Continuation of solutions to elliptic and parabolic equations on hyperplanes and application to inverse source problems

Abstract

We consider continuation of solutions u to elliptic and parabolic equations limited on hyperplane under some symmetry conditions of the coefficients: for two domains γ and Γ on a hyperplane in satisfying γ ⊂ ⊂ Γ, we prove conditional stability estimates of u|Γ by u| γ for an elliptic equation and u|Γ×I by u| γ×(0,T) for a parabolic equation with open interval I ⊂ ⊂ (0, T). The proof is based on the even extension and conditional stability for Cauchy problems for elliptic and parabolic equations. We apply the result to prove the uniqueness for an inverse source problem for a heat equation of determining a spatial factor on Γ only by data on γ. Furthermore we provide characterizations of hyperplanes admitting such continuation. Finally we discuss such continuation for semi-linear elliptic equations.