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.


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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