Weak structural stability of pseudo-monotone equations
Augusto Visintin · Discrete and Continuous Dynamical Systems · 2015
The inclusion $\beta(u) i h$ in $V'$ is studied, assuming that $V$ is a reflexive Banach space,and that $\beta: V \to {\cal P}(V')$ is a generalized pseudo-monotone operator in the sense of Browder-Hess [MR 0365242].A notion of strict generalized pseudo-monotonicity is also introduced.The above inclusion is here reformulated as a minimization problem for a (nonconvex) functional$V \!\times V'\to \mathbf{R} \cup \{+\infty\}$. A nonlinear topology of weak-type is introduced, and related compactness resultsare proved via De Giorgi's notion of$\Gamma$-convergence.The compactness and the convergence of the family of operators $\beta$provide the (weak) structural stability of the inclusion $\beta(u) i h$with respect to variations of $\beta$ and $h$,under the only assumptions that the $\beta$s are equi-coercive andthe $h$s are equi-bounded. These results are then applied to the weak stability of the Cauchy problemfor doubly-nonlinear parabolic inclusions of the form$D_t\partial\varphi(u) + \alpha(u) i h$,$\partial\varphi$ being the subdifferential of a convex lower semicontinuous mapping $\varphi$,and $\alpha$ a generalized pseudo-monotone operator.The technique of compactness by strict convexity is also used in the limit procedure.