Variational composition of a monotone operator and a linear mapping with applications to elliptic PDEs with singular coefficients (Q1868674)

Let \(X\) and \(U\) be two real reflexive Banach spaces, \(A:X\rightarrow U\) be a linear continuous operator, \(A^{\ast}\) the adjoint operator of \(A\) and \(T:U\rightrightarrows U^{\ast}\) be a monotone operator. Then the composite mapping \(A^{\ast}TA:X\rightrightarrows X^{\ast}\) is monotone, but, in general, not maximal monotone, even if \(T\) is. In the paper under review, the authors consider and study the variational composition of \(A\) and \(T,\) which is defined as follows: \[ (A^{\ast}TA)_{v}:=g-\liminf_{\lambda\searrow 0^+} A^{\ast }T_{\lambda}A, \] where the above limit is the Painlevé-Kuratowski inner limit of the graphs and \(T_{\lambda}:=(T^{-1}+\lambda J_{U}^{-1})^{-1}\) is the Yosida regularization of \(T\) (\(J_{U}:U\rightarrow U^{\ast}\) denotes the duality mapping on \(U\), which is assumed to be equipped with its Troyanski norm). It is subsequently established that \((A^{\ast}TA)_{v}=\overline{A^{\ast}TA}\) whenever \(\overline{A^{\ast}TA}\) is maximal monotone. In particular, \((A^{\ast }TA)_{v}\) coincides with the pointwise composition \(A^{\ast}TA,\) provided that the latter is maximal monotone (however, clearly, the former has more chances of being maximal monotone). Variational composition is a natural extension of the variational sum \(\left( T_{1}\underset{v}{+}T_{2}\right) \) of two monotone operators \(T_{1}\) and \(T_{2}\) [see \textit{H. Attouch, J.-B. Baillon} and \textit{M. Théra}, J. Convex Anal. 1, 1-29 (1994; Zbl 0822.47050)]. The authors show that if the variational sum is maximal monotone, then the variational sum coincides with the variational composition of the operators \(T(x_{1},x_{2})=T(x_{1})\times T(x_{2})\) and \(A(x)=(x,x).\) An application is given in the subdifferential chain rule: if \(f:U\rightarrow \mathbb{R}\cup\{+\infty\}\) is a proper lower semi-continuous function on \(U\) and \(A:X\rightarrow U\) is a linear continuous operator with \(\text{dom} (f\circ A)\neq\emptyset,\) then \(\partial(f\circ A)=(A^{\ast}\partial fA)_{v}.\) Variational composition meets also applications in the measurability of multifunctions: given a Hilbert space \(H,\) a complete \(\sigma\)-finite and positive measure space \((\Omega,\mu)\), a convex normal integrand \(f:\Omega\times H\rightarrow\mathbb{R}\cup\{+\infty\}\) and a Carathéodory mapping \(A:\Omega\times H\rightarrow\mathbb{R}\) with \(A_{\omega}:=A(\omega,.)\) linear for all \(\omega\in\Omega,\) the authors show that the family \(\{\partial f_{\omega}\circ A_{\omega}:\omega\in\Omega\}\) is measurable so that \(f\circ A\) is also a convex normal integrand. In the last part of the paper, the authors discuss an interesting application of the previously established subdifferential chain rule: Indeed, consider the lower semi-continuous convex function \(g:H_{0}^{1}(\Omega)\rightarrow \mathbb{R}\cup\{+\infty\}\) defined by \[ g(u)=\begin{cases} \frac{1}{2}\int_{\Omega}\nabla u(\omega)Q(\omega)\nabla u(\omega) d\omega &\text{if }\nabla u\cdot Q\cdot\nabla u\in L^{1}(\Omega)\\ +\infty & \text{otherwise,}\end{cases} \] where \(\Omega\) is an open subset of \(\mathbb{R}^{N}\), and \(Q\in L_{\text{loc}} ^{1}(\Omega;\mathbb{R}^{\mathbb{N}\times\mathbb{N}})\) where \(Q(\omega)\) is a symmetric positive semi-definite matrix (ae). It is easily seen that \(g=I_{f}\circ A\) where \(A(u):=\nabla u\) for all \(u\in H_{0}^{1}(\Omega)\) and \(I_{f}\) is the integral associated to the convex normal integrand \(f(\omega,u)=\frac{1}{2}uQ(\omega)u,\) for all \(\omega\in\Omega\) and all \(u\in L^{2}(\Omega,\mathbb{R}^{N}).\) Consequently, in the particular case where \(Q\in L^{\infty}(\Omega;\mathbb{R}^{\mathbb{N}\times\mathbb{N}})\), one has \(\text{dom}(I_{f})=L^{2}(\Omega,\mathbb{R}^{N})\). In this case, the classical subdifferential chain rule yields the formula \[ \partial g(u)=-\text{div}(Q\nabla u),\tag{1} \] for all \(u\in H_{0}^{1}(\Omega)\) with \(Q\nabla u\in L^{2}(\Omega ;\mathbb{R}^{N}).\) In the general case (where the standard qualification condition \(0\in\text{int}({\text{Im}}(A)-\text{dom }I_{f})\) may fail), the authors show that (1) still holds for all \(u\in H_{0} ^{1}(\Omega)\) such that \(\text{div}(Q\nabla u)\in H^{-1}(\Omega)\) and \(\langle w,-\text{div}(Q\nabla u)\rangle=\int_{\Omega}\nabla wQ\nabla u,\) for all \(w\in\text{dom }g.\) An application of this formula to the existence of solutions of an elliptic PDE associated with the operator \(\partial g\) is presented. The paper is written in a clear and instructive way.