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The singular set for the composite membrane problem. (English) Zbl 1157.35125

Let \(u\) be a solution of \(\Delta u=f\chi\{u\geq 0\}-g\chi\{u<0\}\) in a bounded domain \(D\subset\mathbb R^2\), with \(f, g\in C^{\alpha}(D)\) and satisfying the inequalities \(f>0,\;f+g<0\) on the singular set \(S_u=\{u=| \nabla u| =0\}\). The author proves the following property of \(S_u\): if \(z\in S_u\) is such that \(| \{u<0\}\cap B_r(z)| \geq c_0r^2\), for some positive constant \(c_0\) and for all \(r\) less than a suitable radius \(r_0\), then \(z\) is an isolated point of \(S_u\).
The proof is achieved by means of a blow-up argument combined with the monotonicity of a suitably selected energy functional.

MSC:

35R35 Free boundary problems for PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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