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Regularity of free boundaries in two dimensions. (English) Zbl 0851.35022

The author considers the regularity properties of the free boundary in two dimensions of the following problem: Let \(B\) be the unit disc in \(\mathbb{C}\), \(u\in C^1(B)\) satisfy \(\Delta u= 1\) in \(\Omega(u)= \{z\in B\mid u(z)> 0\}\) and assume that the free boundary \(T(u)= \partial (\Omega(u))\cap B\) contains the origin \(0\).
The main result states that \(0\) is either a regular, a degenerate, a double, or a cusp point of \(T(u)\). Furthermore, the same assertion holds under more general assumptions on \(u\).

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35R35 Free boundary problems for PDEs
30E25 Boundary value problems in the complex plane
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References:

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