## 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

### Keywords:

Schwarz function; singularities of the free boundary
Full Text:

### References:

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