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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

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
Full Text: DOI
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