Overdetermined problems on ring shaped domains. (English) Zbl 0946.35035

Summary: This paper is concerned with overdetermined boundary value problems on ring shaped domains with the overdetermination given on the outer boundary. We investigate first the harmonic case, with various overdeterminations, then we consider more general nonlinear elliptic equations in divergence form. We present an alternative to the classical moving plane method allowing us to have less regularity on the data and to consider more general geometries (ellipsoid, cylinder and cone).


35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory