Structural Stability of Doubly-Nonlinear Flows

Visintin, Augusto

Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 3, pp. 363-391

Résumé

To any maximal monotone operator $\alpha \colon V \to \mathcal{P}(V)$ ($V$ being a real Banach space),in [MR1009594] S. Fitzpatrick associated a lower semicontinuous and convex function $f \colon V \times V' \to \mathbb{R} \cup \{+\infty\}$ such that \begin{equation*} \tag{*} f(v,v') \geq \langle v', v \rangle \quad \forall (v, v'), \qquad f(v,v') = \langle v', v \rangle \iff v' \in \alpha(v).\end{equation*} On this basis, in this work two classes of doubly-nonlinear evolutionary equations are formulated as minimization principles: \begin{equation*} \tag{**} D_{t}\alpha(u) - \operatorname{div} \vec{\gamma}(\nabla u) \ni h, \qquad \alpha(D_{t}u) - \operatorname{div} \vec{\gamma}(\nabla u) \ni h; \end{equation*} here $\alpha$ and $\vec{\gamma}$ are maximal monotone mappings, and one of them is assumed to be cyclically monotone. For associated initial- and boundary-value problems, existence of a solution is proved, as well as the stability with respect to variations of the data and of the operators $D_{t}$, $\nabla$, $\alpha$ and $\vec{\gamma}$.