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

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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### References:

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