Sobolev Spaces in Mathematics III

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 inSobolevsarchiveandmadeavailable to me by Alexander Bukhgeim) he outlined main inverse problems in g- physics:theinverseseismicproblemtheelectromagneticprospectingandthe inverse problem of gravimetry.