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Interior regularity for solutions to obstacle problems. (English) Zbl 0603.49006
The authors study weak solutions $$u\in H^{1,\alpha}(\Omega)$$ of quasilinear variational inequalities of the form $\sum^{n}_{i=1}\int_{\Omega}A_ i(\cdot,u,Du)D_ i\phi +B(\cdot,u,Du)\phi dx\geq 0$ $for\quad all\quad \phi \in \overset\circ H^{1,\alpha}(\Omega)\quad with\quad \phi (x)\geq \psi (x)-u(x)\quad a.e.\quad on\quad \Omega.$ Here $$\psi$$ : $$\Omega\to {\mathbb{R}}$$ is an u.s.c. function and the coefficients A,B satisfy appropriate growth and ellipticity conditions which were introduced by J. Serrin [Acta Math. 111, 247-302 (1964; Zbl 0128.091)]. Working with function spaces similar to De Giorgi classes it is then shown that u is a continuous function if $$\psi$$ has the Lebesgue point property everywhere. Moreover, if the obstacle $$\psi$$ is Hölder continuous then also $$u\in C^{0,\alpha}(\Omega)$$. In a final section the authors prove that a Wiener condition for $$\psi$$ in $$x_ 0\in \Omega$$ is sufficient to get continuity of u at $$x_ 0$$.
Reviewer: M.Fuchs

##### MSC:
 49J40 Variational inequalities 35D10 Regularity of generalized solutions of PDE (MSC2000) 35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators
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##### References:
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