Weak structural stability of pseudo-monotone equations

Augusto Visintin

doi:10.3934/dcds.2015.35.2763

Article Contents

2015, Volume 35, Issue 6: 2763-2796. Doi: 10.3934/dcds.2015.35.2763

This issue Previous Article Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains Next Article

Weak structural stability of pseudo-monotone equations

Augusto Visintin^1,

1.
Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento) - Italia

Received: December 31, 2013

Revised: May 31, 2014

Published: May 2017

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Abstract

The inclusion $\beta(u)\ni h$ 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 results are 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 $\beta(u)\ni h$ with respect to variations of $\beta$ and $h$, under the only assumptions that the $\beta$s are equi-coercive and the $h$s are equi-bounded.
These results are then applied to the weak stability of the Cauchy problem for doubly-nonlinear parabolic inclusions of the form $D_t\partial\varphi(u) + \alpha(u) \ni h$, $\partial\varphi$ 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.

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Mathematics Subject Classification: 35K60, 49J40, 58E.

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