## Sharp estimates of the curvature of some free boundaries in two dimensions.(English)Zbl 1034.30005

Let $$\mu$$ be a positive measure on the interval $$(-1,1) \subset \mathbb R$$ which satisfies $\int_{-1}^1 d\mu > 0 \quad \text{and} \quad \int_{-1}^1 \frac{d\mu(t)}{1-t^2} < \infty\,.$ The Cauchy transform of $$\mu$$ is defined by $f(w) = \int_{-1}^1 \frac{d\mu(t)}{t-w}\,.$ The main result states that $$f$$ is a univalent function in $$\mathbf D^e = \{\, z \in \mathbb C : | z| >1 \,\} \cup \{\infty\}$$, and that $$f$$ maps $$\mathbf D^e$$ onto a bounded domain $$\Omega$$ which can be described as a union of discs centered on the real axis.
Moreover, the authors apply their main result to the obstacle problem, partial balayage, quadrature domains and Hele-Shaw flow moving boundary problems, and they obtain sharp estimates of the curvature of free boundaries appearing in such problems.

### MSC:

 30C20 Conformal mappings of special domains 31A99 Two-dimensional potential theory 35R35 Free boundary problems for PDEs 26A51 Convexity of real functions in one variable, generalizations 76B07 Free-surface potential flows for incompressible inviscid fluids 76D27 Other free boundary flows; Hele-Shaw flows
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