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Gradient flows of non convex functionals in Hilbert spaces and applications. (English) Zbl 1116.34048

Let \(H\) be a separable Hilbert space, \(\partial_l\phi:H\to 2^H\) be a limiting subdifferential (see, e.g. A. Ya. Kruger and B. Sh. Mordukhovich [Dokl. Akad. Nauk BSSR 24, 684–687 (1980; Zbl 0449.49015)]) of a proper, lower semicontinuous functional \(\phi:H\to(-\infty,+\infty]\) wich is not supposed to be convex. The authors study existence and approximation of strong solutions for the gradient flow equation of the following form:
\[ u'(t)+\partial_l\phi(u(t))\ni f(t) \quad\text{ for a.e. } t\in(0,T), \]
\[ u(0)=u_0. \]
Using a variational approximation technique, methods from the theory of Young measures and minimizing movements, some new existence results for the above mentioned equation and new assertions for diffusion problems with quasistationary nonmonotone relations are proved. In the second part of the paper the authors present some examples from the field of ordinary differential inclusions and PDE’s systems.

MSC:

34G25 Evolution inclusions
35A15 Variational methods applied to PDEs
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
80A22 Stefan problems, phase changes, etc.

Citations:

Zbl 0449.49015
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References:

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