Sobolev Spaces in Mathematics III: Applications in Mathematical Physics (International Mathematical Series Book 10)

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) Sobolev spaces become the established and universal language of partial differential equations and mathematical analysis. Among a huge variety of problems where Sobolev spaces are used, the following important topics are in the focus of this volume: boundary value problems in domains with singularities, higher order partial differential equations, local polynomial approximations, inequalities in Sobolev-Lorentz spaces, function spaces in cellular domains, the spectrum of a Schrodinger operator with negative potential and other spectral problems, criteria for the complete integrability of systems of differential equations with applications to differential geometry, some aspects of differential forms on Riemannian manifolds related to Sobolev inequalities, Brownian motion on a Cartan-Hadamard manifold, etc. Two short biographical articles on the works of Sobolev in the 1930's and foundation of Akademgorodok in Siberia, supplied with unique archive photos of S. Sobolev are included. Contributors include: Vasilii Babich (Russia); Yuri Reshetnyak (Russia); Hiroaki Aikawa (Japan); Yuri Brudnyi (Israel); Victor Burenkov (Italy) and Pier Domenico Lamberti (Italy); Serban Costea (Canada) and Vladimir Maz'ya (USA-UK-Sweden); Stephan Dahlke (Germany) and Winfried Sickel (Germany); Victor Galaktionov (UK), Enzo Mitidieri (Italy), and Stanislav Pokhozhaev (Russia); Vladimir Gol'dshtein (Israel) and Marc Troyanov (Switzerland); Alexander Grigor'yan (Germany) and Elton Hsu (USA); Tunde Jakab (USA), Irina Mitrea (USA), and Marius Mitrea (USA); Sergey Nazarov (Russia); Grigori Rozenblum (Sweden) and Michael Solomyak (Israel); Hans Triebel (Germany) 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)

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 20th 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, Lamé 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.

Front Matter....Pages i-xxxv Geometrization of Rings as a Method for Solving Inverse Problems....Pages 5-24 The Ginzburg-Landau Equations for Superconductivity with Random Fluctuations....Pages 25-133 Carleman Estimates with Second Large Parameter for Second Order Operators....Pages 135-159 Sharp Spectral Asymptotics for Dirac Energy....Pages 161-185 Linear Hyperbolic and Petrowski Type PDEs with Continuous Boundary Control → Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with Optimal Decay Rates....Pages 187-276 Uniform Asymptotics of Green's Kernels for Mixed and Neumann Problems in Domains with Small Holes and Inclusions....Pages 277-316 Finsler Structures and Wave Propagation....Pages 317-334 Back Matter....Pages 335-336 This volume, marking the centenary of S.L. Sobolev’s birth, presents the latest the results on some important problems of mathematical physics. The book contains two short biographical articles and unique archive photos of S. Sobolev. 1. Sobolev type inequalities 2. Applications in analysis and partial differential equations 3. Applications in mathematical physics.