An overdetermined problem in potential theory

Dmitry Khavinson, Erik Lundberg and Razvan Teodorescu

Vol. 265 (2013), No. 1, 85–111

DOI: 10.2140/pjm.2013.265.85

Abstract

We investigate a problem posed by L. Hauswirth, F. Hélein, and F. Pacard (Pacific J. Math. 250:2 (2011), 319–334): characterize all the domains in the plane admitting a positive harmonic function that solves simultaneously the Dirichlet problem with null boundary data and the Neumann problem with constant boundary data. Hauswirth et al. suggested that essentially only three possibilities exist: the exterior of a disk, a half-plane, and a nontrivial example they found — the image of the strip |ℑζ| < π∕2 under ζ↦ζ + sinhζ. We partially prove their conjecture, showing that these are indeed the only possibilities if the domain is Smirnov and it is either simply connected or its complement is bounded and connected. We also show the nonexistence in ℝ⁴ of an analogous nontrivial example among axially symmetric domains containing their axis of symmetry.

Keywords

exceptional domain, roof function, vortex dynamics, quadrature domain, null-quadrature domains, non-Smirnov domain, Schwarz function, free boundary

Mathematical Subject Classification 2010

Primary: 31A25, 35R35, 35N25

Secondary: 30C20, 30E25

Milestones

Received: 22 May 2012

Revised: 14 May 2013

Accepted: 19 May 2013

Published: 28 August 2013