On the Mathematical Works of S.L. Sobolev in the 1930s -- Sobolev in Siberia -- Boundary Harnack Principle and the Quasihyperbolic Boundary Condition -- Sobolev Spaces and their Relatives: Local Polynomial Approximation Approach -- Spectral Stability of Higher Order Uniformly Elliptic Operators -- Conductor Inequalities and Criteria for Sobolev-Lorentz Two-Weight Inequalities -- Besov Regularity for the Poisson Equation in Smooth and Polyhedral Cones -- Variational Approach to Complicated Similarity Solutions of Higher Order Nonlinear Evolution Partial Differential Equations -- L q,p -Cohomology of Riemannian Manifolds with Negative Curvature -- Volume Growth and Escape Rate of Brownian Motion on a Cartan—Hadamard Manifold -- Sobolev Estimates for the Green Potential Associated with the Robin—Laplacian in Lipschitz Domains Satisfying a Uniform Exterior Ball Condition -- Properties of Spectra of Boundary Value Problems in Cylindrical and Quasicylindrical Domains -- Estimates for Completely Integrable Systems of Differential Operators and Applications -- Counting Schrödinger Boundstates: Semiclassics and Beyond -- Function Spaces on Cellular Domains.

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)