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A Harnack inequality approach to the regularity of free boundaries. III: Existence theory, compactness, and dependence on X. (English) Zbl 0702.35249

[For part II, cf. Commun. Pure Appl. Math. 42, No.1, 55-78 (1989; Zbl 0676.35086).]
Let \(L=\sum_{i,j}\partial_ i(a_{ij}\partial_ j)\) be a uniformly elliptic operator with Hölder continuous coefficients in a Lipschitz domain \(\Omega \subset {\mathbb{R}}^ n\). The author is concerned with the following free boundary problem:
(1) \(Lu=0\) in \(\Omega^+(u)=\{u>0\},\)
(2) \(Lu=0\) on \(\Omega^-(u)=int\{u\leq 0\},\)
(3) along \(\partial \Omega^+(u)\) the one-sided normal derivatives \(u_{\nu}^{\pm}\) satisfy the relation \(u_{\nu}^+=G(u_ v^- ,x,\nu).\)
Condition (3) has to be interpreted in a weak sense, of course. Under the hypothesis that G is Lipschitz and strictly monotone in the first variable, the author demonstrates the following existence result for the Dirichlet problem: Let continuous boundary data \(\Phi\) be given such that there is a minorant w (in a suitable sense) assuming these data on \(\partial \Omega\) and such that there are supersolutions (in a suitable sense) above w; then there exists a Lipschitz continuous, weak solution u to the free boundary problem with \(u/\partial \Omega =\Phi\). Furthermore, the free boundary \(\partial \Omega^+(u)\) has finite (n-1)-dimensional Hausdorff measure and possesses almost everywhere a normal vector (in the measure theoretic sense).
Reviewer: F.Tomi

MSC:

35R35 Free boundary problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
35Dxx Generalized solutions to partial differential equations
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References:

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