Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems

Abstract

We consider elliptic operators with complex coefficients and we derive microlocal and local Carleman estimates near a boundary, under sub-ellipticity and strong Lopatinskii condition. Carleman estimates are weighted a priori estimates for the solutions of the associated elliptic boundary problem. The weight is of exponential form, exp⁡(τφ) where τ is meant to be taken as large as desired. Such estimates have numerous applications in unique continuation, inverse problems, and control theory. Based on inequalities for interior and boundary differential quadratic forms, the proof relies on the microlocal factorization of the symbol of the conjugated operator in connection with the sign of the imaginary part of its roots. We further consider weight functions of the previous form with moreover φ=exp⁡(γψ), where γ meant to be taken as large as desired, and we derive Carleman estimates where the dependency upon the two large parameters, τ and γ, is made explicit. Applications on unique continuation properties are given.