On increasing stability of the continuation for elliptic equations of second order without (Pseudo)convexity assumptions

Abstract

We derive bounds of solutions of the Cauchy problem for general elliptic partial differential equations of second order containing parameter (wave number) k which are getting nearly Lipschitz for large k. Proofs use energy estimates combined with splitting solutions into low and high frequencies parts, an associated hyperbolic equation and the Fourier-Bros-Iagolnitzer transform to replace the hyperbolic equation with an elliptic equation without parameter k. The results suggest a better resolution in prospecting by various (acoustic, electromagnetic, etc) stationary waves with higher wave numbers without any geometric assumptions on domains and observation sites.