## 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

Zbl 0671.00007