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Discretization of evolution variational inequalities. (English) Zbl 0677.65068

Partial differential equations and the calculus of variations. Essays in Honor of Ennio De Giorgi, 59-92 (1989).
[For the entire collection see Zbl 0671.00007.]
This is a comprehensive work on the problem of discretization of evolution variational inequalities. In the framework of the usual Hilbert triplet \(\{V,H,V'\}\) let K, A be given such that K is a closed convex nonempty subset of V; A is a linear operator from V in \(V'\); there exist M, \(\alpha >0\) and \(\lambda\) real with \(\| Av\| \leq M\| v\|\) \(\forall v\in V\) and \((Av+\lambda v,v)\leq \alpha \| v\|^ 2\) \(\forall v\in K-K.\)
A weak form of the following problem is studied: given initial value \(u_ 0\) in H and a function f(t) with values in \(V'\); find u(t) such that u(t)\(\in K\) a.e. in \(t>0\), \((u'(t)+Au(t)-f(t),\quad u(t)-v)\leq 0\) \(\forall v\in K\), a.e. in \(t>0\), \(u(0)=u_ 0.\) Some existence theorems, stability estimates and limit theorems are stated and proved. Finally, open problems are discussed.
Reviewer: J.Ramik

MSC:

65K10 Numerical optimization and variational techniques
49J40 Variational inequalities

Citations:

Zbl 0671.00007