Increasing stability of the continuation for the Helmholtz type equations

We study the (exponentially ill-posed) Cauchy problem for elliptic equations of second order similar to the Helmholtz equation. Stability in the Cauchy problem determines stability in inverse source and coefficient problems. We show that under some natural a priori bounds stability in this problem is improving when the frequency/wave number is growing. More precisely, we derive stability estimates formed from Lipschitz stable term and logarithmic term which dekays as a power of frequency. We discuss role of convexity conditions.

Prof. Victor Isakov is a famous mathematician working on inverse problems for partial differential equations and related topics. He is now a distinguished professor in the Department of Mathematics and Statistics at Wichita State University. His areas of professional interest include inverse problems of gravimetry, inverse problems of conductivity and their applications to medical imaging and nondestructive testing, inverse scattering problems etc. Prof. Isakov has published about 120 papers in top journals. He is now a editor of several SCI journals, such as <Inverse Problems>, <Inverse Problems and Imaging>, <Applicable Analysis>.