Recovery of the boundary data for a linear second order elliptic problem with a nonlocal boundary condition

Abstract

We study a linear elliptic partial differential equation of second order in a bounded domain ? ? R N , with nonstandard boundary conditions on a part ? of the boundary ??. Here, neither the solution nor its normal derivative are prescribed pointwise. Instead, the average of the solution over ? is given and the normal derivative along ? has to follow a prescribed shape function, apart from an additive (unknown) constant. We prove the well-posedness of the problem and provide a method for the recovery of the unknown boundary data.