Proof. Throughout the proof we set \(\omega^T(x)\doteq u(x,T),\) \(x\in\mathbb{R}\), and we let \(u_l(t), u_r(t)\) denote the one-sided traces of \(u(t,\cdot)\) at \(x=0\).

1. Proof of (i). Given \(x>0\), consider the minimal backward characteristic for the conservation law \(u_t+f_r(u)_x=0\), in the semiplane \(\{x>0\}\), starting from \((x,T)\), defined by \(\vartheta_{x,-}(t)=x-(T-t) \cdot f_r^{\prime}(\omega^T(x-))\). If \(x-T\cdot f_r^{\prime}(\omega^T(x-)) \geq 0\), then \(\vartheta_{x,-}\) is a classical genuine characteristic for \(u\) in the whole interval \([0,T]\), since it never crosses the interface \(x=0\) but at most at \(t=0\). Therefore, according with Definition 6, the map \[\zeta(t)=x-(T-t) \cdot f_r^{\prime}(\omega^T(x-)), \qquad t\in [0,T],\] is an AB-genuine/interface characteristic, and hence by 19 it holds \(\zeta \in C(u,x)\). Otherwise, we have \(x-T\cdot f_r^{\prime}(\omega^T(x-)) < 0\), and thus \(\vartheta_{x,-}\) impacts the interface at the time: \[\label{eq:tau-def-1} \tau_-(x)\doteq T-\frac{x}{f_r^{\prime}(\omega^T(x-))}>0.\tag{34}\] Then, consider the set \[\label{eq:def-E} E \doteq \Big\{ t \in [0, \tau_-(x)] \; \big|\; \text{either}\; u_l(t)>\theta_l\; \;\text{or}\; \;u_r(t)<\theta_r\Big\},\tag{35}\] and let \[\label{eq:def-supE} \overline{\tau}\doteq \sup E,\tag{36}\] where we understand that \(\overline{\tau}=0\) when \(E=\emptyset\). Because of the non-intersection property of classical genuine characteristics in the domains \(\{x<0\}, \{x > 0\}\), and since uniform limit of classical genuine characteristics is a classical genuine characteristic as well (e.g. cfr. [1]), we deduce that \[\label{eq:supE61maxE} \overline{\tau}\in E\qquad\text{if}\quad\; E\neq\emptyset.\tag{37}\] Thus, when \(E\neq\emptyset\), if \(u_l(\overline{\tau})>\theta_l\) we can consider the minimal backward characteristic for the conservation law \(u_t+f_l(u)_x=0\), in the semiplane \(\{x<0\}\), starting from \((0,\overline{\tau})\), defined by \(\vartheta_{\overline{\tau},-}(t)=(t-\overline{\tau}) \cdot f_l^{\prime}(u_l(\overline{\tau}))\). Otherwise, if \(u_r(\overline{\tau})<\theta_r\) we can consider the maximal backward characteristic for the conservation law \(u_t+f_r(u)_x=0\), in the semiplane \(\{x>0\}\), starting from \((0,\overline{\tau})\), defined by \(\vartheta_{\overline{\tau},+}(t)=(t-\overline{\tau}) \cdot f_r^{\prime}(u_r(\overline{\tau}))\). On the other hand, by definition of \(E\), and recalling the interface condition 13 , we find that \[\label{eq:intcond-zeta-1} u_l(t)= A, \quad u_r(t) =B \qquad \forall~t \in \,]\,\overline{\tau}, \tau_-(x)].\tag{38}\] Note in particular that \[\label{eq:trace-ur-tau} \overline{\tau}<\tau_-(x)\quad\Longrightarrow\quad u_r(\tau_-(x))=\omega^T(x-)=B.\tag{39}\] Therefore, the piecewise affine map \[\label{eq:char-int} \zeta(t) = \begin{cases} x-(T-t) \cdot f_r^{\prime}(\omega^T(x-)), & t \in [\tau_-(x), T], \\ 0 , & t \in \,]\overline{\tau} , \tau_-(x)[, \\ (t-\overline{\tau}) \cdot f_l^{\prime}(u_l(\overline{\tau})), & t \in [0, \overline{\tau}],\;\;\text{if} \;u_l(\overline{\tau})>\theta_l, \\ (t-\overline{\tau}) \cdot f_r^{\prime}(u_r(\overline{\tau})), & t \in [0, \overline{\tau}],\;\;\text{if} \;u_r(\overline{\tau})<\theta_r, \end{cases}\tag{40}\] satisfy the conditions of Definition 6, and thus it is an \(AB\)-gic belonging to the set \(\mathcal{C}(u,x)\). Note that it may well happen that \(\overline{\tau} =\tau_-(x)\), in which case there will be in 40 no nontrivial interval where the characteristic is travelling along the interface \(x=0\). Instead, in the case \(\overline{\tau} =0\), the \(AB\)-gic in 40 lies on the interface \(x=0\) in the whole interval \([0,\tau_-(x)]\).

Clearly, the same analysis can be carried out to show that \(\mathcal{C}(u,x) \neq \emptyset\) also for \(x<0\). It remains to consider the case \(x=0\). Notice that this case would follow from (ii) and from (i) for \(x \neq 0\), however for clarity we write the construction explicitly. If we assume that \(\omega^T(0-)>\theta_l\), then the minimal backward characteristic for \(u_t+f_l(u)_x=0\), in the semiplane \(\{x<0\}\), starting from \((0,T)\), is a classical genuine characteristic for \(u\) in the whole interval \(]0,T]\), and hence it it is an \(AB\)-gic belonging to the set \(\mathcal{C}(u,0)\). Similarly, if \(\omega^T(0+)<\theta_r\), then the maximal backward characteristic for \(u_t+f_r(u)_x=0\), in the semiplane \(\{x>0\}\), starting from \((0,T)\), is a classical genuine characteristic for \(u\) in the whole interval \([0,T]\), and hence it is an \(AB\)-gic belonging to the set \(\mathcal{C}(u,0)\). Finally, if \(\omega^T(0-)\leq \theta_l\) and \(\omega^T(0+)\geq \theta_r\), by the interface condition 13 , we deduce that \(\omega^T(0-)=A\), \(\omega^T(0+)=B\). Then, set \[\overline{\tau}=\sup E, \qquad\quad E \doteq \Big\{ t \in [0, T] \; \big|\; \text{either}\; u_l(t)>\theta_l\; \;\text{or}\; \;u_r(t)<\theta_r\Big\}.\] With the same type of analysis as above we find that \(\overline{\tau}\in E\) and that the map \[\label{eq:char-int-2} \zeta(t) = \begin{cases} 0 , & t \in \,]\overline{\tau} , T], \\ (t-\overline{\tau}) \cdot f_l^{\prime}(u_l(\overline{\tau})), & t \in [0, \overline{\tau}],\;\;\text{if} \;u_l(\overline{\tau})>\theta_l, \\ (t-\overline{\tau}) \cdot f_r^{\prime}(u_r(\overline{\tau})), & t \in [0, \overline{\tau}],\;\;\text{if} \;u_r(\overline{\tau})<\theta_r, \end{cases}\tag{41}\] is an \(AB\)-gic belonging to the set \(\mathcal{C}(u,0)\), thus completing the proof of (i).

2. Proof of (ii). The closed graph property of the map \(x \mapsto \mathcal{C}(u, x)\) is equivalent to \[\label{eq:graphclosureC} \Big(x_n \to x, \quad \zeta_n \in \mathcal{C}(u,x_n),\quad \zeta_n \to \zeta \quad \text{uniformly}\Big) \quad \Longrightarrow \quad \zeta \in \mathcal{C}(u,x).\tag{42}\] Then, let \(\{x_n\}_n\) be a sequence converging to \(x\geq 0\), and consider a sequence of \(AB\)-gic \(\zeta_n\in \mathcal{C}(u,x_n)\), that converge uniformly to some \(\zeta\in \mathrm{Lip}([0,T]\; ; \mathbb{R})\). By Remark 7, for every \(n\) there will be \(0\leq\tau_{1,n}\leq\tau_{2,n}\leq T\), such that \[\label{interf-gen-char-cond-n1} \zeta_n(t)=0,\qquad f_l(u_l(t))=f_r(u_r(t))=\gamma \qquad\;\forall~t\in [\tau_{1,n},\,\tau_{2,n}],\tag{43}\] and such that the restriction of \(\zeta_n\) to \(]0, \tau_{1,n}[\) and to \(]\tau_{2,n}, T[\) is either a classical genuine characteristic for \(u_t+f_l(u)_x=0\)  on \(\{x<0\}\), or it is a classical genuine characteristic for \(u_t+f_r(u)_x=0\)  on \(\{x>0\}\). This, in particular, implies that \[\label{eq:der-char-n} \dot{\zeta}_n(t)= \begin{cases} \dfrac{x_n}{T-\tau_{2,n}} \quad&\forall~t \in \, ]\tau_{2,n}, T[\,, \\ \noalign{\medskip} -\dfrac{\zeta_n(0)}{\tau_{1,n}} \quad&\forall~t \in \, ]0,\tau_{1,n}[\,. \end{cases}\tag{44}\] Possibly considering a subsequence we can assume that \(\{\tau_{i,n}\}_n\) converge to some \(\tau_i\in [0,T]\), \(i=1,2\), with \(\tau_1\leq \tau_2\). Suppose that \(\tau_1>0\), \(\tau_2<T\). The cases where \(\tau_1=0\), or/and \(\tau_2=T\) can be treated with entirely similar and simpler arguments. Up to extracting a further subsequence we may also assume that \(x_n>0\) for all \(n\), and that \[\label{interf-gen-char-cond-n2} \zeta_n(t)<0\qquad \forall~t\in [0, \tau_{1,n}[\,,\qquad\qquad \zeta_n(t)>0\qquad \forall~t\in \,]\tau_{2,n},\,T], \quad\forall~n\,.\tag{45}\] Again, the cases where \(\zeta_n(t)>0\) for all \(t\in [0, \tau_{1,n}[\,\), or/and \(x_n<0\), \(\zeta_n(t)<0\) for all \(t\in \,]\tau_{2,n},\,T]\), can be analyzed in an entirely similar way. By the uniform convergence of \(\zeta_n\) to \(\zeta\) and since \(\tau_{i,n}\to \tau_i\), \(i=1,2\), it follows from 43 that \[\label{interf-gen-char-cond-1} \zeta(t)=0,\qquad f_l(u_l(t))=f_r(u_r(t))=\gamma \qquad\;\forall~t\in \,]\tau_{1},\,\tau_{2}[\,.\tag{46}\] and \[\label{interf-gen-char-cond-2} \zeta(t)\leq 0\quad \;\forall~t\in [0, \tau_{1}],\qquad\quad \zeta(t)\geq 0\quad\;\forall~t\in [\tau_{2},\,T].\tag{47}\] Moreover, we have \[\label{interf-gen-char-cond-T} \zeta(T)=x,\tag{48}\] since \(x_n\to x\) and \(x_n=\zeta_n(T)\to \zeta(T)\). Note also that, because of 44 , there holds \[\label{eq:der-char} \dot{\zeta}(t)=\lim_n \dot{\zeta}_n(t)= \begin{cases} \dfrac{x}{T-\tau_{2}} \quad&\forall~t \in \, ]\tau_{2}, T[\,, \\ \noalign{\medskip} -\dfrac{\zeta(0)}{\tau_{1}} \quad&\forall~t \in \, ]0,\tau_{1}[\,. \end{cases}\tag{49}\]

Now, if we assume that \(x>0\), it follows from 49 that that \(\zeta(t)>0\) for all \(t\in\,]\tau_2, T]\). On the other hand, since uniform limit of classical genuine characteristics is a classical genuine characteristic as well, we deduce that the restriction of \(\zeta\) to \(\,]\tau_2, T]\) is a classical genuine characteristic for \(u_t+f_r(u)_x\).

Next, if we assume that \(x=0\), then the uniform convergence of \(\zeta_n\) to \(\zeta\), together with 45 , 49 , imply that \[\label{eq:limchar-der-2} \zeta(t)=\dot{\zeta}(t)=0\qquad \forall~t \in \, ]\tau_{2}, T[\,,\tag{50}\] and \[\label{eq:limchar-der-3} f'_r(u_r(t))= \lim_n f'_r(u(\zeta_n(t),t))=\lim_n \dot{\zeta}_n(t)=0 \qquad \forall~t \in \, ]\tau_{2}, T[\,.\tag{51}\] In turn, 51 implies that \(u_r(t)=\theta_r=B\) for all \(t \in \, ]\tau_{2}, T[\), and that \((A,B)\) is a critical connection. On the other hand, because of the interface condition 13 , it follows that \(f_l(u_l(t))=f_r(u_r(t))=\gamma\) for all \(t \in \, ]\tau_{2}, T[\), which proves that the restriction of \(\zeta\) to the interval \([\tau_2, T]\) satisfies the condition (ii) of Definition 6. With entirely similar arguments one can show that the restriction of \(\zeta\) to the interval \([0,\tau_1]\) satisfies the condition (i) or (ii) of Definition 6, which, together with 46 , completes the proof that \(\zeta\) is an \(AB\)-gic belonging to the set \(\mathcal{C}(u,x)\). This completes the proof of 42 whenever \(\{x_n\}_n\) is a sequence converging to \(x\geq 0\). The case where the limit point \(x\) of \(\{x_n\}_n\) is non positive can be treated in entirely similar way.

3. Proof of (iii). Let \(\{x_n\}_n\) be a sequence converging to \(x\in\mathbb{R}\), and let \(\{y_n\}_n\) be a sequence of elements of \(\mathcal{C}_0(u,x_n)\) converging to some point \(y\in\mathbb{R}\). Then, there will be a sequence of \(AB\)-gic \(\zeta_n\in \mathcal{C}(u,x_n)\), such that \(y_n=\zeta_n(0)\) for all n. Observe that by Definition 6 it follows that \[\label{eq:boundchar-derchar-n} |\zeta_n(t)|\leq |x_n|+LT,\qquad |\dot{\zeta}_n(t)|\leq L,\qquad\forall~t\in[0,T],\quad \forall~n\,,\tag{52}\] for some constant \(L>0\) depending on \(\|u\|_{\boldsymbol{L}^\infty}\). Hence, applying Ascoli-Arzelà Theorem, we deduce that up to a subsequence \(\{\zeta_{n}\}_n\) converges uniformly to some \(\zeta\in \mathrm{Lip}([0,T]\; ; \mathbb{R})\). Thus, in particular we have \[\label{eq:initial-char} \zeta(0)=\lim_m\zeta_n(0)=\lim_n y_n=y.\tag{53}\] Then, in view of property (ii) established at previous point, we find that \(\zeta\in \mathcal{C}(u,x)\), and 53 implies \(y\in \mathcal{C}_0(u,x)\), completing the proof of (iii).

4. Proof of (iv). Given \(x_1<x_2\), let \(y_1=\max \mathcal{C}_0(u, x_1)\), and consider \(\zeta_1 \in \mathcal{C}(u, x_1)\) such that \(\zeta_1(0) = y_1\). Choose any \(\zeta_2 \in \mathcal{C}(u, x_2)\) and define \[\zeta(t) \doteq \max \{\zeta_1(t), \, \zeta_2(t)\} \qquad t \in [0,T].\] Observe that, by definition the maximum of two \(AB\)-gic is still an \(AB\)-gic, and \(\zeta(T) = x_2\), so that one has \(\zeta \in \mathcal{(}u,x_2)\). Moreover: \[\max \mathcal{C}_0(u, x_1) =y_1 = \zeta_1(0) \leq \zeta(0) \leq \max \mathcal{C}_0(u, x_2).\] This proves (iv), and thus concludes the proof of the proposition. ◻

Proof. To fix the ideas, we will assume that, letting \(\mathsf{L}, \mathsf{R}\) be the quantities defined in ?? , there holds \(\mathsf{L}=0\), \(\mathsf{R}\in \,]0, T\cdot f'_r(B)[\) , and that \(\omega^T\) fulfills the conditions (i)‘-(ii)’ of [1] for a non critical connection \((A,B)\), which in particular require \[\begin{align} \tag{54} \omega^T(x-)&\geq \omega^T(x+) \qquad\forall~x\neq 0, \\ \tag{55} \omega^T(x)&\geq B\qquad\qquad \forall~x\in\,]0,\mathsf{R}[\,. \end{align}\] The cases where \(\omega^T\) belongs to other classes of reachable profiles described in [1] can be analyzed with entirely similar arguments. Throughout the proof we let \(u_l(t), u_r(t)\), and \(u^*_l(t), u^*_r(t)\), denote the one-sided traces of \(u(t,\cdot)\) and \(u^*(t,\cdot)\), respectively, at \(x=0\).

1. Relying on the fact that any sequence \(\{\zeta_n\}_n\) of \(AB\)-gic (for \(u^*\) and \(u\)) admits a subsequence uniformly convergent to some \(\zeta\in \mathrm{Lip}([0,T]\; ; \mathbb{R})\) (see point 3. of the proof of Proposition 14), and since the map \(x\to \mathcal{C}(u^*, x) \cap \mathcal{C}(u, x)\) has closed graph by Proposition 14-(ii), it will be sufficient to show that \(\mathcal{C}(u^*, x) \cap \mathcal{C}(u, x) \neq \emptyset\) holds for all point \(x\) of continuity for \(\omega^T\). Moreover, for every point \(x\in\,]\!-\infty, 0[\;\cup\;]\mathsf{R}, +\infty[\)  of continuity for \(\omega^T\), there exists a unique \(AB\)-gic for \(u^*\) and \(u\) that reaches the point \(x\) at time \(T\), which is a classical genuine characteristic \(\vartheta_x\) for \(u^*\) and \(u\) (since it never crosses the interface \(x=0\), but at most at \(t=0\), by definition ?? ). Thus we have \(\mathcal{C}(u^*, x) \cap \mathcal{C}(u, x)=\{\vartheta_x\}\) for all point \(x\in\,]\!-\infty, 0[\;\cup\;]\mathsf{R}, +\infty[\)  of continuity for \(\omega^T\). As a consequence, in order to establish ?? it will be sufficient to show \[\label{eq:Cuu9442-2} \mathcal{C}(u^*, x) \cap \mathcal{C}(u, x) \neq \emptyset\qquad\quad\text{for all x\in\,]0, \mathsf{R}[\, of continuity for \omega^T}.\tag{56}\] To this end, given any \(x\in\,]0, \mathsf{R}[\,\) of continuity for \(\omega^T\), we consider the \(AB\)-gic \(\zeta\in \mathcal{C}(u, x)\) defined in 40 , with \(\overline{\tau}\) as in 36 and \[\label{eq:tau-def-2} \tau(x) \doteq T-\frac{x}{f_r^{\prime}(\omega^T(x))},\tag{57}\] in place of \(\tau_-(x)\). We will show that \(\zeta\) also belongs to \(\mathcal{C}(u^*, x)\). Note that by definition of \(\mathsf{R}\) at ?? we have \(\tau(x) >0\).

2. We determine here explicitly the \(AB\)-entropy solution \(u^*\) defined by ?? ?? when \(\omega^T\) satisfies the conditions (i)‘-(ii)’ of [1] for a non critical connection, with \(\mathsf{L}=0\), \(\mathsf{R}\in \,]0, T\cdot f'_r(B)[\) . These conditions require in particular that \[\begin{align} \tag{58} \omega^T(0-)&\geq\pi(\omega^T(0+)), \\ \tag{59} \omega^T(x)&\geq B,\qquad \forall~x\in\,]0, \mathsf{R}[\,, \\ \tag{60} \omega^T(\mathsf{R}+)&\leq \boldsymbol{u}\,, \end{align}\] where \[\pi(u)\doteq ({f_l}_{\mid [\theta_l,+\infty)})^{-1} \circ f_r(u),\qquad u\in\mathbb{R},\] and \(\boldsymbol{u}\doteq \boldsymbol{u}[\mathsf{R}, B, f_r]\) is the quantity defined in [1] that satisfies \[\label{eq:prop-constraint-1} B>\boldsymbol{u} >\overline{B}, \qquad\quad f'_r(\boldsymbol{u})<\mathsf{R}/T,\tag{61}\] (with \(\overline{B}\) defined as in 14 ). Because of condition 59 , according with the analysis in [1] the solution \(u^*\) contains a shock curve starting at the interface \(x=0\) and reaching the point \(\mathsf{R}\) at time \(T\), which is parametrized by a map \(\gamma : [\boldsymbol{\tau}, T]\to [0,\infty[\,\), with the properties that \(\gamma(\boldsymbol{\tau})=0\), \(\gamma(T)=\mathsf{R}\), where \(\boldsymbol{\tau}\doteq \boldsymbol{\tau}[\mathsf{R}, B, f_r]\) is a quantity defined as in [1]. The curve \(t\to (\gamma(t), t)\) is the location of a shock for the conservation law \(u_t+f_r(u)_x=0\), which connects the left state \(B\) with the right states \((f'_r)^{-1}\big(\big({\gamma(t)-\mathsf{R}+T \cdot f'_r(\boldsymbol{u} )}\big)/{t}\big)\), \(t\in [\boldsymbol{\tau}, T]\). On the right of \(\gamma(t)\) there is a rarefaction wave, connecting the left state \(\overline{B}\) with the right state \(\boldsymbol{u}\), and centered at the point \((\mathsf{R}-T \cdot f'_r(\boldsymbol{u}), 0)\). Moreover, there holds \[\label{eq:tau-id-1} \boldsymbol{\tau}=\dfrac{T\cdot f'_r(\boldsymbol{u})-\mathsf{R}}{f'_r(\,\overline{B})}.\tag{62}\] Following the procedure described in [1], in order to define \(u^*\) we introduce some notations for the polygonal lines along which \(u^*\) takes constant values in each region \(\{x<0\}\), \(\{x>0\}\) (that correspond to \(AB\)-gic for \(u^*\)). We define \[\begin{align} \vartheta_{\rm 0, -}(t) &\doteq (t-T)\cdot f'_l(\omega^T(0-)), \\ \vartheta_{\rm 0, +}(t) &\doteq (t-T)\cdot f'_l\big(\pi(\omega^T(0+))\big), \\ \vartheta_{\rm R, -}(t) &\doteq \begin{cases} \mathsf{R}-(T-t) \cdot f'_r\big(\omega^T(\mathsf{R}-), \;\;&\text{if}\quad \tau_-(\mathsf{R})\leq t\leq T, \\ \noalign{\smallskip} \big(t-\tau_-(\mathsf{R})\big) \cdot f'_l\circ\pi(\omega^T(\mathsf{R}-)), \;\; &\text{if}\quad 0\leq t\leq \tau_-(\mathsf{R}), \end{cases} \\ \vartheta_{\rm R, +}(t) &\doteq \mathsf{R}-(T-t) \cdot f'_r\big(\omega^T(\mathsf{R}+)\big), \end{align}\] and, for every \(y \in \,]-\infty, 0\,[\,\cup \,]\mathsf{R}, +\infty[\) , we define \[\label{eq:pollines} \vartheta_{y, \pm}(t) \doteq \!\begin{cases} y-(T-t) \cdot f_l^{\prime}\big(\omega^T(y\pm)\big), & \text{ if \;y < 0, \quad 0\leq t \leq T},\\ \noalign{\smallskip} y-(T-t) \cdot f'_r\big(\omega^T(y\pm)\big), & \text{ if \;0< y < {\mathsf{R}}, \quad \tau_{\pm}(y) \leq t \leq T},\\ \noalign{\smallskip} \big(t-\tau_{\pm}(y)\big) \cdot f'_l\circ\pi(\omega^T(y\pm)), & \text{ if \;0< y < {\mathsf{R}}, \quad 0 \leq t < \tau_{\pm}(y),}\\ \noalign{\smallskip} y-(T-t) \cdot f'_r\big(\omega^T(y\pm)\big), & \text{ if \;y> \mathsf{R}, \quad 0\leq t \leq T}, \end{cases}\tag{63}\] where \[\label{eq:tau-def-pm} \begin{align} \tau_{\pm}(y)&\doteq T-\frac{y}{f'_r(\omega^T(y\pm))},\qquad y>0. \end{align}\tag{64}\] Moreover, letting \(\{y_n\}_n\) denote the (at most) countably many discontinuity points of \(\omega^T\) in the intervals \(]-\infty, 0]\), \(]\mathsf{R}, +\infty[\), we set \[\begin{align} \mathcal{I}^n_{0} &=\,]x_n^-,x_n^+[\,, \quad x_n^\pm=\vartheta_{y_n,\pm}(0), \quad y_n\in \,]-\infty, 0]\,, \\ \mathcal{I}^n_{\mathsf{R}}&=\,]x_n^-,x_n^+[\,, \quad x_n^\pm=\vartheta_{y_n,\pm}(0), \quad y_n\in \,]\mathsf{R},+\infty[\,, \end{align}\] (here we consider the possibility of a jump of \(\omega^T\) in \(x=0\) when \(\omega^T(0-)>\pi(\omega^T(0+))\)). The intervals \(\mathcal{I}^n_{0}\), \(\mathcal{I}^n_{\mathsf{R}}\), consist of the starting points of compression waves in \(u^*\) that generate a shock at \((y_n, T)\).

Next, we introduce the polygonal lines connecting two points \((z,0)\), \((y,T)\) (that correspond to compression fronts for \(u^*\) generating a shock at the point \((y,T)\)) defined by \[\eta_{y,z}\doteq \begin{cases} y -(T-t)\cdot \frac{(y-z)}{T},\quad &\text{if}\quad y\in\,]\!-\infty, 0]\,\cup\,]\mathrm{R}, +\infty[\,,\quad 0\leq t \leq T, \\ \noalign{\smallskip} y -(T-t)\cdot f'_r(u_{y,z}) \quad &\text{if}\quad 0<y<\mathsf{R}, \quad T-{y}/{f'_r(u_{yz})}\leq t\leq T, \\ \noalign{\smallskip} \big(t-T+{y}/{f'_r(u_{yz})}\big)\cdot f'_l\circ \pi(u_{y,z}) \quad &\text{if}\quad 0<y<\mathsf{R}, \quad 0\leq t<T-{y}/{f'_r(u_{yz})}, \end{cases}\] where \(u_{y,z}\) is the unique constant \(u\geq (f'_r)^{-1}(y/T)\) satisfying \[\Big(\frac{y}{f'_r(u)}-T\Big)\cdot f'_l\circ \pi(u)=z\] (see [1]). Finally, we set \[\begin{align} r_-(t)&\doteq \mathsf{R}-T \cdot f'_r(\boldsymbol{u})+ t \cdot f'_r(\,\overline{B}), \quad\;t\in [0,\boldsymbol{\tau}], \\ \noalign{\smallskip} r_+(t)&\doteq \mathsf{R}-(T-t) \cdot f'_r(\boldsymbol{u}),\quad\;t\in [0,T]. \end{align}\] Then, the function \(u^*\) defined by ?? ?? is given by \[\label{eq:u9442-ex3} u^*(x,t)= \begin{cases} \omega^T(y\pm), & \text{if \;x=\vartheta_{y,\pm}(t) \;for some \;y \in \,]-\infty, 0[\, \cup \,]\mathsf{R}, +\infty[\,},\\ \noalign{\smallskip} \omega^T(y\pm), & \text{if \; x=\vartheta_{y,\pm}(t)>0 \;for some \;y \in \,]0, \mathsf{R}[ \,},\\ \noalign{\smallskip} \pi (\omega^T(y\pm)), & \text{if \; x=\vartheta_{y,\pm}(t)<0 \;for some \;y \in \,]0,\mathsf{R}[ \,},\\ \noalign{\smallskip} (f_r^{\prime})^{-1}\big(\frac{y_n-z}{T}\big), & \text{if \; x = \eta_{y_n, z}(t) \;for some \;z \in \mathcal{I}^n_{\mathsf{R}}},\\ \noalign{\smallskip} (f_l^{\prime})^{-1}\big(\frac{y_n-z}{T}\big), & \text{if \; x = \eta_{y_n, z}(t) \;for some \;z \in \mathcal{I}^n_{0}},\\ \noalign{\smallskip} \;B & \text{if}\; \left\{ \begin{align} &\vartheta_{\rm R, -}(t)\leq x<\gamma(t), \;\;t\in [\tau_-(\mathsf{R}), T], \\ \noalign{\smallskip} &0<x<\gamma(t), \;\;t\in [\boldsymbol{\tau}, \tau_-(\mathsf{R})], \end{align} \right. \\ \noalign{\smallskip} \;\overline{A} & \text{if}\;\;\vartheta_{\rm R, -}(t)\leq x<0, \;\;t\in [0,\tau_-(\mathsf{R})], \\ \noalign{\smallskip} \;\overline{B} & \text{if}\;\;0<x\leq r_-(t), \;\;t\in [0, \boldsymbol{\tau}], \\ \noalign{\smallskip} (f'_r)^{-1}\Big(\dfrac{x-\mathsf{R}+T \cdot f'_r(\boldsymbol{u} )}{t}\Big) \;\;&\text{if}\; \left\{ \begin{align} &\gamma(t)<x<r_+(t), \;\;t\in [\boldsymbol{\tau}, T], \\ \noalign{\smallskip} &r_-(t)<x<r_+(t), \;\;t\in [0,\boldsymbol{\tau}], \end{align} \right. \\ \noalign{\smallskip} (f'_r)^{-1}\big(\frac{\mathsf{R}-x}{T-t}\big) & \text{if}\;\; r_+(t)\leq x\leq \vartheta_{\rm R, +}(t), \;\;t\in [0, T[\,. \end{cases}\tag{65}\]

Observe that the left and right traces of \(u^*\) satisfy \[\label{eq:ul42cond1} \begin{align} u^*_l(t)&\geq \overline{A},\qquad u^*_r(t)\geq B, \qquad \forall~t\in\,]\tau_-(\mathsf{R}),T], \\ u^*_l(t)&= \overline{A},\qquad u^*_r(t)= B, \qquad \forall~t\in\,]\boldsymbol{\tau},\tau_-(\mathsf{R})], \\ u^*_l(t)&= \overline{A},\qquad u^*_r(t)= \overline{B}, \qquad \forall~t\in\,]0,\boldsymbol{\tau}]. \end{align}\tag{66}\] Moreover, since \(x\) is a point of continuity for \(\omega^T\), it follows that the restriction of \(\zeta\) to \(\,]\tau(x), T]\) is a (classical) genuine characteristic both for \(u\) and \(u^*\) with slope \(f'_r(\omega^T(x))>0\) so that, recalling 13 , there holds \[\label{eq:ur42cond1} u_r(\tau(x))=\omega^T(x)=u^*_r(\tau(x)) >\theta_r, \qquad\quad f_l(u_l(\tau(x)))=f_r(u^*_r(\tau(x))).\tag{67}\] Now, we will distinguish two cases according with the position of \(\overline{\tau}\) with respect to the time \(\tau_-(\mathsf{R})\) defined as in 64 . Note that by definition of \(\mathsf{R}\) at ?? we have \(\tau_-(\mathsf{R}) \geq 0\).

3. Assume that \(\overline{\tau} \geq \tau_-(\mathsf{R})\), and suppose first that \(\tau(x)=\overline{\tau}> \tau_-(\mathsf{R})\). Note that, since \(\overline{\tau}>0\) is an element of the set \(E\) in 35 , and because of  66 , 67 , we have \(u_l(\tau(x))=u^*_l(\tau(x))>\theta_l\). Therefore, also the restriction of \(\zeta\) to \([0,\tau(x)[\,\) is a (classical) genuine characteristic both for \(u\) and \(u^*\). Hence, when \(\tau(x)=\overline{\tau}>\tau_-(\mathsf{R})\), the map \(\zeta\) in 40 is an \(AB\)-gic also for \(u^*\), proving that \(\zeta\in \mathcal{C}(u^*, x) \cap \mathcal{C}(u, x)\).

Next, consider the subcase \(\tau(x)>\overline{\tau}\geq \tau_-(\mathsf{R})\), and observe that by 38 , 39 , 67 , we have \[\label{eq:urtrace} u_l(t)=A,\quad u_r(t)=B\qquad\forall~t\in\,]\,\overline{\tau}, \tau(x)], \qquad\quad u^*_r(\tau(x))=B\,.\tag{68}\] Moreover, we claim that \[\label{eq:clu42trace} \begin{align} \omega^T(z)&=B,\qquad\forall~z\in [x,\,\overline{x}[\,, \quad \overline{x} \doteq (T-\overline{\tau})\cdot f'_r(B), \\ \noalign{\smallskip} u_r^*(t)&=B, \qquad\forall~t\in \,]\,\overline{\tau},\tau(x)]\,. \end{align}\tag{69}\] Note that the first equality in 69 implies the second one by tracing the backward (genuine) characteristics for \(u^*\) at time \(T\), from points \(z\in [x,\,\overline{x}[\,\). In order to prove the first equality in 69 , we trace the minimal backward characteristic \(\vartheta_{z,-}\) for the solution \(u\), at time \(T\), from points \(z \in [x, \overline{x}]\). Observe that \(\vartheta_{z,-}\) impacts the interface \(x=0\) at time \(\tau_-(z)\doteq T-{z}/{f'_r(\omega^T(z-))}\). Moreover, since \(\omega^T(\mathsf{R}-)\geq B\) because of 55 , we deduce from \(\overline{\tau}\geq \tau_-(\mathsf{R})\) that \[\label{eq:barx} \overline{x} \leq \mathsf{R}.\tag{70}\] Furthermore, we have \[\label{eq:omega-B-1} \tau_-(z)\geq T-\frac{z}{f'_r(B)}\geq \overline{\tau }\qquad\;\; \forall~z \in [x,\, \overline{x}],\tag{71}\] since \(\omega^T(z-) \geq B\) by virtue of 55 . On the other hand, by 57 we know that the (genuine) characteristic \(\vartheta_x\) for \(u\), starting at time \(T\) from the point \(x\), reaches the interface \(x=0\) at time \(\tau(x)\). Since \(\vartheta_x\), \(\vartheta_{z,-}\) are (classical) genuine characteristics that cannot cross in the domain \(\{x>0\}\), it follows that \[\label{eq:omega-B-2} \tau(x)\geq \tau_-(z) \qquad\;\; \forall~z \in [x,\, \overline{x}]\,.\tag{72}\] Combining together 71 , 72 , we deduce that, for every \(z\in [x,\,\overline{x}]\), the minimal backward characteristics \(\vartheta_{z,-}\) reaches the interface \(x=0\) at time \(\tau_-(z)\in [\,\overline{\tau},\tau(x)]\). Hence, because of 68 , we find that for all \(z\in [x,\,\overline{x}]\) there holds \(\omega^T(z-) = u_r(\tau_-(z))=B\), and this yields the first equality in 69 , concluding the proof of claim 69 .

Relying on 69 we will show now that \(\zeta\) satisfies the condition of an \(AB\)-gic also for \(u^*\) on the interval \([0, \tau(x)]\). To this end, observe that 66 69 together imply \[\label{eq:cru42trace-2} u_l^*(t)=\overline{A}, \qquad\forall~t\in \,]\,\overline{\tau},\tau(x)]\,.\tag{73}\] Hence, because of 69 , 73 , \(\zeta\) satisfies the condition (ii) of Definition 6 of an \(AB\)-gic for \(u^*\) on the interval \(\,]\overline{\tau}, \tau(x)]\). Moreover, let \(\vartheta_{t_n}^*\) denote the (classical genuine) backward characteristic for \(u^*\), on the region \(\{x<0\}\), starting at time \(t_n\in \,]\,\overline{\tau},\tau(x)]\,\) from \(x=0\), for a sequence \(t_n\downarrow \overline{\tau}\). Note that, because of 73 , all \(\vartheta_{t_n}^*\) have slope \(f_l'(\,\overline{A})\). Thus \(\{\vartheta_{t_n}^*\}_n\) converges uniformly to a function \(\vartheta^*: [0,\overline{\tau}]\to\mathbb{R}\) that is as well a (classical) genuine characteristic for \(u^*\) with slope \(f_l'(\,\overline{A})\) and such that \(\vartheta^*(\overline{\tau})=0\). This in turn implies that \[\label{eq:cluu42trace-1} u^*_l(\overline{\tau}) = \overline{A}\,.\tag{74}\]

Next, we will prove that \[\label{eq:cluu42trace-2} u_l(\overline{\tau}) = \overline{A}\,.\tag{75}\] To this end observe that 54 , 55 , 69 , 70 together imply \(\omega^T(\overline{x}-)= B=\omega^T(\overline{x}+)\). This means that the characteristic \(\vartheta_{\overline{x}}\) starting at time \(T\) from \(\overline{x}\), and reaching \(x=0\) at time \(\overline{\tau}\), is a (classical) genuine characteristic for \(u\) (on the semiplane \(\{x>0\}\)), and hence we deduce that \[\label{eq:crutrace-2} u_r(\overline{\tau})=\omega^T(\overline{x})=B.\tag{76}\] Recalling 37 and the definition 35 of the set \(E\), we derive from 76 and from condition (2) of Definition 2 that \[\label{eq:ltracebartau} u_l(\overline{\tau})>\theta_l.\tag{77}\] In turn, condition 77 , together with 68 , implies 75 by a blow-up argument as in [1]. Namely, we can consider the blow ups of \(u\) at the point \((0, \overline{\tau}\,)\): \[\label{eq:bup-def} u_n(x,t) \doteq u\big(x/n,\, \overline{\tau }+{(t-\overline{\tau})}/{n}\big) \qquad x\in\mathbb{R}, \;t\geq 0\,,\;\;n\in\mathbb{N},\tag{78}\] and observe that, because of 68 , the left and right traces of \(u_n(\cdot, t)\) at \(x=0\) satisfy \[\label{eq:tracesit-3} (u_{n,l}(t), u_{n,r}(t))=(A,B)\quad\qquad\forall~t\in\big]\overline{\tau}, \, \overline{\tau }+n\big(\tau(x)-\overline{\tau}\big)\big[\,.\tag{79}\] When \(n\to\infty\), up to a subsequence, the blow ups \(u_n(\cdot, t)\) converge in \({\boldsymbol{L}^1_{loc}}\) to a limiting \(AB\) entropy solution \(v(\cdot, t)\), for all \(t>0\), and there holds \[\label{eq:indata-blup} v(x, \overline{\tau}\,)= \begin{cases} u_l(\overline{\tau}), & \text{if x < 0},\\ u_r(\overline{\tau}), & \text{if x > 0}, \end{cases}\tag{80}\] \[\label{eq:tracesit} v(0-,t)\in\{A,\overline{A}\,\},\qquad\qquad v(0+,t)\in\{B,\overline{B}\,\},\qquad\forall~t>\overline{\tau}\,.\tag{81}\] Then, by a direct inspection we find that, if an \(AB\) entropy solution of a Riemann problem for 1 , with initial datum 80 at time \(\overline{\tau}\), enjoys the properties  77 , 81 , it follows that the left initial datum at time \(\overline{\tau}\) must be \[v(x,\overline{\tau})=u_l(\overline{\tau}) = \overline{A},\qquad \forall~x<0,\] thus proving 75 .

The two conditions  equalities 74 , 75 and the definition 40 imply that the restriction of \(\zeta\) to \([0, \overline{\tau}[\,\) is a (classical) genuine characteristic both for \(u\) and \(u^*\) with slope \(f'_l(\,\overline{A})>0\). Therefore we can conclude that \(\zeta\) satisfies the condition of an \(AB\)-gic also for \(u^*\) on the interval \([0, \tau(x)]\), and hence on the whole interval \([0,T]\) by the analysis in the point 2. This completes the proof that \(\zeta\in \mathcal{C}(u^*, x) \cap \mathcal{C}(u, x)\) when \(\overline{\tau}\geq \tau_-(\mathsf{R})\).

4. Assume that \(\overline{\tau} <\tau_-(\mathsf{R})\), with \(\tau_-(\mathsf{R})\) as in 64 . If we suppose that 77 holds, since 68 is still verified we can deduce as above that 75 holds as well, and then we conclude that \(\zeta\in \mathcal{C}(u^*, x) \cap \mathcal{C}(u, x)\) with the same arguments of point 3.

Therefore, let us assume that \(u_l(\overline{\tau}) \leq \theta_l\) and that \(\zeta(t) > 0\) for all \(t\in [0, \overline{\tau}[\) . Observe that because of 37 , and by definition 35 of the set \(E\), we have \[\label{eq:rtracebartau} u_r(\overline{\tau})<\theta_r.\tag{82}\] Then, relying on 68 , we deduce with the same blow up argument of above that \[\label{eq:cruu42trace-2} u_r(\overline{\tau}) = \overline{B}\,.\tag{83}\] On the other hand, if we show that \[\label{eq:tau-bar-tau-ineq} \overline{\tau }\leq \boldsymbol{\tau},\tag{84}\] it would follow from 66 that \[\label{eq:cruu42trace-3} u^*_r(\overline{\tau})= \overline{B}\,.\tag{85}\] The two conditions 83 , 85 and the definition 40 imply that the restriction of \(\zeta\) to \([0, \overline{\tau}[\,\) is a (classical) genuine characteristic both for \(u\) and \(u^*\) with slope \(f'_r(\,\overline{B})<0\). Therefore, if 84 holds, we can conclude that \(\zeta\) satisfies the condition of an \(AB\)-gic also for \(u^*\) on the interval \([0, \tau(x)]\), and hence on the whole interval \([0,T]\) by the analysis in the point 2. Hence, in order to completes the proof that \(\zeta\in \mathcal{C}(u^*, x) \cap \mathcal{C}(u, x)\) when \(\overline{\tau}<\tau_-(\mathsf{R})\), it remains to establish 84 .

By contradiction, assume that \(\overline{\tau }> \boldsymbol{\tau}\). Define the curve

\[\xi(t) \, \doteq \, \inf \Big\{ R >0 \; \big |\; x-t \, f_r^{\prime}(u(x,t)) \geq 0 \;\;\;\forall~R>0\Big\} \qquad t \in [\overline{\tau}, T].\] Notice that, because of 83 , we have \(\xi(\overline{\tau}) =0\), while the definition ?? yields \(\xi(T) = \mathsf{R}\). But now using a comparison argument between \(\xi(t)\) and the map \(\gamma(t)\) defining the shock curve of \(u^*\) at point 2, we obtain as in [1] that \(\xi(t)<\gamma(t)\) for all \(t\in [\overline{\tau}, T]\). Thus, we find in particular that \(\xi(T) <\gamma(T)=\mathsf{R}\), which gives a contradiction. This concludes the proof of the proposition. ◻

Proof. By the analysis in [1] we deduce that \(\mathbb{R}\) can be partitioned as the union of sets containing points of three types:

• starting points of compression fronts (possibly refracted by the interface \(x=0\)) which meet together generating a shock at time \(T\);

• starting points of classical genuine characteristics or of polygonal lines made of two segments consisting of classical genuine characteristics in each semiplane \(\{x<0\}\), \(\{x>0\}\), which reach at time \(T\) a point of continuity of \(\omega^T\).

• starting points \(y\) of polygonal lines \(\xi:[0,T] \to \mathbb{R}\) with \(\xi(0) = y\), composed of three segments of the form \[\xi(t) = \begin{cases} y+t \, f^\prime(u(t,\xi(t)), \xi(t)), & \text{if 0 \leq t \leq t_1},\\ 0 , & \text{if t_1 \leq t \leq t_2},\\ (t-t_2) f^\prime(u(t,\xi(t)), \xi(t)) & \text{if t_2 \leq t \leq T} \end{cases}\] where \[f(u(t, 0\pm)) = \gamma \qquad \forall t \in (t_1, t_2).\] Notice that these polygonal lines may belong to the near-interface regions \(\Delta_{\mathsf{L}}, \Gamma_{\mathsf{R}}\) defined in ([1], §5.4.4).

In all cases they are starting points of segments or of polygonal lines which are \(AB\)-gics for \(u^*\), and the result follows. ◻

Proof. Let \(0\leq\tau_1\leq\tau_2\leq T\), be the partition of \([0,T]\) for \(\zeta\in \mathcal{C}(u^*,x)\) given by Remark 7, so that there holds \[\label{interf-gen-char-cond-5} \zeta(t)=0,\qquad f_l(u^*_l(t))=f_r(u^*_r(t))=\gamma \qquad\;\forall~t\in [\tau_1,\,\tau_2].\tag{91}\] To fix the ideas we assume that \[\label{interf-gen-char-cond-n4} \zeta(t)<0\qquad \forall~t\in [0, \tau_1[\,,\qquad\quad \zeta(t)>0\qquad \forall~t\in \,]\tau_2,\,T].\tag{92}\] The cases where \(\zeta(t)>0\) for all \(t\in [0, \tau_1[\,,\) or \(\zeta(t)<0\) for all \(t\in \,]\tau_2,\,T]\) are entirely similar. Then, setting \[\label{eq:ggstardef} \begin{align} g(t)&\doteq \begin{cases} f_r(u(\zeta(t)+,t))-\dot{\zeta}(t)\, u(\zeta(t)+,t) \quad &\text{if}\quad t\in \,]\tau_2,\,T], \\ \noalign{\smallskip} f_r(u_r(t)) \quad &\text{if}\quad t\in [\tau_1, \tau_2], \\ \noalign{\smallskip} f_l(u(\zeta(t)+,t))-\dot{\zeta}(t)\, u(\zeta(t)+,t) \quad &\text{if}\quad t\in [0,\tau_1[\,, \end{cases} \\ \noalign{\medskip} g^*(t)&\doteq \begin{cases} f_r(u^*(\zeta(t),t))-\dot{\zeta}(t)\, u^*(\zeta(t),t) \quad &\text{if}\quad t\in \,]\tau_2,\,T], \\ \noalign{\smallskip} \;\gamma \quad &\text{if}\quad t\in [\tau_1, \tau_2], \\ \noalign{\smallskip} f_l(u^*(\zeta(t),t))-\dot{\zeta}(t)\, u^*(\zeta(t),t) \quad &\text{if}\quad t\in [0,\tau_1[\,, \end{cases} \end{align}\tag{93}\] we write \[\label{eq:Fg1} \mathcal{F}_t(\zeta, u)=\int_t^T g(t)\mathop{}\!\mathrm{d}t, \qquad\quad \mathcal{F}_t(\zeta, u^*)=\int_t^T g^*(t)\mathop{}\!\mathrm{d}t,\qquad\forall~t\in [0,T].\tag{94}\] Note that, because of 88 , in the definition of \(g\) we may equivalently take \(u(\zeta(t)-,t)\) instead of \(u(\zeta(t)+,t)\), while in the definition of \(g^*\) we take \(u^*\) continuous at \((\zeta(t),t)\) since \(\zeta\) is a classical genuine characteristic for \(u^*\) when \(t\in \,]0,\tau_1[\, \cup\,]\tau_2, T[\) . Observe that, since \(u\) is an \(AB\)-entropy solutions to 1 , by the interface condition 13 we have \[\label{eq:gest1} f_r(u_r(t))\geq \gamma \qquad \forall~t\in [\tau_1, \tau_2].\tag{95}\] On the other hand, because of the convexity of \(f_r\) there holds \[\label{eq:convexest} f_r(v) - f_r^{\prime}(w)\,v \geq f_r(w) - f_r^{\prime}(w)\,w, \quad \forall \; v,w \in \mathbb{R}.\tag{96}\] Moreover, since \(\zeta\) is an \(AB\)-gic for \(u^*\), and because of 92 , note that the restriction of \(\zeta\) to \([0,\tau_1[\,\) is a classical genuine characteristic for \(u^*\) as solution of \(u_t+f_l(u)_x=0\), and the restriction of \(\zeta\) to \(\,]\tau_2, T]\) is a classical genuine characteristic for \(u^*\) as solution of \(u_t+f_r(u)_x=0\). Hence, it follows that \[\label{eq:charspeed-1} \dot{\zeta}(t)= \begin{cases} f'_l(u^*(\zeta(t),t))\quad &\text{if}\;\quad t \in \,]0,\tau_1[\,, \\ \noalign{\smallskip} \;0 &\text{if}\;\quad t \in \,]\tau_1, \tau_2[\,, \\ \noalign{\smallskip} f'_r(u^*(\zeta(t),t))\quad &\text{if}\;\quad t \in \,]\tau_2,T[\,. \end{cases}\tag{97}\] Thus, 96 97 together imply \[\label{eq:gest2} \begin{align} f_l(u(\zeta(t)+,t))-\dot{\zeta}(t)\, u(\zeta(t)+,t)&\geq f_l(u^*(\zeta(t),t))- \dot{\zeta}(t)\, u^*(\zeta(t),t) \qquad\forall~t\in \,]0, \tau_1[\,, \\ \noalign{\smallskip} f_r(u(\zeta(t)+,t))-\dot{\zeta}(t)\, u(\zeta(t)+,t)&\geq f_r(u^*(\zeta(t),t))- \dot{\zeta}(t)\, u^*(\zeta(t),t) \qquad\forall~t\in \,]\tau_2,T[\,. \end{align}\tag{98}\] Therefore, from 95 , 98 we deduce that \[\label{eq:gest3} g(t)\geq g^*(t)\qquad\forall~t\in [0,T],\tag{99}\] which, because of 94 , yields 89 .

Concerning 90 , if \(\zeta \in \mathcal{C} (u^*,x) \cap \mathcal{C}(u,x)\), then the inequality 89 is verified also when \(u\) and \(u^*\) switch their places, so that we have \(\mathcal{F}_t(u,\zeta) \geq \mathcal{F}_t(u^*,\zeta)\) for all \(t\in [0,T]\), thus proving \[\label{eq:Fest3} \zeta \in \mathcal{C} (u^*,x) \cap \mathcal{C}(u,x) \quad \Longrightarrow\quad \mathcal{F}(u,\zeta) = \mathcal{F}(u^*,\zeta).\tag{100}\] Next, given \(\zeta\in \mathcal{C}(u^*,x)\), assume that \[\label{eq:Fest4} \mathcal{F}(\zeta, u) = \mathcal{F}(\zeta, u^*).\tag{101}\] Since, by the above analysis we have 95 , it follows from  101 that \(g(t)=g^*(t)\) for a.e. \(t\in [0,T]\). Because of 93 , 97 , this in particular implies that, for a.e. \(t\in [0,T]\), there holds \[\label{eq:gest5} f_r(u_r(t))=\gamma\qquad\text{if} \quad t\in [\tau_1,\tau_2]\,,\tag{102}\] and \[\label{eq:gest4} \begin{align} &f_l(u(\zeta(t)+,t))-f'_l(u^*(\zeta(t),t))\, u(\zeta(t)+,t)= \\ &\qquad\quad= f_l(u^*(\zeta(t),t))-f'_l(u^*(\zeta(t),t))\, u^*(\zeta(t),t) \qquad\text{if}\quad t\in \,]0,\tau_1[\,, \\ \noalign{\medskip} &f_r(u(\zeta(t)+,t))-f'_r(u^*(\zeta(t),t))\, u(\zeta(t)+,t)= \\ &\qquad\quad = f_r(u^*(\zeta(t),t))-f'_r(u^*(\zeta(t),t))\, u^*(\zeta(t),t) \qquad\text{if}\quad t\in \,]\tau_2,T[\,. \end{align}\tag{103}\] Since \(f_l, f_r\) are strictly convex functions, we deduce from 103 that \(u(\zeta(t)+,t)=u^*(\zeta(t),t)\) for a.e. \(t\in \,]0,\tau_1[\, \cup\,]\tau_2, T[\) . If we repeat the same analysis taking \(u(\zeta(t)-,t)\) instead of \(u(\zeta(t)+,t)\) in the definition 93 of \(g\), we find that also \(u(\zeta(t)-,t)=u^*(\zeta(t),t)\) for a.e. \(t\in \,]0,\tau_1[\, \cup\,]\tau_2, T[\) . This shows that the restriction of \(\zeta\) to \([0,\tau_1[\,\) and to \(\,]\tau_2, T]\) is a classical genuine characteristic for \(u\) as well, as solution of \(u_t+f_l(u)_x=0\), and of \(u_t+f_r(u)_x=0\), respectively. Hence, by Remark 7 we deduce that \(\zeta\) is an \(AB\)-igc also for \(u\), which means that \(\zeta\in \mathcal{C}(u,x)\), completing the proof of \[\label{eq:Fest5} \mathcal{F}(\zeta, u) = \mathcal{F}(\zeta, u^*) \quad \Longrightarrow\quad \zeta \in \mathcal{C} (u^*,x) \cap \mathcal{C}(u,x),\tag{104}\] and thus concluding the proof of the Lemma. ◻

Proof. Observe that, by property (1) of Definition 3, \(u\) is a weak distributional solution to 1 , 3 . Moreover, by Remark 4, \(u(t,\cdot)\) is a function of locally bounded variation on \(\{x<0\}\), \(\{x>0\}\), and it admits left and right strong traces at \(x = 0\), for all \(t>0\). Thus, we can recover the equality 105 recalling definition 87 , applying the divergence theorem to the vector field \((f(x,u), u)\) on each domain \(\Delta\cap \{x<\rho\}\), \(\Delta\cap\{x>\rho\}\), with \(\Delta\doteq\{(x,t)\; | \; \alpha(t)\leq x\leq \beta(t), \;t\in[t_0, T] \}\), and then taking the limit as \(\rho\to 0\). ◻

Proof of Theorem 9. Given \(\omega^T \in \mathcal{A}^{[AB]}(T)\), let \(u^*\) be the \(AB\)-entropy solution defined by ?? ?? .

1. We will show that if \(u_0 \in \mathcal{I}_T^{[AB]}(\omega^T)\), then for every point \(\overline{x}\in\mathbb{R}\) there exists \(\overline{y} \in \mathcal{C}_0(u^*,\overline{x})\) such that there hold ?? . The proof of ?? is entirely similar. By Proposition 15, choose \(\zeta_{\overline{x}} \in \mathcal{C}(u^*,\overline{x}) \cap \mathcal{C}(u,\overline{x})\), with \(u \doteq \mathcal{S}^{\, [A B]+}u_0(\cdot)\), and set \(\overline{y} \doteq \zeta_{\overline{x}}(0)\). Then, consider any \(y < \min \mathcal{C}_0(u^*,\overline{x})\). By Lemma 1, and because of Proposition 14-(iv), there will be some \(x< \bar x\), and some \(\zeta_x \in \mathcal{C}(u^*,x)\), such that \(y = \zeta_x(0)\). Hence, applying Lemma 2, we deduce that \[\label{Fineq} \mathcal{F}(u,\zeta_x) \geq \mathcal{F}(u^*, \zeta_x).\tag{106}\] Moreover, since \(\zeta_{\bar x} \in \mathcal{C}(u^*,\overline{x}) \cap \mathcal{C}(u,\overline{x})\), by the second part of Lemma 2 we have \[\label{Feq} \mathcal{F}(u^*,\zeta_{\bar x}) = \mathcal{F}(u, \zeta_{\bar x}).\tag{107}\] On the other hand, applying Lemma 3 to the solution \(u^*\) with \(\alpha = \zeta_x, \beta = \zeta_{\overline{x}}\), \(\tau=0\), and recalling Remark 17, one obtains \[\label{eq:Fineq2} \int_x^{\bar x} \omega^T(\xi) \mathop{}\!\mathrm{d}\xi - \int_y^{\bar y} u_0^*(\xi) \mathop{}\!\mathrm{d}\xi = \mathcal{F}(u^*,\zeta_x)-\mathcal{F}(u^*,\zeta_{\bar x}).\tag{108}\] With the same arguments, applying Lemma 3 to the solution \(u\), and relying on 106 , 107 , we find \[\label{eq:Fineq3} \int_x^{\overline{x}} \omega^T(\xi) \mathop{}\!\mathrm{d}\xi-\int_y^{\overline{y}} u_0(\xi) \mathop{}\!\mathrm{d}\xi = \mathcal{F}(u, \zeta_x) -\mathcal{F}(u, \zeta_{\overline{x}})\geq \mathcal{F}(u^*,\zeta_x)-\mathcal{F}(u^*,\zeta_{\overline{x}}).\tag{109}\] Combining 108 , 109 we deduce \[\label{eq:Fineq1} - \int_y^{\bar y} u_0(\xi) \mathop{}\!\mathrm{d}\xi \geq -\int_y^{\overline{y}} u_0^*(\xi) \mathop{}\!\mathrm{d}\xi,\tag{110}\] which yields ?? .

2. Now we prove that if \(u_0 \in \mathbf{L}^{\infty}(\mathbb{R})\) satisfies ?? , ?? , then \({\mathcal{S}}_{\it T}^{\, [{\it A B}\,]+}u_0 = \omega^T\). Namely, we are going to prove that under the conditions ?? , ?? , there hold \[\label{Lebesgueeq} \int_{x_1}^{x_2} \big(\omega^T(x) - {\mathcal{S}}_{\it T}^{\, [{\it A B}\,]+}u_0(x)\big) \mathop{}\!\mathrm{d}x = 0, \quad\;\forall~x_1< x_2,\tag{111}\] which clearly implies that \({\mathcal{S}}_{\it T}^{\, [{\it A B}\,]+}u_0 = \omega^T\).

Towards a proof of 111 we will first show that \[\label{Lebesguegeq} \int_{x_1}^{x_2} \big(\omega^T(x) - {\mathcal{S}}_{\it T}^{\, [{\it A B}\,]+}u_0(x)\big) \mathop{}\!\mathrm{d}x \geq 0, \quad\;\forall~x_1< x_2,\tag{112}\] distinguishing two cases.
\(\max \mathcal{C}_0(u,x_1) \geq \min \mathcal{C}_0(u^*,x_2)\) (see Figure 11, right). Then we can choose \(\zeta_1 \in \mathcal{C}(u,x_1)\) and \(\zeta_2 \in \mathcal{C}(u^*,x_2)\) such that \(\zeta_1(0)\geq \zeta_2(0)\). By continuity there will be a point \(\tau \in [0,T[\,\) such that \(\zeta_1(\tau) = \zeta_2(\tau)\), \(\zeta_1(t) < \zeta_2(t)\) for all \(t\in\,]\tau, T]\). Applying Lemma 3 to the solution \(u^*\), with the curves \(\alpha=\zeta_1\), \(\beta=\zeta_2\), and using the first part of Lemma 2 for \(\zeta_1\), we obtain \[\int_{x_1}^{x_2} \omega^T(x) \mathop{}\!\mathrm{d}x = \mathcal{F}_{\tau}(u^*,\zeta_1)- \mathcal{F}_{\tau}(u^*,\zeta_2) \geq \mathcal{F}_{\tau}(u,\zeta_1)- \mathcal{F}_{\tau}(u^*,\zeta_2).\] Next, applying again Lemma 3 to the solution \(u\), with the curves \(\alpha=\zeta_1\), \(\beta=\zeta_2\), and then Lemma 2 for \(\zeta_2\), we obtain \[\int_{x_1}^{x_2} {\mathcal{S}}_{\it T}^{\, [{\it A B}\,]+}u_0(x) \mathop{}\!\mathrm{d}x = \mathcal{F}_{\tau}(u,\zeta_1)- \mathcal{F}_{\tau}(u,\zeta_2) \leq \mathcal{F}_{\tau}(u,\zeta_1)- \mathcal{F}_{\tau}(u^*,\zeta_2).\] Taking the difference of the above two inequalities, we derive 112 . Note that in this case we are not using the conditions ?? , ?? to establish 112 .
\(\max \mathcal{C}_0(u,x_1) < \min \mathcal{C}_0(u^*,x_2)\) (see Figure 11, left). Choose any \(\zeta_1 \in \mathcal{C}_0(u,x_1)\), and set \(y \stackrel{\cdot}{=} \zeta_1(0)\). Since \(y <\min \mathcal{C}_0(u^*,x_2)\), invoking condition ?? we find that there exists \(\zeta_2 \in\mathcal{C}(u^*,x_2)\) such that, setting \(y_2=\zeta_2(0)\), there holds \[\label{condleqy952} \int_y^{y_2} u_0(x) \mathop{}\!\mathrm{d}x \leq \int_y^{y_2}u^*_0(x) \mathop{}\!\mathrm{d}x\,.\tag{113}\] Note that, since \(\max \mathcal{C}_0(u,x_1) < \min \mathcal{C}_0(u^*,x_2)\), we have \(\zeta_1(t)<\zeta_2(t)\) for all \(t\in [0,T]\). Hence, applying Lemma 3 to \(u^*\), with the curves \(\alpha=\zeta_1\), \(\beta=\zeta_2\), and using the first part of Lemma 2 for \(\zeta_1\), we obtain \[\begin{align} \int_{x_1}^{x_2} \omega^T(x) \mathop{}\!\mathrm{d}x &= \mathcal{F}(u^*,\zeta_1)- \mathcal{F}(u^*,\zeta_2) + \int_y^{y_2}u^*_0(x) \mathop{}\!\mathrm{d}x \\ &\geq \mathcal{F}(u,\zeta_1)- \mathcal{F}(u^*,\zeta_2)+ \int_y^{y_2}u^*_0(x) \mathop{}\!\mathrm{d}x\,. \end{align}\] Next, applying Lemma 3 to \(u\), with the curves \(\alpha=\zeta_1\), \(\beta=\zeta_2\), and using the first part of Lemma 2 for \(\zeta_2\), we obtain \[\begin{align} \int_{x_1}^{x_2} {\mathcal{S}}_{\it T}^{\, [{\it A B}\,]+}u_0(x) \mathop{}\!\mathrm{d}x &= \mathcal{F}(u,\zeta_1)- \mathcal{F}_{\tau}(u,\zeta_2)+\int_y^{y_2}u_0(x) \mathop{}\!\mathrm{d}x \\ &\leq \mathcal{F}(u,\zeta_1)- \mathcal{F}(u^*,\zeta_2)+\int_y^{y_2}u_0(x) \mathop{}\!\mathrm{d}x\,. \end{align}\] Taking the difference of the above two inequalities, and using 113 , we derive \[\int_{x_1}^{x_2} \omega^T(x) \mathop{}\!\mathrm{d}x-\int_{x_1}^{x_2} \mathcal{S}_t^{\, [A B]+}u_0(x) \mathop{}\!\mathrm{d}x \geq \int_y^{y_2}u^*_0(x) \mathop{}\!\mathrm{d}x-\int_y^{y_2}u_0(x) \mathop{}\!\mathrm{d}x \geq 0\] which proves 112 also in Case 2.

The proof of the opposite inequality of 112 is entirely symmetric and is accordingly omitted. Thus the proof of 111 is completed, and this concludes the proof of the Theorem. ◻

Proof. Given \(\omega^T \in \mathcal{A}^{[AB]}(T)\), let \(u^*\) be the \(AB\)-entropy solution defined by ?? ?? . We prove the Theorem point by point, in order.

1. Proof of (i). First assume that \(\left|\mathcal{C}_0(u^*,x)\right| = 1\) for every \(x \in \mathbb{R}\). We will show that any initial data \(u_0\in \mathcal{I}_T^{[AB]}(\omega^T)\) satisfies \[\label{eq:initidatumsingleton} \int_{y_1}^{y_2} u_0(x) \mathop{}\!\mathrm{d}x = \int_{y_1}^{y_2} u_0^* (x) \mathop{}\!\mathrm{d}x, \quad \forall~y_1 < y_2,\tag{114}\] and this uniquely identifies \(u_0\) as an element of \(L^{\infty}(\mathbb{R})\), thus proving that \(\mathcal{I}_T^{[AB]}(\omega^T)=\{u_0^*\}\). Given any two points \(y_1 < y_2\), by Lemma 1 and because of Proposition 14-(iv), there exist \(x_1<x_2\), and \(\zeta_i \in \mathcal{C}(u^*,x_i)\), \(i=1,2\), such that \(\zeta_i(0)=y_i\), \(i=1,2\). Then, applying ?? of Theorem 9 we find \[\int_{y_1}^{y_2} u_0(x) \mathop{}\!\mathrm{d}x \leq \int_{y_1}^{y_2} u_0^* (x) \mathop{}\!\mathrm{d}x\,.\] Next, if we exchange the role of \(y_1\) and \(y_2\), applying this time ?? of Theorem 9 we find the opposite inequality \[\int_{y_1}^{y_2} u_0(x) \mathop{}\!\mathrm{d}x \geq \int_{y_1}^{y_2} u_0^* (x) \mathop{}\!\mathrm{d}x\,.\] Combining together the above two inequalities we obtain 114 .

Conversely, assume that \(\mathcal{I}_T^{[AB]}(\omega^T) = \{u_0^*\}\), and by contradiction suppose that there is some \(\widetilde{x} \in \mathbb{R}\) such that \(\left|\mathcal{C}_0(u^*,\widetilde{x}\,)\right| \neq 1\). Using the characterization of Theorem 9 we will then show that there exist infinitely many initial data \(u_0 \neq u^*_0\) such that \({\mathcal{S}}_{\it T}^{\, [{\it A B}\,]+}u_0 = \omega^T\). To this end, set \[\mathrm{conv} \, \mathcal{C}_0(u^*, \widetilde{x}\,)\doteq [\min \mathcal{C}_0(u^*, \widetilde{x}\,),\, \max \mathcal{C}_0(u^*, \widetilde{x}\,)]\,,\] and let \(\mathbf{L}^{\infty}(\mathrm{conv} \, \mathcal{C}_0(u^*, \widetilde{x}\,))\) denote the space of \(\mathbf{L}^{\infty}(\mathbb{R})\) function with essential support in \(\mathrm{conv} \, \mathcal{C}_0(u^*, \widetilde{x}\,))\). Note that \(\mathrm{conv} \, \mathcal{C}_0(u^*, \widetilde{x}\,)\) is a non trivial interval because \(\left|\mathcal{C}_0(u^*,\widetilde{x}\,)\right| \neq 1\), and hence \(\mathbf{L}^{\infty}(\mathrm{conv} \, \mathcal{C}_0(u^*, \widetilde{x}\,))\) is an infinite dimensional space. Next, consider the infinite dimensional cone \(\mathrm{V}_0 \subset \mathbf{L}^{\infty}(\mathrm{conv} \, \mathcal{C}_0(u^*, \widetilde{x}\,))\) consisting of all \(v_0\in \mathbf{L}^{\infty}(\mathrm{conv} \, \mathcal{C}_0(u^*, \widetilde{x}\,))\) that satisfy \[\label{eq:v950props} \begin{align} \int_{y}^{\max \mathcal{C}_0(u^*, \widetilde{x}\,)} v_0(x) \mathop{}\!\mathrm{d}x &\leq 0, \qquad \forall \; y \in \mathrm{conv}\; \mathcal{C}_0(u^*, \widetilde{x}\,),\\ \int_{\min \mathcal{C}_0(u^*, \widetilde{x}\,)}^y v_0(x) \mathop{}\!\mathrm{d}x &\geq 0, \qquad \forall \; y \in \mathrm{conv}\; \mathcal{C}_0(u^*, \widetilde{x}\,). \end{align}\tag{115}\] Note that 115 in particular imply \[\label{eq:V-0int-eq} \int_{\min \mathcal{C}_0(u^*, \widetilde{x}\,)}^{\max \mathcal{C}_0(u^*, \widetilde{x}\,)} v_0(x) \mathop{}\!\mathrm{d}x = 0\,.\tag{116}\] We will show that \[\label{eq:Vcone-1} V \doteq u_0^* + V_0 \subset \mathcal{I}_T^{[AB]}(\omega^T).\tag{117}\] Relying on Theorem 9 this is equivalent to prove that, for any \(v_0\in V_0\), and for every \(\overline{x}\in\mathbb{R}\), there exists \(\overline{y} \in \mathcal{C}_0(u^*, \overline{x})\) such that ?? , ?? hold for \(u_0\doteq u_0^*+v_0\). We will verify only ?? , the proof of the other inequality being entirely symmetric.

Then, consider first any \(\overline{x} \leq \widetilde{x}\), and choose \(\overline{y} = \min \mathcal{C}_0(u^*, \overline{x})\). Observe that, for every \(y < \min \mathcal{C}_0(u^*, \overline{x})\), we have \(u_0 = u_0^*\) on the interval \([y,\, \overline{y}]\) since \(\overline{x} \leq \widetilde{x}\), together with Proposition 14-(iv), implies \[\overline{y} \leq \min \mathcal{C}_0(u^*, \widetilde{x}),\] and hence \(v_0 =0\) on \([y,\, \overline{y}]\), because the essential support of \(v_0\) is contained in \(\mathrm{conv} \, \mathcal{C}_0(u^*, \widetilde{x}\,))\). This implies that, for every \(y < \min \mathcal{C}_0(u^*, \overline{x})\), we have \[\label{eq:V-0int-eq-2} \int_y^{\overline{y}} u_0(x) \mathop{}\!\mathrm{d}x = \int_y^{\overline{y}} u_0^*(x) \mathop{}\!\mathrm{d}x\,,\tag{118}\] which proves ?? as an equality.

Next, consider any \(\overline{x} > \widetilde{x}\), and choose \(\overline{y} = \max \mathcal{C}_0(u^*, \overline{x})\). Then, for every \(y < \min \mathcal{C}_0(u^*, \overline{x})\), one of the following three cases occurs:

1. If \(y \in \,]\max \mathcal{C}_0(u^*, \tilde{x}), \min \mathcal{C}_0(u^*, \bar x)[\,\), then ?? holds again as an equality, because \(u_0\) coincides with \(u_0^*\) in the interval \([y,\, \overline{y}]\) as in the case \(\overline{x} \leq \widetilde{x}\) considered above, and thus 118 is verified.

2. If \(y \in \mathrm{conv} \; \mathcal{C}_0(u^*, \tilde{x})\), then by 115 we have \[\begin{align} \int_y^{\overline{y}} u_0(x) \mathop{}\!\mathrm{d}x &= \int_y^{\max \mathcal{C}_0(u^*, \widetilde{x})} u_0(x) \mathop{}\!\mathrm{d}x + \int _{\max \mathcal{C}_0(u^*, \widetilde{x})}^{\overline{y}} u^*_0(x) \mathop{}\!\mathrm{d}x \\ &\leq \int_y^{\overline{y}} u_0^*(x) \mathop{}\!\mathrm{d}x\,, \end{align}\] which proves ?? .

3. If \(y < \min \mathcal{C}_0(u^*, \tilde{x})\), we obtain ?? relying on 116 , since \[\begin{align} \int_y^{\overline{y}}u_0(x) \mathop{}\!\mathrm{d}x &= \!\int_y^{\min \mathcal{C}_0(u^*, \widetilde{x})} \!u_0^*(x) \mathop{}\!\mathrm{d}x +\! \int_{\min \mathcal{C}_0(u^*, \widetilde{x})}^{\max \mathcal{C}_0(u^*, \widetilde{x})} \!u_0(x) \mathop{}\!\mathrm{d}x +\! \int_{\max \mathcal{C}_0(u^*, \widetilde{x})}^{\bar y} \!u_0^*(x) \mathop{}\!\mathrm{d}x \\ &= \int_y^{\bar y}u^*_0(x) \mathop{}\!\mathrm{d}x\,. \end{align}\]

Thus, for all \(u_0=u_0^*+v_0\), \(v_0\in V_0\), and for every \(\overline{x}\in\mathbb{R}\), there exists \(\overline{y} \in \mathcal{C}_0(u^*, \overline{x})\) such that ?? , ?? hold. Hence 117 is verified, which contradicts the assumption \(\mathcal{I}_T^{[AB]}(\omega^T) = \{u_0^*\}\), and thus completes the proof of the first part of property (i).

Finally, observe that if \(x\) is a point of discontinuity for \(\omega^T\), then one can consider the \(AB\)-gics \(\vartheta_{x,-}, \vartheta_{x,+}: [0,T]\to\mathbb{R}\) that are the minimal and maximal \(AB\)-gics for \(u^*\) reaching at time \(T\) the point \(x\) (e.g. see point 2 of the proof of Proposition 15). Since \(\vartheta_{x,-}(0)\neq \vartheta_{x,+}(0)\) if \(x\neq 0\), and because \(\{\vartheta_{x,-}(0),\,\vartheta_{x,+}(0)\}\subset\mathcal{C}_0(u^*, \tilde{x})\), this implies \(\left|\mathcal{C}_0(u^*,x\,)\right| \neq 1\), thus proving by contradiction that if \(\mathcal{I}_T^{[AB]}(\omega^T)\) is a singleton, then \(\omega^T\) must be continuous at any point \(x\neq 0\). This concludes the proof of property (i).

2. Proof of (ii). To prove that the set \(\mathcal{I}_T^{[AB]}(\omega^T)-u_0^*\) is a linear cone, we will show that, for every \(u_0 \in \mathcal{I}_T^{[AB]}(\omega^T)\) and \(\lambda \geq 0\), it holds \(u_0^* + \lambda(u_0 -u_0^*) \in \mathcal{I}_T^{[AB]}(\omega^T)\). To see this, applying Theorem 9 it’s sufficient to prove that, given any \(\overline{x}\in\mathbb{R}\), there exists \(\overline{y} \in \mathcal{C}_0(u^*, \overline{x})\) such that  ?? , ?? hold with \(u_0^* + \lambda(u_0 -u_0^*)\) in place of \(u_0\). Since \(u_0 \in \mathcal{I}_T^{[AB]}(\omega^T)\), by Theorem 9 we know that there is some \(\overline{y}\in \mathcal{C}_0(u^*, \overline{x})\) such that ?? holds. Then, for all \(y < \min \mathcal{C}_0(u^*,\overline{x})\), one finds \[\int_y^{\overline{y}} \big(u_0^*(x)+\lambda(u_0(x)-u_0^*(x))\big) \mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}} \big(u_0^*(x)+\lambda(u_0^*(x)-u_0^*(x))\big) \mathop{}\!\mathrm{d}x = \int_y^{\overline{y}} u_0^*(x) \mathop{}\!\mathrm{d}x\,.\] This proves that ?? is verified with \(u_0^* + \lambda(u_0 -u_0^*)\) in place of \(u_0\). The proof that also ?? holds, is entirely symmetric.

Next, we prove that \(u_0^*\) is an extremal point of \(\mathcal{I}_T^{[AB]}(\omega^T)\). Assume by contradiction that there exist \(u_{0,i} \in \mathcal{I}_T^{[AB]}(\omega^T)\), \(u_{0,i}\neq u_0^*\), \(i = 1,2\), and \(\lambda \in \,]0,1[\,\), such that \[\label{eq:uo42convexcomb} u^*_0 = \lambda u_{0,1} + (1-\lambda) u_{0,2}\,.\tag{119}\] Take any \(\overline{x} \in \mathbb{R}\) for which \(\mathcal{C}_0(u^*, \overline{x})\) is a singleton (one can choose \(\overline{x}\) as a point of continuity for \(\omega^T\) belonging to the set \(]-\infty, \mathsf{L}[\, \cup\,]\mathsf{R}, +\infty[\), with \(\mathsf{L}, \mathsf{R}\) as in ?? ), and call \(\overline{y}\) the unique element of \(\mathcal{C}_0(u^*, \overline{x})\). Because of 119 it holds \[\label{convexinteq} \int_{\bar y}^y u_0^*(x) \mathop{}\!\mathrm{d}x= \lambda \int_{\bar y}^y u_{0,1}(x) \mathop{}\!\mathrm{d}x+ (1-\lambda) \int_{\bar y}^y u_{0,2}(x) \mathop{}\!\mathrm{d}x, \qquad \forall \; y \in \mathbb{R}\,.\tag{120}\] Then, since \(u_{0,1}, u_{0,2}\) are different from \(u_0^*\), there must be some \(y \in \mathbb{R}\) such that one of the following three cases occurs:

1. \[\label{convexinteq-2} \int_{\overline{y}}^y u_{0,1}(x) \mathop{}\!\mathrm{d}x \neq \int_{\overline{y}}^y u_0^*(x) \mathop{}\!\mathrm{d}x \qquad \text{and} \qquad \int_{\overline{y}}^y u_{0,2}(x) \mathop{}\!\mathrm{d}x = \int_{\overline{y}}^y u_0^*(x) \mathop{}\!\mathrm{d}x.\tag{121}\]

2. \[\int_{\overline{y}}^y u_{0,1}(x) \mathop{}\!\mathrm{d}x = \int_{\overline{y}}^y u_0^*(x) \mathop{}\!\mathrm{d}x \qquad \text{and} \qquad \int_{\overline{y}}^y u_{0,2}(x) \mathop{}\!\mathrm{d}x \neq \int_{\overline{y}}^y u_0^*(x) \mathop{}\!\mathrm{d}x.\]

3. \[\int_{\overline{y}}^y u_{0,1}(x) \mathop{}\!\mathrm{d}x \neq \int_{\overline{y}}^y u_0^*(x) \mathop{}\!\mathrm{d}x \qquad \text{and} \qquad \int_{\overline{y}}^y u_{0,2}(x) \mathop{}\!\mathrm{d}x \neq \int_{\overline{y}}^y u_0^*(x) \mathop{}\!\mathrm{d}x.\]

Assume that Case 1 holds, with \(y \neq \overline{y}\). Then, applying conditions ?? , ?? of Theorem 9 to \(u_0^1\), we find that \[\label{convexinteq-3} \int_{\overline{y}}^y u_{0,1}(x) \mathop{}\!\mathrm{d}x > \int_{\overline{y}}^y u_0^* \mathop{}\!\mathrm{d}x.\tag{122}\] But 122 , together with the equality in 121 , is in contradiction with 120 . The analysis of the other two cases is entirely similar, thus it is omitted. This proves that \(u_0^*\) is an extremal point of \(\mathcal{I}_T^{[AB]}(\omega^T)\) (and of course it is unique since \(u_0^*\) is the vertex of the affine cone \(\mathcal{I}_T^{[AB]}(\omega^T)\)), and thus concludes the proof of property (ii).

3. Proof of (iii). We first show that, if condition 27 is verified than the set \(\mathcal{I}_T^{[AB]}(\omega^T)\) is convex. Given \(u_{0,1}, u_{0,2} \in \mathcal{I}_T^{[AB]}(\omega^T)\), and \(\lambda \in \,]0,1[\,\), let \(\overline{x} \in \,]\mathsf{L},\, \mathsf{R}[\,\) be a point of continuity for \(\omega^T\). By Theorem 9, and because \(\mathcal{C}_0(u^*,\overline{x})\) is a singleton \(\{\overline{y}\}\), we know that there hold \[\label{ybar12} \int_y^{\overline{y}} u_{0,1}(x) \mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}}u_0^*(x) \mathop{}\!\mathrm{d}x , \qquad \int_y^{\overline{y}} u_{0,2}(x) \mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}}u_0^*(x) \mathop{}\!\mathrm{d}x, \quad \forall \; y < \overline{y}.\tag{123}\] Then, using 123 , we derive \[\label{eq:convexity-ineq} \int_y^{\overline{y}} \big(\lambda u_{0,1}(x) + (1-\lambda)u_{0,2}(x)\big)\mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}} u^*(x) \mathop{}\!\mathrm{d}x, \qquad \forall \; y < \overline{y}, \quad\forall~\lambda \in \,]0,1[\,,\tag{124}\] so that \(\lambda u_{0,1}+(1-\lambda) u_{0,2}\) satisfies ?? for all \(\lambda \in \,]0,1[\,\), and \(\overline{y}\in \mathcal{C}_0(u^*,\overline{x})\) with \(\overline{x} \in \,]\mathsf{L},\, \mathsf{R}[\,\) of continuity for \(\omega^T\). The proof that also ?? holds for the same \(\overline{y}\) is entirely similar and is accordingly omitted. Next, consider a point \(\overline{x} \in \,]-\infty,\,\mathsf{L}[\;\cup\;]\mathsf{R}, +\infty[\,\) of continuity for \(\omega^T\). Notice that by definition ?? the classical backward characteristics starting from \((\overline{x}, T)\) never cross the interface \(x=0\) at positive times. Therefore the unique \(AB\)-gic reaching the point \(\overline{x}\) at time \(t=T\) is a classical genuine characteristic starting say at \(\overline{y}\) at time \(t=0\). Hence \(\mathcal{C}_0(u^*,\overline{x})=\{\overline{y}\}\), and we can proceed as above to show that \(\lambda u_{0,1}+(1-\lambda) u_{0,2}\) satisfies ?? , ?? for all \(\lambda \in \,]0,1[\,\), also when \(\overline{y}\in \mathcal{C}_0(u^*,\overline{x})\) with \(\overline{x} \in \,]-\infty,\,\mathsf{L}[\;\cup\;]\mathsf{R}, +\infty[\,\) of continuity for \(\omega^T\). Then, by Remark 18 we can conclude that \(\lambda u_{0,1}+(1-\lambda) u_{0,2}\in \mathcal{I}_T^{[AB]}(\omega^T)\), for all \(\lambda \in \,]0,1[\,\).

Now assume that condition ?? is verified. By the above analysis it is clear that in order to prove the convexity of \(\mathcal{I}_T^{[AB]}(\omega^T)\) it is sufficient to show that, for any \(\overline{x} \in \,]\mathsf{L},\, \mathsf{R}[\,\) of continuity for \(\omega^T\) there exists \(\overline{y} \in \mathcal{C}_0(u^*, \overline{x})\) such that there holds \[\label{eq:convexity-ineq-2} \int_y^{\overline{y}} \big(\lambda u_{0,1}(x) + (1-\lambda)u_{0,2}(x)\big)\mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}} u^*(x) \mathop{}\!\mathrm{d}x, \qquad \forall \; y < \min \mathcal{C}_0(u^*,\overline{x})\,, \quad\forall~\lambda \in \,]0,1[\,.\tag{125}\] The problem in this case is the following. Since \(\mathcal{C}_0(u^*, \overline{x})\) may not be a singleton, by Theorem 9 we know that there will be in general \(\overline{y}_i\in \mathcal{C}_0(u^*, \overline{x})\), \(i=1,2\), \(\overline{y}_1\neq \overline{y}_2\), such that there hold \[\label{ybar12-2} \int_y^{\overline{y}_1} u_{0,1}(x) \mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}_1}u_0^*(x) \mathop{}\!\mathrm{d}x , \qquad \int_y^{\overline{y}_2} u_{0,2}(x) \mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}_2}u_0^*(x) \mathop{}\!\mathrm{d}x, \quad \forall \; y < \min \mathcal{C}_0(u^*,\overline{x}).\tag{126}\] Here the choice of \(\overline{y}_i\) depends on the initial datum \(u_{0,i}\in \mathcal{C}_0(u^*, \overline{x})\), \(i=1,2\). Hence, we cannot rely on 126 to derive immediately the existence of \(\overline{y} \in \mathcal{C}_0(u^*, \overline{x})\) such that 125 holds. However, thanks to the assumption ?? we can show that, for every point \(\overline{x} \in \,]\mathsf{L},\, \mathsf{R}[\,\) of continuity for \(\omega^T\), there exists \(\overline{y} \in \mathcal{C}_0(u^*, \overline{x})\), independent on the initial datum \(u_0\) taken in consideration, such that ?? , ?? are verified. In fact, given any point \(\overline{x} \in \,]\mathsf{L},\, \mathsf{R}[\,\) of continuity for \(\omega^T\), let \(\{\overline{x}_n\}_n\) be a sequence of points in \(\mathcal{X}(\omega^T)\) such that \(\overline{x}_n\to \overline{x}\). Letting \(\{\overline{y}_n\}=\mathcal{C}_0(u^*, \overline{x}_n)\), we may assume that, up to a subsequence, \(\{\overline{y}_n\}_n\) converges to some point \(\overline{y}\in\mathbb{R}\). Since \(x\mapsto \mathcal{C}_0(u^*, x)\) has closed graph by Proposition 14 it follows that \(\overline{y}\in \mathcal{C}_0(u^*, \overline{x})\). Hence, applying Theorem 9 we find that, for any \(y < \min \mathcal{C}_0(u^*,\overline{x})\), and for \(n\) sufficiently large, there hold \[\label{ybar12-3} \int_y^{\overline{y}_n} u_{0,1}(x) \mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}_n}u_0^*(x) \mathop{}\!\mathrm{d}x , \qquad \int_y^{\overline{y}_n} u_{0,2}(x) \mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}_n}u_0^*(x) \mathop{}\!\mathrm{d}x\,.\tag{127}\] taking the limit as \(n\to\infty\) in 127 we derive \[\int_y^{\overline{y}} u_{0,1}(x) \mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}}u_0^*(x) \mathop{}\!\mathrm{d}x , \qquad \int_y^{\overline{y}} u_{0,2}(x) \mathop{}\!\mathrm{d}x \leq \int_y^{\overline{y}}u_0^*(x) \mathop{}\!\mathrm{d}x, \quad \forall \; y < \min \mathcal{C}_0(u^*,\overline{x})\,,\] which yields 125 . This completes the proof of property (iii), and thus concludes the proof of the theorem. ◻