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Evolution equations with lack of convexity. (English) Zbl 0545.46029
Let \(\Omega\) be an open subset of a real Hilbert space H, whose norm and scalar product are denoted by \(| \cdot |\) and (\(\cdot | \cdot)\). If \(f:\quad \Omega \to {\mathbb{R}}\cup \{+\infty \}\) is a function, set \[ \partial^-f(u)=\{\alpha \in H:\lim \quad \inf_{v\to u}\frac{f(v)-f(u)-(\alpha | v-u)}{| v-u|}\geq 0\},\quad if\quad f(u)<+\infty; \]
\[ \partial^-f(u)=\emptyset,\quad if\quad f(u)=+\infty. \] If f is lower semicontinuous (with respect to the norm topology), f is said to have a \(\phi\)-monotone subdifferential, if there exists a continuous function \(\phi:\quad \Omega \times {\mathbb{R}}^ 2\to {\mathbb{R}}^+\) such that \[ (\alpha -\beta | u-v)\geq - [\phi(u,f(u),| \alpha |)+\phi(v,f(v),| \beta |)]\quad | u-v|^ 2 \] whenever \(\partial^-f(u)\neq \emptyset\), \(\partial^-f(v)\neq \emptyset\), \(\alpha \in \partial^-f(u)\), \(\beta \in \partial^-f(v)\). In this paper some general properties of this class of functions are studied and some theorems of existence, uniqueness, regularity and convergence, concerning the associated evolution equation \(U'(t)\in -\partial^-f(U(t))\) are proved.

MSC:
46G05 Derivatives of functions in infinite-dimensional spaces
46A50 Compactness in topological linear spaces; angelic spaces, etc.
58D25 Equations in function spaces; evolution equations
49J27 Existence theories for problems in abstract spaces
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