \(\Omega\) and integer \(m \ge 1\), \(C_0^\infty(\Omega;{\mathrm X})\) – the set of all infinitely differentiable \({\mathrm X}\)-valued functions with compact support in \(\Omega\); \(C^m(\Omega;{\mathrm X}), \; C^m(\overline{\Omega};{\mathrm X})\) – the set of all \(m\)-th order differentiable \({\mathrm X}\)-valued functions on \(\Omega\) (or \(\overline{\Omega}\)).
\(\delta_{nk}\) denotes the Kronecker delta.
\(C\) denotes a positive constant which may change from line to line.
Let \({\mathrm X}\) and \({\mathrm Y}\) be Banach spaces. The algebraic tensor product \({\mathrm X}\otimes {\mathrm Y}\) is a vector space equipped with a bilinear map \[\label{otimes} \otimes: {\mathrm X}\times {\mathrm Y}\to {\mathrm X}\otimes {\mathrm Y}, \qquad (x,y) \mapsto x \otimes y,\tag{1}\] satisfying the following universal property: for every bilinear map \(\tau : {\mathrm X}\times {\mathrm Y}\to {\mathrm Z}\), with \({\mathrm Z}\) a Banach space, there exists a unique linear map \(\tilde{\tau} : {\mathrm X}\otimes {\mathrm Y}\to {\mathrm Z}\) such that \[\tilde{\tau}(x \otimes y) = \tau(x,y).\]
Moreover, the algebraic tensor product \({\mathrm X}\otimes {\mathrm Y}\) is the vector space generated by elementary tensors \(x \otimes y\), with \(x \in {\mathrm X}\) and \(y \in {\mathrm Y}\), that is, every element \(u\in {\mathrm X}\otimes{\mathrm Y}\) can be written in the form \[u = \sum_{i=1}^n x_i \otimes y_i, \qquad n \in \mathbb{N}.\]
Let \((\mathcal{S},\mathcal{A},\mu)\) be a measure space, where \(\mathcal{A} \subset 2^{\mathcal{S}}\) is a \(\sigma\)-algebra and \(\mu\) is a measure on \(\mathcal{A}\). A \(\mu\)-simple \({\mathrm X}\)-valued function is a function of the form \[f = \sum_{k=1}^n \chi_{A_k} \otimes x_k, \qquad n \in \mathbb{N}\] where for each \(1 \leq k \leq n\), \(x_k \in {\mathrm X}\), \(A_k \in \mathcal{A}\) with \(\mu(A_k) < \infty\) and \(\chi_{A_k}\) is the indicator function of \(A_k\).
For a \(\mu\)-simple function we define \[\int_\mathcal{S} f \, d\mu := \sum_{k=1}^n \mu(A_k)\otimes x_k.\]
A function \(f:\mathcal{S}\to{\mathrm X}\) is strongly \(\mu\)-measurable if there exists a sequence \(\{f_j\}_{j\in\mathbb{N}}\) of \(\mu\)-simple functions converging to \(f\), \(\mu\)-almost everywhere.
In particular, a strongly \(\mu\)-measurable function is Bochner integrable with respect to \(\mu\) if there exists a sequence \(\{f_j\}_{j\in\mathbb{N}}\) of \(\mu\)-simple functions on \(\mathcal{S}\) and with values in \({\mathrm X}\) such that \[\lim_{j\to\infty} \int_{\mathcal{S}} \| f - f_j\|_{\mathrm X}\, d\mu = 0.\]
The Bochner integral of \(f\) with respect to \(\mu\) is \[\int_\mathcal{S} f \, d\mu := \lim_{j\to\infty} \int_\mathcal{S} f_j \, d\mu.\]
The Bochner space \(L^p(\mathcal{S}; {\mathrm X})\), \(1 \le p < \infty\) is the Banach space of strongly \(\mu\)-measurable \({\mathrm X}\)-valued functions \(f:\mathcal{S}\to{\mathrm X}\) such that the Bochner norm \[\label{Bnorm} \|f\|_{L^p(\mathcal{S}; {\mathrm X})} = \left( \int_\mathcal{S} \|f\|_{{\mathrm X}}^p \, d\mu \right)^{1/p} < \infty.\tag{2}\]
We denote \(L^p(\mathcal{S};\mathbb{K})\), with \(\mathbb{K}\) being the real or complex camp, simply by \(L^p(\mathcal{S})\) and by \(L^p(\mathcal{S})\otimes {\mathrm X}\) the algebraic tensor product of the Banach spaces \(L^p(\mathcal{S})\) and \({\mathrm X}\).
For simplicity, we write the elementary tensor \(f \otimes x\) in the form \[\label{eq-product} x f := f \otimes x, \qquad x\in {\mathrm X}, \;f\in L^p(\mathcal{S}).\tag{3}\]
It is well known that the algebraic tensor product \(L^p(\mathcal{S})\otimes {\mathrm X}\) is dense in \(L^p(\mathcal{S};{\mathrm X})\) with respect to the Bochner norm (see, e.g., [15]). In particular, its closure in \(L^p(\mathcal{S};{\mathrm X})\) coincides with \(L^p(\mathcal{S};{\mathrm X})\). The following classical results ensure the density of simple functions in a Bochner space (cf. [14], [15]).
Lemma 1. [15] The set of \(\mu\)-simple \({\mathrm X}\)-valued functions is dense in \(L^p(\mathcal{S};{\mathrm X})\) for \(1 \le p < \infty\). In particular, \(L^p(\mathcal{S}) \otimes {\mathrm X}\) is dense in \(L^p(\mathcal{S};{\mathrm X})\).
We also recall the notion of UMD spaces (see [15], [16]).
Definition 2. A Banach space \({\mathrm X}\) is said to have the property of unconditional martingale difference* (UMD property) if for all \(p \in (1,\infty)\) there exists a constant \(\beta \geq 0\) depending on \(p\) and \({\mathrm X}\) such that the following holds: whenever \((\mathcal{S},\mathcal{A},\mu)\) is a \(\sigma\)-finite measure space, \(\{\mathcal{F}_k\}_{k=0}^n\) is a \(\sigma\)-finite filtration, and \(\{f_k\}_{k=0}^n\) is a finite martingale in \(L^p(\mathcal{S};{\mathrm X})\), then for every choice of scalars \(\{\xi_k\}_{k=1}^n\) with \(|\xi_k| = 1\) for all \(k=1,\dots,n\), we have \[\left\| \sum_{k=1}^n \xi_k\, d f_k \right\|_{L^p(\mathcal{S};{\mathrm X})} \leq \beta \left\| \sum_{k=1}^n d f_k \right\|_{L^p(\mathcal{S};{\mathrm X})},\] where \(d f_k = f_k - f_{k-1}\) denotes the martingale difference.*
Let \(\vec{f} = \{f_k\}_{k \in \mathbb{N}}\) be a family of functions in \(L^p(\mathcal{S})\). The \(\otimes\)-span \(L_\otimes[\vec{f}]\) is defined by \[L_\otimes[\vec{f}] = \left\{ g \in L^p(\mathcal{S}; {\mathrm X}) : \exists\, m \in \mathbb{N}, \{x_k\}_{k=1}^m\subset {\mathrm X}, \{f_{k}\}_{k=1}^m\subset \vec{f}, \text{ such that } g = \sum_{k=1}^m x_k f_k \right\}.\]
Definition 3. A system \(\vec{f}=\{f_k\}_{k\in {\mathbb{N}}}\subset L^p(\mathcal{S})\) is said to be \(\otimes\)-complete* in \(L^p(\mathcal{S};{\mathrm X})\) if \[\overline{L_\otimes[\vec{f}]} = L^p(\mathcal{S};{\mathrm X}),\] where the closure is taken in \(L^p(\mathcal{S};{\mathrm X})\).*
Definition 4. A system of operators \(\{t_n\}_{n\in {\mathbb{N}}} \subset [L^p(\mathcal{S};{\mathrm X});{\mathrm X}]\) is \(\otimes\)-biorthogonal* to \(\vec{f}\) if \[\label{biortog} t_n(x f_k)=\delta_{nk}x, \qquad \forall \, x\in {\mathrm X}, \quad \forall \, n,k\in {\mathbb{N}}.\tag{4}\] *
The following results relate the concepts just described and extend some previous results obtained in [3], [17], [18] to the framework of Bochner and UMD spaces.
Lemma 5. Every \(\otimes\)-complete system \(\vec{f}=\{f_k\}_{k\in {\mathbb{N}}}\subset L^p(\mathcal{S})\) admits at most one \(\otimes\)-biorthogonal system.
Proof. Assume that \(\vec{f}\) admits two \(\otimes\)-biorthogonal systems \[\{t_n^i\}_{n\in {\mathbb{N}}} \subset [L^p(\mathcal{S};{\mathrm X});{\mathrm X}], \quad i=1,2\] and set \(t_n:=t_n^1-t_n^2\). Then, for every \(x\in {\mathrm X}\) and every \(n,k\in {\mathbb{N}}\), \[\label{eq-tnk} t_n(x f_k) =t_n^1(x f_k)-t_n^2(x f_k) =\delta_{nk}x-\delta_{nk}x =0.\tag{5}\]
Let \(g\in L^p(\mathcal{S};{\mathrm X})\) be arbitrary. Since \(\vec{f}\) is \(\otimes\)-complete, for every \(\varepsilon>0\) there exist \(m\in {\mathbb{N}}\), \(\{x_k\}_{k=1}^{m}\subset {\mathrm X}\), and \(\{f_k\}_{k=1}^{m}\subset \vec{f}\) such that \[\left\| g-\sum_{k=1}^{m} x_k f_k \right\|_{L^p(\mathcal{S};{\mathrm X})}<\varepsilon.\] By linearity of \(t_n\) and 5 , we obtain \[\begin{align} \|t_n(g)\|_{{\mathrm X}} &= \left\| t_n \left( g - \sum_{k=1}^{m} x_k f_k + \sum_{k=1}^{m} x_k f_k\right) \right\|_{{\mathrm X}} \\ &\leq \left\| t_n\left(g-\sum_{k=1}^{m} x_k f_k\right) \right\|_{{\mathrm X}} \\ &\leq \|t_n\|_{[L^p(\mathcal{S};{\mathrm X});{\mathrm X}]} \left\| g-\sum_{k=1}^{m} x_k f_k \right\|_{L^p(\mathcal{S};{\mathrm X})} < \varepsilon \end{align}\] Since \(\varepsilon>0\) is arbitrary, it follows that \(t_n(g)=0\) for all \(g\in L^p(\mathcal{S};{\mathrm X})\).
Since \(\vec{f}\) is \(\otimes\)-complete, the space \(L_\otimes[\vec{f}]\) is dense in \(L^p(\mathcal{S};{\mathrm X})\). Therefore, by continuity of \(t_n\), we conclude that \(t_n\equiv 0.\) Hence, \(t_n^1=t_n^2\) for all \(n\in {\mathbb{N}}\) that proves the uniqueness. ◻
Definition 6. A system \(\vec{f}\subset L^p(\mathcal{S})\) is said to form a \(\otimes\)-basis for \(L^p(\mathcal{S};{\mathrm X})\) if, for every \(g\in L^p(\mathcal{S};{\mathrm X})\), there exists a unique sequence \(\{x_k\}_{k\in N}\subset {\mathrm X}\) such that \[g=\sum_{k=1}^{\infty} x_k f_k,\] where the series converges in \(L^p(\mathcal{S};{\mathrm X})\). Equivalently, every \(g\) admits a representation of the algebraic tensor form \[g=\sum_{k=1}^\infty f_k \otimes x_k,\] with convergence in \(L^p(\mathcal{S};{\mathrm X})\).
Completely analogously to the classical notion of a basis, the following criterion holds.
Theorem 7. The system \(\vec{f}\subset L^p(\mathcal{S})\) forms a \(\otimes\)-basis for \(L^p(\mathcal{S};{\mathrm X})\) if and only if the following conditions are satisfied:
\(\vec{f}\) is \(\otimes\)-complete in \(L^p(\mathcal{S};{\mathrm X})\);
\(\vec{f}\) admits a \(\otimes\)-biorthogonal system \[\{t_n\}_{n\in N} \subset [L^p(\mathcal{S};{\mathrm X});{\mathrm X}]\] satisfying 4 ;
the sequence of projection operators \(\{P_m\}_{m\in N} \subset [L^p(\mathcal{S};{\mathrm X})]\), defined by \[P_m(g)=\sum_{k=1}^{m} f_k\otimes t_k(g), \qquad g\in L^p(\mathcal{S};{\mathrm X}),\] is uniformly bounded, that is, \[\sup_{m\in {\mathbb{N}}} \|P_m\|_{[L^p(\mathcal{S};{\mathrm X})]} < \infty.\]
The proof is a direct application of Lemma 5 and is analogous to the classical one.
Let \(\Omega\subset\mathbb{R}^n, n\geq 1,\) be an open and connected domain. The Bochner-Sobolev space \(W_p^m(\Omega; {\mathrm X})\), \(1 \leq p < \infty, \;m \in \mathbb{N}\) is defined by \[W_p^m(\Omega; {\mathrm X}) = \left\{u \in L^p(\Omega; {\mathrm X}) \;: \;\partial^\alpha u \in L^p(\Omega; {\mathrm X}), \;|\alpha|\leq m \right\}.\] It is a Banach space equipped with the norm \[\| u \|_{W_p^m(\Omega; {\mathrm X})} = \sum_{|\alpha|\leq m} \| \partial^\alpha u \|_{L^p(\Omega; {\mathrm X})},\] called the Bochner–Sobolev norm.
A function \(f : \Omega \to {\mathrm X}\) is said to be locally integrable, that is \(f \in L^1_{\mathrm{loc}}(\Omega; {\mathrm X})\), if it is Bochner integrable on every compact subset of \(\Omega\).
We also recall the following Bochner-valued version of the fundamental theorem of calculus [15].
Lemma 8. [15] Let \(g \in L^1_{\mathrm{loc}}({\mathbb{R}};{\mathrm X})\), \(a \in {\mathbb{R}}\), and define \(f:\mathbb{R}\to{\mathrm X}\) by \[f(x) = \int_a^x g(s) \, ds, \qquad x\in \mathbb{R}.\] Then the weak derivative \(\partial f\) and almost everywhere derivative \(f'\) of \(f\) both exist in \(L^1_{\mathrm{loc}}({\mathbb{R}};{\mathrm X})\) and satisfy \(\partial f = f' = g\).
We say that a domain \(\Omega\subset \mathbb{R}^n,\) \(n\geq 2\), satisfies the segment condition if, for every \(z \in \partial\Omega\), there exist an open neighbourhood \(U_z\) of \(z\) and a vector \(h \in {\mathbb{R}}^n \setminus \{0\}\) such that \[\xi + th \in \Omega, \qquad \forall \, \xi \in U_z \cap \overline{\Omega}, \;\forall \, t \in (0,1).\]
The following result is a Bochner–valued extension of the classical density of smooth functions for Sobolev spaces (see [15]). Note that the segment condition is implied by many other boundary conditions such as \(\Omega\) having Lipschitz boundary.
Proposition 9. Let \(\Omega\) satisfy the segment condition, let \(m \in {\mathbb{N}}\), and let \(1\leq p <\infty\). Then the set \[\left\{f\big|_{\Omega} \;: \;f \in C_0^\infty({\mathbb{R}}^n;{\mathrm X})\right\},\] is dense in \(W^m_p(\Omega;{\mathrm X})\).
Let \(S \subset \partial\Omega\) be an \((n-1)\)-dimensional surface. We denote by \(C^\infty_{0;S}(\Omega;{\mathrm X})\) the set of all infinitely differentiable \({\mathrm X}\)-valued functions on \(\Omega\) that vanish in a neighbourhood of \(S\), i.e., \[C^{\infty}_{0;S}(\Omega;{\mathrm X}) = \left\{ u \in C^{\infty}(\Omega;{\mathrm X}) :\; \exists \;\text{an open set } U \supset S \text{ such that } u|_{U \cap \Omega}=0 \right\}.\]
The closure of \(C^{\infty}_{0;S}(\Omega;{\mathrm X})\) in \(W^1_p(\Omega;{\mathrm X})\) is denoted by \[\mathring{W}^1_{p;S}(\Omega;{\mathrm X}) := \overline{C^\infty_{0;S}(\Omega;{\mathrm X})}^{\,W_p^1(\Omega;{\mathrm X})}.\]
Now we give the notion of trace operator and trace space. For this end, we use the following theorem, proved in [22].
Theorem 10. [22] Let \(\Omega \subset {\mathbb{R}}^n\) be a open domain with uniform Lipschitz boundary, \(n\geq 2\), and let \(1\leq p < \infty\). Then there exists a linear and continuous bounded operator \[\Gamma \in [W^1_p(\Omega;{\mathrm X}); L^p(\partial\Omega;{\mathrm X})],\] such that \[\Gamma u = u\big|_{\partial\Omega}\qquad \forall \, u \in W^1_p(\Omega;{\mathrm X}) \cap C(\overline{\Omega};{\mathrm X}).\]
Moreover, for every \(u \in W^1_p(\Omega;{\mathrm X})\), we have \[u \in \mathring{W}^1_p(\Omega;{\mathrm X}) \iff \Gamma u =0.\]
As a consequence we obtain the following corollary.
Corollary 11. Let \(\Omega \subset {\mathbb{R}}^n\) be a connected domain and let \(S \subset \partial\Omega\) be a measurable subset, with respect to the \((n-1)\)-dimensional measure, which is a Lipschitz surface. Let \(1\leq p < \infty\). Then there exists a bounded linear operator \[\Gamma_S \in [W^1_p(\Omega;{\mathrm X}); L^p(S;{\mathrm X})],\] such that \[\Gamma_S u = u\big|_{S}, \qquad \forall \, u \in W^1_p(\Omega;{\mathrm X}) \cap C(\overline{\Omega};{\mathrm X}).\]
Moreover, \[u \in \mathring{W}^1_{p;S}(\Omega;{\mathrm X}) \iff \Gamma_S u =0.\]
Let \(n=1\) and let the domain be \(I=(0,2\pi)\). We consider the Bochner space \(L^p(I;{\mathrm X})\) endowed with the norm \[\label{eq-Boch-norm} \| f \|_{L^p(I;{\mathrm X})} =\left( \int_I \| f(x) \|_{{\mathrm X}}^p \, dx\right)^{\frac{1}{p}},\tag{6}\] and the corresponding Bochner–Sobolev space \(W^2_{p}(I;{\mathrm X})\), endowed with the norm \[\| f\|_{W^2_{p}(I;{\mathrm X})} =\sum_{k=0}^2 \| f^{(k)} \|_{L^{p}(I;{\mathrm X})},\] where \(f^{(k)}\) denotes the \(k\)-th weak derivative of \(f \in W^2_p(I;{\mathrm X})\).
Consider the exponential system \(\{e^{inx}\}_{n\in\mathbb{Z}}\). An \({\mathrm X}\)-valued trigonometric polynomial is a function \(P : I \to {\mathrm X}\) of the form \[P(x) = \sum_{k=-n}^n a_k e^{ikx}, \qquad x \in I,\] for some \(n \in \mathbb{N}\) and coefficients \(a_k \in {\mathrm X}\). We denote by \(\mathcal{P}({\mathrm X})\) the collection of all such polynomials.
The following result is a vector-valued extension of the classical density theorem for trigonometric polynomials (cf. [15]).
Proposition 12. Let \({\mathrm X}\) be a Banach space. Then \(\mathcal{P}({\mathrm X})\) is dense in \(L^p(I;{\mathrm X})\) for all \(1 \le p < \infty\).
Moreover, the following result characterizes the Riesz projections in UMD spaces (see, for instance, [18], [19], [21] where it is proved for exponential Fourier series).
Theorem 13 ([18]). Let \({\mathrm X}\) be a UMD space and let \(p \in (1,\infty)\). Then the exponential system \(\{e^{inx}\}_{n \in \mathbb{Z}}\) forms a \(\otimes\)-basis in \(L^p(I;{\mathrm X})\), and for every \(m \in \mathbb{Z}\) the Riesz projections \[R_m^+ f(x)= \sum_{n=m}^{\infty} \hat{f}(n)e^{inx}, \qquad R_m^- f(x)= \sum_{n=-\infty}^{m-1} \hat{f}(n)e^{inx},\] where \[\hat{f}(n)= \frac{1}{2\pi}\int_0^{2\pi} f(x)e^{-inx}\,dx, \qquad n \in \mathbb{Z},\] are bounded linear operators on \(L^p(I;{\mathrm X})\).
Since the exponential functions can be expressed in terms of sine and cosine functions, Theorem 13 yields the following result.
Corollary 14. Let \({\mathrm X}\) be a UMD space and let \(p \in (1,\infty)\). Then the trigonometric system \[\mathcal{T}=\{1,\cos(nx),\sin(nx)\}_{n\in\mathbb{N}},\] forms a \(\otimes\)-basis in \(L^p(I;{\mathrm X})\) in the sense that \[f(x)=\lim_{n\to \infty}S_nf(x)=\lim_{n\to\infty}\sum_{k=1}^{n} \bigl( \ell_0^c(f)+\ell_k^c(f)\cos(kx)+\ell_k^s(f)\sin(kx)\bigr),\] where \(\{\ell_k^c(f),\ell_k^s(f)\}_{k\in\mathbb{N}}\) are the corresponding Fourier coefficients given by \[\label{eq-ell} \begin{align} \ell_0^c(f)&= \frac{1}{2\pi} \int_I f(x) \, dx,\quad \ell_k^c(f)=\frac{1}{\pi}\int_I f(x)\cos(kx)\, dx,\\ \ell_k^s(f)&=\frac{1}{\pi}\int_I f(x)\sin(kx)\, dx, \end{align}\qquad{(1)}\] and the integrals are understood in the Bochner sense.
Let us note that in this case, the system \(\otimes\)-biorthonormal of \(\mathcal{T}\) coincides with \(\mathcal{T}\) up to multiplication by suitable constants. More precisely, it is the system \[\mathcal{T}^* = \left\{\frac{1}{2\pi},\frac{1}{\pi}\cos(nx),\frac{1}{\pi}\sin(nx)\right\}_{n\in\mathbb{N}},\] and the Fourier coefficients are computed with respect to \(\mathcal{T}^*\).
We introduce the following notion.
Definition 15. The \(\otimes\)-basis \(\mathcal{T}\) in \(L^p(I;{\mathrm X})\), with \(p \in (1,\infty)\), is said to have the \(\otimes\)-Riesz property if the projection operators \[S_n^c f(x) := \sum_{k=0}^n \ell_k^c(f) \cos(kx), \quad S_n^s f(x) := \sum_{k=1}^n \ell_k^s(f) \sin(kx), \qquad n\in{\mathbb{N}},\] are uniformly bounded in \(L^p(I;{\mathrm X})\).
For further details on these and related topics, we refer to the monographs [15], [23] and the papers [13], [17]–[19], [21], [24]. The following result holds.
Proposition 16. Let \({\mathrm X}\) be a UMD space. Then the system \(\mathcal{T}\) forms a basis of \(L^p(I;{\mathrm X})\) for \(p \in (1,\infty)\) and possesses the \(\otimes\)-Riesz property.
Proof. Suppose that \(f\) is extended to \(2\pi\)-periodic function. Then, by periodicity, it suffices to work in \(L^p((-\pi,\pi);{\mathrm X})\).
We consider the following \(\otimes\)-spans: \[\label{eq-Lspan} \begin{align} L^c_p((-\pi,\pi);{\mathrm X}) &:=\overline{L_{\otimes}[\{\cos(nx)\}_{n\in{\mathbb{N}}_0}]},\\ L^s_p((-\pi,\pi);{\mathrm X}) &:=\overline{L_{\otimes}[\{\sin(nx)\}_{n\in{\mathbb{N}}}]}, \end{align}\tag{7}\] where the closure is taken in \(L^p((-\pi,\pi);{\mathrm X})\).
Let \(f\in L^p((-\pi,\pi);{\mathrm X})\) and decompose it into even and odd parts: \[f(x)=f^{+}(x)+f^{-}(x), \qquad f^{\pm}(x)=\frac{f(x)\pm f(-x)}{2}.\] Clearly \(f^{\pm}\in L^p((-\pi,\pi);{\mathrm X})\), \(f^+\) is even, and \(f^-\) is odd.
Since \(\mathcal{T}\) forms a \(\otimes\)-basis in \(L^p((-\pi,\pi);{\mathrm X})\), by Corollary 14, we have the Fourier expansion \[f(x)= \ell_0^c(f)+\sum_{n=1}^{\infty} \bigl(\ell_n^c(f)\cos(nx)+\ell_n^s(f)\sin(nx)\bigr),\] with convergence in \(L^p((-\pi,\pi);{\mathrm X})\), that is, \[\|S_nf-f\|_{L^p((-\pi,\pi);{\mathrm X})} \to 0 \qquad \text{ as } n\to\infty.\]
Since \(f^-\) is odd, we have \(\ell_k^c(f^-)=0\), and therefore \(\ell_k^c(f)=\ell_k^c(f^+)\). Hence, \[S_n^cf(x) = \ell_0^c(f^+) + \sum_{k=1}^{n} \ell_k^c(f^+)\cos(kx)=S_n f^+(x).\]
Similarly, since \(f^+\) is even, \(\ell_k^s(f^+)=0,\) and thus \(\ell_k^s(f)=\ell_k^s(f^-)\). Therefore, \[S_n^sf(x)=S_nf^-(x).\]
It follows that \[\label{eq-BS} \lim_{n\to \infty}S_n^cf(x)= f^{+}(x), \quad \lim_{n\to \infty}S_n^sf(x)= f^{-}(x) \quad \text{ in } \; L^p((-\pi,\pi);{\mathrm X}).\tag{8}\] Hence, \[\sup_{n\in {\mathbb{N}}} \|S_n^c f\|_{L^p((-\pi,\pi);{\mathrm X})}< \infty, \quad \sup_{n\in {\mathbb{N}}} \|S_n^s f\|_{L^p((-\pi,\pi);{\mathrm X})} < \infty.\]
By the Banach–Steinhaus theorem, the families \(\{S_n^c\}_{n\in {\mathbb{N}}}\) and \(\{S_n^s\}_{n\in {\mathbb{N}}}\) are uniformly bounded in \(L^p((-\pi,\pi);{\mathrm X})\). This completes the proof. ◻
We now adapt the above definitions to functions defined on the unbounded strip \(\Pi=I\times(0,\infty)\subset {\mathbb{R}}^2.\)
In what follows, we consider functions possessing mixed regularity property. To this end, we introduce the mixed–norm Bochner space \(L^{p,1}(\Pi;{\mathrm X})\) defined by \[\| u \|_{L^{p,1}(\Pi;{\mathrm X})} = \int_0^{\infty} \left(\int_0^{2\pi} \| u(x,y) \|_{{\mathrm X}}^p \, dx \right)^\frac{1}{p} dy=\int_0^\infty \|u(\cdot,y)\|_{L^p(I;{\mathrm X})}\, dy.\]
The mixed-norm Bochner–Sobolev space \(W^2_{p,1}(\Pi;{\mathrm X})\) is defined as the space of all functions whose weak derivatives up to order \(2\) belong to \(L^{p,1}(\Pi;{\mathrm X})\). It is equipped with the norm \[\| u \|_{W^2_{p,1}(\Pi;{\mathrm X})} = \sum_{|\alpha|\leq 2} \| \partial^\alpha u\|_{L^{p,1}(\Pi;{\mathrm X})}.\]
Let \(u \in W^1_{p,1}(\Pi;{\mathrm X})\). Then \(u \in W^1_{p,1}(\Pi_\xi;{\mathrm X})\) for every \(\xi >0\). Applying Corollary 11, we deduce that for every \(\xi>0\) there exists a linear continuous operator \[\Gamma_{J_0^\xi} \in [W^1_{p,1}(\Pi_\xi;{\mathrm X}); L^1(J_0^\xi;{\mathrm X})],\] such that \(\Gamma_{J_0^\xi} u = u|_{J_0^\xi}\) for all \(u \in W^1_{p,1}(\Pi_\xi;{\mathrm X}) \cap C(\overline{\Pi}_\xi;{\mathrm X})\).
By continuity, the compatibility condition \[\big(\Gamma_{J_0^{\xi_2}} u\big)\big|_{J_0^{\xi_1}} = \Gamma_{J_0^{\xi_1}}u, \quad \text{ a.e. on } J_0^{\xi_1}, \;0 < \xi_1 < \xi_2\] holds. Consequently, there exists a global trace operator \[\begin{align} \Gamma_{J_0}: W_{p,1}^1(\Pi;{\mathrm X}) &\to L^1_{\mathrm{loc}}(J_0;{\mathrm X}), \\ u &\mapsto \Gamma_{J_0} u, \end{align}\] such that \(\Gamma_{J_0} u = u|_{J_0}\) for all \(u \in W_{p,1}^1(\Pi;{\mathrm X}) \cap C(\overline{\Pi};{\mathrm X})\). Moreover \[\Gamma_{J_0} \in [W^1_{p,1}(\Pi_{\xi};{\mathrm X}); L^1(J^\xi_0;{\mathrm X})], \quad \forall \, \xi > 0.\]
We call \(\Gamma_{J_0}\) the trace operator on the boundary part \(J_0 \subset \partial\Pi\).
Analogously, one defines the trace operator \(\Gamma_{J_{2\pi}}\) corresponding to \(J_{2\pi} \subset \partial\Pi\).
For simplicity, we set \(J_1 := J_0\) and \(J_2 := J_{2\pi}\).
Proposition 17. There exists a trace operator \[\begin{align} \Gamma_{J_k}: W_{p,1}^1(\Pi;{\mathrm X}) &\to L^1_{\mathrm{loc}}(J_k;{\mathrm X}), \\ u &\mapsto \Gamma_{J_k} u,\quad k=1,2, \end{align}\] such that \(\Gamma_{J_k} u = u|_{J_k}\) for all \(u \in W_{p,1}^1(\Pi;{\mathrm X}) \cap C(\overline{\Pi};{\mathrm X})\).
Moreover, for every \(\xi>0\), \[\Gamma_{J_k} \in [W^1_{p,1}(\Pi_{\textcolor{red}{\xi}};{\mathrm X}); L^1(J_k^\xi;{\mathrm X})], \qquad k=1,2.\]
In order to study the existence of solution to problem 13 we adapt the classical Fourier method, based on the separation of variables for the Laplace equation (see [25]). Representing the solution in the form \(u(x,y)=\varphi(x)\psi(y)\), we obtain two second-order linear ordinary differential equations.
Since we are working in Bochner spaces, endowed with \(\otimes\)-product structure, the classical Fourier method must be modified to fit our framework.
The first generalization concerns the Sturm–Liouville problem associated with the function \(\varphi(x)\) and leads to the following spectral problem \[\label{SPE} \begin{cases} \varphi''(x) + \lambda \varphi(x) = 0, & x \in I,\\ \varphi(0) = \varphi(2\pi),\quad \varphi'(0) = 0. & \end{cases}\tag{9}\]
Let us note that the conditions in \(x=0\) and \(x=2\pi\) reflect the periodicity of the solution \(u(x,y)\) with respect to the variable \(x\), together and the absence of flux through the boundary part \(J_0\subset \partial \Pi\) in the \(x\)-direction, namely \(\partial_x u(0,y)=0\).
To solve problem 9 , we apply the generalized spectral method. To this end, we establish the \(\otimes\)-basis property of the system of root functions associated with this spectral problem.
Direct calculations, following standard approach (see for instance [25]) yields the set of all eigenvalues and the corresponding eigenfunctions: \[\label{eq-eigen} \{\lambda_n = n^2\}_{n=0}^\infty,\qquad \left\{\varphi_n^c(x) = \cos(nx)\right\}_{n=0}^\infty.\tag{10}\]
Since the system of eigenfunctions is not \(\otimes\)-complete in \({\mathrm X}\), we additionally need to determine the corresponding associated functions, which we denote by \(\varphi_n^s\).
These functions are defined as solutions of the second characteristic equation \[(\mathcal{L}-\lambda_n \mathcal{I})\varphi^s_n(x)=\varphi_n^c(x),\qquad n=1,2,\ldots\] where \(\mathcal{L}\) is the operator \(-\partial^2\).
Substituting the explicit form of \(\varphi_n^c\), we obtain \[\varphi(x)'' + n^2 \varphi(x) = -\cos(nx), \qquad x\in I.\] Up to multiplicative constants, a corresponding family of associated functions is given by \[\varphi_n^s(x) = x \sin(nx), \quad n=1,2,\ldots.\]
Thus, we consider the system of root functions \[\label{eq-system} \big\{\varphi_0^c = 1,\; \varphi_n^c(x) = \cos(nx),\; \varphi_n^s(x) = x \sin(nx) \big\}_{n\in \mathbb{N}}.\tag{11}\]
It is well known (see [20], [25]) that the system 11 is not orthogonal. Therefore, we introduce a \(\otimes\)-biorthonormal system in order to compute the expansion coefficients (e.g. [3], [12]).
Let us consider \[\label{eq-system2} \left\{ v_0^c(x) = \frac{2\pi - x}{2\pi^2},\; v_n^c(x) = \frac{2\pi - x}{\pi^2}\cos(nx),\; v_n^s(x) = \frac{1}{\pi^2}\sin(nx) \right\}.\tag{12}\]
This system is constructed precisely to replace orthogonality by biorthogonality. Therefore, taking into account the \(\otimes\)-theory introduced in Subsection 2.2, we are able to establish that 11 forms a \(\otimes\)-basis in \(L^p(I;{\mathrm X})\) with unique \(\otimes\)-biorthogonal system 12 .
Theorem 18. Let \({\mathrm X}\) be a UMD space and let \(p\in(1,\infty)\). Then the system \(\{\varphi_n^c;\varphi^s_n\}_{n\in{\mathbb{N}}_0}\), defined by 11 is \(\otimes\)-complete in \(L^p(I;{\mathrm X})\) and admits a \(\otimes\)-biorthogonal system \(\{v_n^c,v_n^s\}_{n\in \mathbb{N}_0}\) defined by 12 . Moreover the associated projectors satisfy the \(\otimes\)-Riesz property. Namely, there exists a constant \(C>0\) such that \[\begin{align} \left\| \sum_{k=0}^n v_k^c(f) \varphi_k^c \right\|_{L^p(I;{\mathrm X})} &\leq C \| f \|_{L^p(I;{\mathrm X})}, \\ \left\| \sum_{k=1}^n v_k^s(f) \varphi_k^s \right\|_{L^p(I;{\mathrm X})} &\leq C \| f \|_{L^p(I;{\mathrm X})}, \end{align}\] for all \(f \in L^p(I;{\mathrm X})\) and all \(n \in \mathbb{N}\), where \[v_k^c(f)=\int_0^{2\pi} f(x) v_k^c(x)\, dx, \qquad v_k^s(f)=\int_0^{2\pi} f(x) v_k^s(x)\, dx.\]
Proof. First, by construction we have \[\{v_n^c;v^s_n\}\subset [L^p(I;{\mathrm X});{\mathrm X}],\] and the \(\otimes\)-biorthogonality of the systems \(\{\varphi_n^c;\varphi^s_n\}\) and \(\{v_n^c,v_n^s\}\) follows from the scalar case (see [3], [17]).
Next, since the system \(\{\varphi_n^c,\varphi^s_n\}\) forms a basis in the scalar space \(L^p(I)\) by [3], and since the algebraic tensor product \(L^p(I) \otimes {\mathrm X}\) is dense in \(L^p(I;{\mathrm X})\), it follows that \(\{\varphi_n^c,\varphi^s_n\}\) is \(\otimes\)-complete in \(L^p(I;{\mathrm X})\).
It remains to prove the uniform boundedness in \(L^p(I;{\mathrm X})\) of the projectors \[P_n^c(f) = \sum_{k=0}^n v_k^c(f) \varphi_k^c, \quad P_n^s(f) = \sum_{k=0}^n v_k^s(f) \varphi_k^s, \qquad n\in {\mathbb{N}}.\]
To this end, we reduce the problem to the trigonometric system. Let \[g(x)=(2\pi-x)f(x), \qquad x \in I.\] Then, the relation between the coefficients is given by \[v_0^c(f) = \frac{1}{\pi} \,\ell_0^c(g), \quad v_k^c(f) = \frac{1}{\pi} \, \ell_k^c(g), \quad v_k^s(f) = \frac{1}{\pi} \, \ell_k^s(f),\] and the \(\ell\)-coefficients are computed using ?? (see for instance [3], [17]).
Thus, up to multiplicative constants, the operators \(P_n^c\) and \(P_n^s\) coincide with the cosine and sine Fourier partial sum operators applied to \(g\) and \(f\), respectively.
Since \({\mathrm X}\) is UMD space, the trigonometric system \(\mathcal{T}\) has the \(\otimes\)-Riesz property in \(L^p(I,{\mathrm X})\) by Proposition 16. Therefore, \[\begin{align} \|P_n^c(f)\|_{L^p(I;{\mathrm X})}&+ \|P_n^s(f)\|_{L^p(I;{\mathrm X})} \leq C\left\| \sum_{k=0}^n \ell_k^c(g)\cos(kx) \right\|_{L^p(I;{\mathrm X})} \\ &+ C\left\| \sum_{k=1}^n\ell_k^s(f)\sin(kx) \right\|_{L^p(I;{\mathrm X})} \leq C (\|g\|_{L^p(I;{\mathrm X})}+ \|f\|_{L^p(I;{\mathrm X})})\\ &\leq C\|f\|_{L^p(I;{\mathrm X})}, \end{align}\] since the function \(2\pi -x\) is bounded on \(I\). The estimate is uniform with respect to \(n.\)
This completes the proof. ◻
Consider the following \({\mathrm X}\)-valued nonlocal boundary value problem \[\label{BVP}\begin{cases} \Delta u = 0, & \text{ in } \Pi,\\ \Gamma_I u = f(x), \quad & \text{ on } I,\\ \Gamma_{J_0} u(0,y) = \Gamma_{J_{2\pi}} u(2\pi,y),& y\in (0,\infty),\\ \Gamma_{J_0} (\partial_x u )= 0,& \text{ on } J_0, \end{cases}\tag{13}\] where \(f \in L^p(I;{\mathrm X})\) is a given function.
By a solution of problem 13 , we mean a function \(u \in W_{p,1}^2(\Pi;{\mathrm X}),\) such that the Laplace equation holds almost everywhere in \(\Pi\), and the boundary conditions on \(\partial \Pi\) are satisfied in the trace sense.
Theorem 19 (Uniqueness). Let \(f \in W_p^2(I;{\mathrm X})\), with \(1 < p < \infty\). If the problem 13 admits a solution in \(W_{p,1}^2(\Pi;{\mathrm X})\), then the solution is unique.
Proof. It suffices to prove uniqueness in the case \(f = 0\). Consider the homogeneous problem \[\label{E3463} \begin{cases} \Delta u = 0, & \text{in } \Pi,\\ \Gamma_I u =0, & \text{on } I,\\ \Gamma_{J_0}u=\Gamma_{J_0}(\partial_x u)=0, & \text{on } J_0,\\ \Gamma_{J_{2\pi}}u=0, & \text{on } J_{2\pi}. \end{cases}\tag{14}\]
Let \(u \in W_{p,1}^2(\Pi;{\mathrm X})\) be a solution of 14 . For any \(v \in {\mathrm X}^*\), define \[u_v(x,y) := v(u(x,y)).\] Then \(u_v \in W_{p,1}^2(\Pi)\) and satisfies \(\Delta u_v = 0\) in \(\Pi,\) together with the corresponding homogeneous boundary conditions.
By classical regularity results for scalar harmonic functions (see [26]), it follows that \(u_v\) is smooth in \(\Pi\). Moreover, by the uniqueness result for the corresponding scalar problem (see [7], [11]), we obtain \(u_v \equiv 0\) in \(\Pi.\)
Since \(v(u(x,y)) = 0\) for all \(v \in {\mathrm X}^*\) and all \((x,y)\in \Pi\), the Hahn–Banach theorem implies that \(u(x,y)=0\) for all \((x,y)\in\Pi.\) Hence \(u \equiv 0\), and the proof is complete. ◻
The following result ensures the existence of a generalized solution and provides a suitable a priori estimate.
Theorem 20 (Existence). Let \({\mathrm X}\) be a UMD space and let \(f \in W_p^2(I;{\mathrm X}),\) \(1<p<\infty\), satisfy \(f(0) = f(2\pi)= f'(0)=0\), and \(\int_0^{2\pi}f(x)(2\pi-x)\, dx=0.\) Then the boundary value problem 13 admits a unique solution \(u \in W_{p,1}^2(\Pi; {\mathrm X})\). Moreover, there exists a constant \(C > 0\), such that \[\|u\|_{W_{p,1}^2(\Pi; {\mathrm X})} \leq C \|f\|_{W_{p}^2(I; {\mathrm X})}.\]
Proof. In analogy with the scalar case, we are looking for a solution in the form \[u(x,y)=\varphi(x)\theta(y).\] The function \(\varphi\) solves the spectral problem 9 , where \[\varphi(0)=f(0),\quad \varphi(2\pi)=f(2\pi),\quad \varphi'(0)=f'(0).\] The function \(\theta(y)\) satisfies the problem \(\theta''(y)=\lambda_n \theta,\) where \(\lambda_n\) are the eigenvalues of the problem 9 given by 10 . Since \(u(x,y)\) must remain bounded in \(\Pi\), we obtain the solutions \(\theta_n(y)= e^{-ny}\).
We therefore consider the formal series \[\label{E3464} u(x,y) = u_0(y) + \sum_{n=1}^{\infty} \left( u_n(y) \cos(nx) + v_n(y)\, x \sin(nx) \right), \quad (x,y) \in \Pi,\tag{15}\] where the coefficients are calculated with respect to the system 12 . Formally deriving and substituting in the problem (see for instance [3]) we obtain the following explicit representation for the coefficients \[\begin{align} u_0(y) &= \frac{1}{2\pi^2} \int_0^{2\pi} f(x)(2\pi - x) \,dx;\\ u_n(y) &= \frac{e^{-ny}}{\pi^2} \int_0^{2\pi} f(x)(2\pi - x) \cos(nx)\,dx + \frac{ye^{-ny}}{2\pi} \int_{0}^{2\pi} f(x)\sin (nx)\,dx, \quad n \in \mathbb{N}; \\ v_n(y) &= \frac{e^{-ny}}{\pi^2} \int_0^{2\pi} f(x) \sin(nx)\,dx , \quad n \in \mathbb{N}. \end{align}\]
We aim to prove that \(u \in W_{p,1}^2(\Pi; {\mathrm X})\). We begin with the second term in 15 . Set \[u_1(x,y) = \sum_{n=1}^{\infty} v_n(y) x \sin(nx).\] Formally differentiating termwise, we obtain \[\begin{align} \partial^2_x u_1(x,y) &= 2 \sum_{n=1}^{\infty} n v_n(y) \cos(nx) -\sum_{n=1}^{\infty} n^2 v_n(y) \, x \sin(nx),\\ \partial^2_y u_1 (x,y)&= \sum_{n=1}^{\infty} v''_n(y)\, x \sin(nx). \end{align}\] Define \[u_2(x,y) = \sum_{n=1}^{\infty} n^{2} v_n(y)\, x \sin(nx),\] for which we are going to prove that \(u_2 \in L^{p,1}(\Pi; {\mathrm X})\). Using the notation ?? we may write \(v_n(y) = \ell_n^s(f) e^{-ny}\).
Since \(f \in W_p^2(I; {\mathrm X})\) and \({\mathrm X}\) is a UMD space, it follows that \(f, f' \in C(\overline{I}; {\mathrm X})\) (see, e.g., [7], [15]). Hence integration by parts yields \[\begin{align} \ell_n^s(f) = \frac{1}{\pi n^2} \int_0^{2\pi} f''(x) \sin(nx) dx=\frac{1}{n^2} \ell_n^s(f''). \end{align}\]
Thus \[u_2(x,y) = \sum_{n=1}^{\infty} \ell_n^s(f'')\, x \sin(nx) e^{-ny}.\]
To estimate \(\|u_2\|_{L^{p;1}(\Pi; {\mathrm X})}\), we distinguish two cases.
I. \(p > 2\). Then, \(p' \in (1, 2)\). By the \({\mathrm X}\)-valued Hausdorff–Young theorem, we obtain \[\begin{align} \left( \int_0^{2\pi} \| u_2(x,y) \|_{{\mathrm X}}^p\, dx \right)^{1/p} &\leq C \left( \sum_{n=1}^{\infty} \| \ell_n^s(f'') e^{-n y}\|_{{\mathrm X}}^{p'} \right)^{1/p'}\\ &\leq C \sum_{n=1}^{\infty} \| \ell_n^s(f'') e^{-ny} \|_{{\mathrm X}} . \end{align}\] In the last step, we used the inequality \(\left( \sum_{n=1}^{\infty} |a_n| \right)^\alpha \leq \sum_{n=1}^{\infty} |a_n|^\alpha,\) which holds for all \(\alpha \in (0, 1]\). From the above estimate, it follows that \[\begin{align} \| u_2 \|_{L^{p;1}(\Pi; {\mathrm X})} &\leq C \sum_{n=1}^{\infty} \| \ell_n^s(f'') \|_{{\mathrm X}} \int_0^{+\infty} e^{-ny} \,dy =C \sum_{n=1} ^{\infty} \frac{\| \ell_n^s(f'') \|_X}{n} \\ &\leq C \left(\sum_{n=1} ^{\infty} \| \ell_n^s(f'') \|_{{\mathrm X}}^2\right)^{\frac{1}{2}} = C \| f'' \|_{L^2(I;{\mathrm X})} \leq C \| f'' \|_{L^p(I;{\mathrm X})}. \end{align}\]
II. \(1 < p \leq 2\). Then \[\left( \int_0^{2\pi} \| u_2(x,y) \|_{{\mathrm X}}^p\, dx \right)^{1/p} \leq C \left( \int_0^{2\pi} \| u_2(x,y) \|_{{\mathrm X}}^{p{'}} \, dx \right)^{1/p{'}}.\]
Applying again the \({\mathrm X}\)-valued Hausdorff–Young inequality, we obtain \[\begin{align} \left( \int_0^{2\pi} \| u_2(x,y) \|_{{\mathrm X}}^p\, dx \right)^{1/p} &\leq C \left( \sum_{n=1}^{\infty} \| \ell_n^s(f'') e^{-ny}\|_X^{p} \right)^{1/p}\\ &\leq C \sum_{n=1}^{\infty} \| \ell_n^s(f'') \|_X e^{-ny} \end{align}\] and the desired estimate follows as in Case I.
Arguing analogously, we estimate the remaining terms in the representation of \(u(x,y)\) and obtain \[\| u \|_{W_{p;1}^2(\Pi; {\mathrm X})} \leq C \| f \|_{W^2_p(I; {\mathrm X})}.\]
It is straightforward to verify that \(u\) satisfies 13 , since the series 15 can be differentiated termwise.
We now verify the boundary conditions in 13 . First, we show that \(\Gamma_I u = f\). Since \[\Gamma_{I} \in \left[ W_{p,1}^2(\Pi; {\mathrm X}), W^1_p(I; {\mathrm X})\right],\] it follows that \(u_m \to u\) in \(W_{p,1}^2(\Pi; {\mathrm X})\) implies \(\Gamma u_m \to \Gamma u\) in \(W^1_p(I; {\mathrm X})\).
Consider \[u_m(x,y) = u_0(y) + \sum_{n=1}^m \left( u_n(y) \cos(nx) + v_n(y) x \sin(nx) \right), \quad \text{ in } \Pi.\] Then \[\Gamma_I u_m(x,y) = \Gamma_I u_0(y) + \sum_{n=1}^m \left[ \Gamma_I (u_n(y) \cos(nx)) + \Gamma_I (v_n(y) x \sin(nx)) \right].\]
If \(u \in C^2(\overline{\Pi}; H)\), then \(\Gamma_S u = u|_S\), for every half–line \(S \subset \overline{\Pi}\). It is easy to verify that \[\left\{ u_n(y) \cos(nx), \;v_n(y) \,x \sin(nx) \right\} \subset C^2( \overline{\Pi}; H).\]
Using the explicit expressions for the coefficients \(\{u_n, v_n\}_{n\in\mathbb{N}}\), we obtain \[\label{E3465} \begin{align} \Gamma_I u_m =&\, \frac{1}{2\pi} \int_0^{2\pi} f(t) (2\pi - t)\, dt\\ +&\, \sum_{n=1}^m \Big( \frac{1}{\pi^2} \int_0^{2\pi} f(t) (2\pi - t) \, dt \cos(nx)\\ &+ \frac{1}{\pi} \int_0^{2\pi} f(t) \sin (nt) \,dt \; x\sin(nx)\Big), \quad m\in {\mathbb{N}}. \end{align}\tag{16}\]
Since by Theorem 18, the system 11 forms a \(\otimes\)-basis in \(L^p(I; {\mathrm X})\), it follows that the right-hand side of 16 converges to \(f\) in \(L^p(I; {\mathrm X})\).
On the other hand, \(\Gamma_I u_m \to \Gamma_I u\) in \(L^p(I; {\mathrm X})\) as \(m\to \infty\). Hence \(\Gamma_I u = f\). The remaining boundary conditions are verified analogously. ◻
The research of B. Bilalov and S. Sadigova is supported by the Azerbaijan Science Foundation-Grant no. AEF-MGC-2024-2(50)-16/02/1-M-02
P. Salerno and L. Softova are members of INDAM-GNAMPA. The research of L. Softova is partially supported by the project "AI Magister" CUP B47H22004440001 and the FARB 300396FRB25SOFTO.
The authors declare that they have no conflict of interest.
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