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Victor Isakov This volume contains various results on partial di?erential equations where Sobolev spaces are used. Their selection is motivated by the research int- ests of the editor and the geographicallinks to the places where S. L. Sobolev worked and lived: St. Petersburg, Moscow, and Novosibirsk. Most of the papers are written by leading experts in control theory and inverse pr- lems. Another reason for the selection is a strong link to applied areas. In my opinion, control theory and inverse problems are main areas of di?er- tial equations of importance for some branches of contemporary science and engineering. S. L. Sobolev, as many great mathematicians, was very much motivated by applications. He did not distinguished between pure and - plied mathematics, but, in his own words, between "good mathematics and bad mathematics. " While he possessed a brilliant analytical technique, he most valued innovative ideas, solutions of deep conceptual problems, and not mathematical decorations, perfecting exposition, and "generalizations. " S. L.
Sobolev himself never published papers on inverse problems or c- trol theory, but he was very much aware of the state of art and he monitored research on inverse problems. In particular, in his lecture at a Conference on Di?erentialEquationsin1954(found inSobolev'sarchiveandmadeavailable to me by Alexander Bukhgeim), he outlined main inverse problems in g- physics:theinverseseismicproblem,theelectromagneticprospecting,andthe inverse problem of gravimetry.
Contents
Geometrization of Rings as a Method for Solving Inverse Problems.- The Ginzburg-Landau Equations for Superconductivity with Random Fluctuations.- Carleman Estimates with Second Large Parameter for Second Order Operators.- Sharp Spectral Asymptotics for Dirac Energy.- Linear Hyperbolic and Petrowski Type PDEs with Continuous Boundary Control ? Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with Optimal Decay Rates.- Uniform Asymptotics of Green's Kernels for Mixed and Neumann Problems in Domains with Small Holes and Inclusions.- Finsler Structures and Wave Propagation.
