A simple proof of the regularity theorem for the variational inequality of the obstacle problem. (English) Zbl 0612.49005

Gustafsson, Björn

Nonlinear Anal., Theory Methods Appl. 10, 1487-1490 (1986).

The author gives a new simple proof of the following well-known theorem: Let \(n<p<\infty\), \(p\geq 2\). Suppose \(f\in L^ p(\Omega)\), \(\psi \in H^{2,p}(\Omega)\) and let \(u\in H^ 1_ 0(\Omega)\) be the solution of \[ u\in K,\quad a(u,v-u)\geq <f,v-u>\text{ for all } v\in K, \] where \(K=\{v\in H^ 1_ 0(\Omega):\) \(v\geq \psi \}\), \(a(u,v)=\int_{\Omega}\sum_{i}\frac{\partial u}{\partial x_ i}\frac{\partial v}{\partial x_ i}dx\) and \(\Omega \in {\mathbb{R}}^ n\) is a bounded open subset. Then \[ (*)\quad f\leq -\Delta u\leq \max \{-\Delta \psi,f\}. \] In particular \(u\in H^{2,p}(\Omega)\) and \(u\in C^{1,\alpha}({\bar \Omega})\) with \(\alpha =1-\frac{n}{p}\). The idea of proof is to consider a new variational inequality for which the estimate (*) is fulfilled a priori, and then show that this solution also solves the original problem.

Reviewer: V.Mustonen

MSC:

49J40	Variational inequalities
35D10	Regularity of generalized solutions of PDE (MSC2000)
35J85	Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35J20	Variational methods for second-order elliptic equations

Keywords:

regularity; variational inequality

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