AN OVERDETERMINED PROBLEM IN POTENTIAL THEORY

作者：Khavinson Dmitry^*; Lundberg Erik; Teodorescu Razvan

来源：Pacific Journal of Mathematics, 2013, 265(1): 85-111.

DOI：10.2140/pjm.2013.265.85

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

We investigate a problem posed by L. Hauswirth, F. Helein, 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 vertical bar(sic)zeta vertical bar %26lt; pi/2 under zeta bar right arrow zeta + sin zeta. 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 R-4 of an analogous nontrivial example among axially symmetric domains containing their axis of symmetry.

• 出版日期2013-9