On some overdetermined free boundary problems

Overdetermined boundary value problems for the ∞-Laplacian

We consider overdetermined boundary value problems for the ∞-Laplacian in a domain Ω of Rn and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞-Laplacian, the normalized or game-theoretic ∞-Laplacian and the limit of the p-Laplacian as p→∞ are considered and provide different answers. Mathematics Subject Classification (2000). 35R35, 4...

Boundary Regularity and Compactness for Overdetermined Problems

1 Let D be either the unit ball B1(0) or the half ball B + 1 (0), let f be a strictly positive and continuous function, and let u and Ω ⊂ D solve the following overdetermined problem: ∆u(x) = χ Ω (x)f(x) in D, 0 ∈ ∂Ω, u = |∇u| = 0 in Ω, where χ Ω denotes the characteristic function of Ω, Ωc denotes the set D \Ω, and the equation is satisfied in the sense of distributions. When D = B 1 (0), then...

On Free Boundary Problems

Overdetermined Boundary Value Problems, Quadrature Domains and Applications

We discuss an overdetermined problem in planar multiply connected domains Ω. This problem is solvable in Ω if and only if Ω is a quadrature domain carrying a solid-contour quadrature identity for analytic functions. At the same time the existence of such quadrature identity is equivalent to the solvability of a special boundary value problem for analytic functions. We give a complete solution o...

Approximate radial symmetry for overdetermined boundary value problems