Stability of a class of action functionals depending on convex functions (Q6102523)

The authors consider a fixed metric space \((X,d)\), a function \( f:X\rightarrow \mathbb{R}\cup \{+\infty \}\), and the functional defined on \( C([0,1],X)\) as: \(\Theta _{x_{0},x_{1}}^{f}(\gamma )=\int_{0}^{1}\left\vert \overset{.}{\gamma }\right\vert ^{2}(t)+\left\vert \partial f\right\vert (\gamma (t))dt\), if \(\gamma \in AC([0,1],X)\), \(\gamma (0)=x_{0}\), \(\gamma (1)=x_{1}\), and \(\Theta _{x_{0},x_{1}}^{f}(\gamma )=\infty \), otherwise, where \(x_{0}\) and \(x_{1}\) are given points in \(X\), \(\left\vert \partial f\right\vert \) is the descending slope of \(f\) defined as: \(\left\vert \partial f\right\vert (x)=+\infty \), if \(x\in X\setminus D(f)\), \(\left\vert \partial f\right\vert (x)=0\), if \(x\in D(f)\) is isolated and \(\left\vert \partial f\right\vert (x)=\limsup_{y\rightarrow x}\frac{(f(y)-f(x))^{-}}{ d(y,x)}\), otherwise, \(D(f)\) being the effective domain of \(f\): \(D(f)=\{x\in X:f(x)<+\infty \}\). The purpose of the paper is to describe the \(\Gamma \)-convergence of \(\Theta _{x_{0},x_{1}}^{f^{h}}(\gamma )\) when \(f^{h}\) Mosco converges to \(f\). The first main result proves that if \((X,d)\) is an Hadamard space, \(\{f_{h}\}_{h}\) and \(f\) are proper, \(\lambda \)-convex (for \( \lambda \in \mathbb{R}\)) and lower semi-continuous functions from \(X\) to \( \mathbb{R}\cup \{+\infty \}\), such that \(f^{h}\) Mosco-converges to \(f\), \( \{x_{0}^{h}\}_{h}\) and \(\{x_{1}^{h}\}_{h}\subseteq X\) are two sequences with \(x_{0}^{h}\rightarrow x_{0}\) and \(x_{0}^{h}\rightarrow x_{1}\), and \( \limsup_{h}\left\vert \partial f^{h}\right\vert (x_{0}^{h})<+\infty \) and \( \limsup_{h}\left\vert \partial f^{h}\right\vert (x_{1}^{h})<+\infty \), then \( \Theta _{x_{0},x_{1}}^{f^{h}}\) \(\Gamma \)-converges to \(\Theta _{x_{0},x_{1}}^{f}(\gamma )\) with respect to the \(C([0,1],X)\) topology. For the proof, the authors use properties of the descending slope, of the resolvent operator, and of Hadamard spaces. The \(\Gamma \)-\(\liminf\) inequality is proved applying a lower semicontinuity argument. The \(\Gamma \)-\(\limsup\) inequality is proved through the construction of an appropriate test-function and using properties of the descending slope and of the resolvent operator associated to \(f^{h}\): \(J_{\tau }^{f^{h}}x=\operatorname{argmin}\left\{ f^{h}(\cdot )+\frac{d(\cdot ,x)^{2}}{2}\right\} \). The second main result proves the same \(\Gamma \)-convergence result assuming that \((X,d)\) is a proper geodesic metric space, and \(\{f_{h}\}_{h}\) is a sequence of lower semicontinuous functions \(X\rightarrow \mathbb{R}\cup \{+\infty \}\), such that \(f^{h}\) \(\Gamma \)-converges to a proper and lower semicontinuous function \(f:X\rightarrow \mathbb{R}\cup \{+\infty \}\) and \(f^{h}\) and \(f\) satisfy further hypotheses. For the proof, the authors here replace the resolvent operators with their continuous version given by gradient flow trajectories. The paper ends with two examples.