A coercive James's weak compactness theorem and nonlinear variational problems
Let \(E\) be a Banach space, \(E^{\ast }\) its dual and \(\partial f:E\rightrightarrows E^{\ast }\) denote the Fenchel-Moreau subdifferential of the function \(f:E\rightarrow \mathbb{R}\cup \{+\infty \}\), that is, \(x^{\ast }\in \partial f(x_{0}),\) if and only if \(x_{0}\) is a (global) maximizer of the function \(x^{\ast }-f\) on \(E\). Provided \(\partial f(E)=E^{\ast }\) (equivalently, for all \(x^{\ast }\in E^{\ast }\), the function \(x^{\ast }-f\) attains its supremum in \(E\)), the authors show that the function \(f\) is \(1\)-coercive (that is, \(f(x)/\|x\|\) tends to \(+\infty \) as \(\|x\|\) goes to \(+\infty \)) if and only if the subdifferential \(\partial f\) maps bounded subsets of \(E\) to bounded subsets of \(E^{\ast }\), and under this assumption, the sublevel sets \([f\leq c],\) \(c\in \mathbb{R}\), of \(f\) are relatively weakly compact. This result yields the following generalization of the classical James theorem: if \(A\) is a weakly closed subset of \(E\), \(\psi :A\rightarrow \mathbb{R}\) is a bounded function and for all \(x^{\ast }\in E^{\ast }\) the function \(x^{\ast }-\psi \), defined in \(A\), attains its supremum in \(A,\) then \(A\) is weakly compact. (Taking in particular \(A\) to be the unit ball of \(E\) and \(\psi \) any constant function there, we obtain the classical characterization of reflexivity of \(E\) via norm attaining functionals.) In the last part of the paper, the authors revisit some results from mathematical finance as well as surjectivity results for monotone coercive functionals, and come up with the assertion that reflexivity is the unique possible framework for the development of existence results for a wide class of nonlinear variational problems.
scientific article; zbMATH DE number 5987793
scientific article