An Overdetermined Problem for an Elliptic Equation

An Overdetermined Problem for an Elliptic Equation

We consider the following overdetermined boundary value problem: ∆u+ λu+ μ = 0 in Ω, u = 0 on ∂Ω and ∂u/∂n = c on ∂Ω, where c 6= 0, λ and μ are real constants and Ω ⊂ R is a smooth bounded convex open set. We first show that it may happen that the problem has no solution. Then we study the existence of solutions for a wide class of domains. 2010 Mathematics Subject Classification: 35J05, 35R30.

On an Eighth Order Overdetermined Elliptic Boundary Value Problem

We consider the overdetermined boundary value problem for the 4-harmonic operator, Δ4 = Δ(Δ3) , and show that if the solution of the problem exists, then the domain must be an open N -ball (N 2) . As a consequence of overdetermined problems mean value results are obtained for harmonic, biharmonic, triharmonic and 4-harmonic functions. Mathematics subject classification (2010): 35J25, 35P15, 35B50.

A Note on Some Overdetermined Elliptic Problem

Ω 7−→ λ1(Ω) , under the volume constraint Vol(Ω) = α (where α ∈ (0,Vol(M)) is fixed) are called extremal domains. Smooth extremal domains are characterized by the property that the eigenfunctions associated to the first eigenvalue of the Laplace-Beltrami operator have constant Neumann boundary data [2]. In other words, a smooth domain is extremal if and only if there exists a positive function ...

Numerical Solution of a Cauchy Problem for an Elliptic Equation

Remarks on an overdetermined boundary value problem

We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator to a general quasilinear operator and remove a strong ellipticity assumption in [9] and a growth assumption in [5] on the diffusion coefficient A, as well as a starshapedness assumption on Ω in [4]. Mathematics Subject Classification (2000). 35J65, 35B35