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.
The mathematical works of S.L.Sobolev were strongly motivated by particular problems coming from applications. In his celebrated book Applications of Functional Analysis in Mathematical Physics, 1950 and other works, S.Sobolev introduced general methods that turned out to be very influential in the study of mathematical physics in the second half of the XXth century. This volume, dedicated to the centenary of S.L. Sobolev, presents the latest results on some important problems of mathematical physics describing, in particular, phenomena of superconductivity with random fluctuations, wave propagation, perforated domains and bodies with defects of different types, spectral asymptotics for Dirac energy, Lam\'e system with residual stress, optimal control problems for partial differential equations and inverse problems admitting numerous interpretations. Methods of modern functional analysis are essentially used in the investigation of these problems.
Contributors include: Mikhail Belishev (Russia); Andrei Fursikov (Russia), Max Gunzburger (USA), and Janet Peterson (USA); Victor Isakov (USA) and Nanhee Kim (USA); Victor Ivrii (Canada); Irena Lasiecka (USA) and Roberto Triggiani (USA); Vladimir Maz'ya (USA-UK-Sweden) and Alexander Movchan (UK); Michael Taylor (USA)
