About the Book
Sobolev spaces and inequalities are fundamental tools in the theory of partial differential equations, analysis, differential geometry, mathematical physics, etc. Introduced 70 years ago, they turned out to be extremely useful in many different settings and continue to attract the attention of new generations of mathematicians. Recent advantages in the theory of Sobolev spaces and in applications are presented by globally recognized specialists in topics covering Sobolev-type spaces of functions in metric spaces, various aspects of Sobolev-type inequalities, boundary value problems for differential operators, spectral problems, approximations, optimal control, important problems of mathematical physics, analysis, partial differential equations, geometry, etc.
The book is dedicated to the centenary of S.L. Sobolev and includes biographical articles supplied with the bibliography of Sobolev's works in the 1930s and archive photos of Sobolev previously unpublished in the English-language literature.
Table of Contents:
Volume I
My Love Affair with the Sobolev Inequality, D.R. Adams.- Maximal Functions in Sobolev Spaces, D. Aalto, J. Kinnunen.- Hardy Type Inequalities Via Riccati and Sturm–Liouville Equations, S. Bobkov, F. Götze.- Quantitative Sobolev and Hardy Inequalities and Related Symmetrization Principles, A. Cianchi.- Inequalities of Hardy–Sobolev Type in Carnot–Carathéodory Spaces, D. Danielli et al.- Sobolev Embeddings and Hardy Operators, D.E. Edmunds, W.D. Evans.- Sobolev Mappings between Manifolds and Metric Spaces, P. Hajlasz.- A Collection of Sharp Dilation Invariant Integral Inequalities for Differentiable Functions, V. Maz'ya, T. Shaposhnikova.- Optimality of Function Spaces in Sobolev Embeddings, L. Pick.- On the Hardy–Sobolev–Maz'ya Inequality and Its Generalizations, Y. Pinchover, K. Tintarev.- Sobolev Inequalities in Familiar and Unfamiliar Settings, L. Saloff-Coste.- A Universality Property of Sobolev Spaces in Metric Measure Spaces, N. Shanmugalingam.- Cocompact Imbeddings and Structure of Weakly Convergent Sequences, K. Tintarev.
Volume II
On the Mathematical Works of S.L. Sobolev in the 1930s, V. Babich.- Sobolev in Siberia, Y. Reshetnyak.- Boundary Harnack Principle and the Quasihyperbolic Boundary Condition, H. Aikawa.- Sobolev Spaces and their Relatives: local Polynomial Approximation Approach, Y. Brudnyi.- Spectral Stability of Higher Order Uniformly Elliptic Operators, V. Burenkov, P.D. Lamberti.- Conductor Inequalities and Criteria for Sobolev - Lorentz Two - Weight Inequalities, S. Costea, V. Maz'ya.- Besov Regularity for the Poisson Equation in Smooth and Polyhedral Cones, S. Dahlke, W. Sickel.- Variational Approach to Complicated Similarity Solutions of Higher Order Nonlinear Evolution PartialDifferential Equations, V. Galaktionov et al.- Lq,p-Cohomology of Riemannian Manifolds with Negative Curvature, V. Gol'dshtein, M. Troyanov.- Volume Growth and Escape Rate of Brownian Motion on a Cartan–Hadamard Manifold, A. Grigor'yan, E. Hsu.- Sobolev Estimates for the Green Potential Associated with the Robin–Laplacian in Lipschitz Domains Satisfying a Uniform Exterior Ball Condition, T. Jakab et al.- Properties of Spectra of Boundary Value Problems in Cylindrical and Quasicylindrical Domains, S. Nazarov.- Estimates for Completeley Integrable Systems of Differential Operators and Applications, Y. Reshetnyak.- Counting Schrödinger Boundstates: Semiclassics and Beyond, G. Rozenblum, M. Solomyak.- Function Spaces on Cellular Domains, H. Triebel.
Volume III
Geometrization of Rings as a Method for Solving Inverse Problems, M. Belishev.- The Ginzburg–Landau Equations for Superconductivity with Random Fluctuations, A. Fursikov et al.- Carleman Estimates with Second Large Parameter for Second Order Operators, V. Isakov, N. Kim.- Sharp Spectral Asymptotics for Dirac Energy, V. Ivrii.- Linear Hyperbolic and Petrowski Type PDEs with Continuous Boundary Control - Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with Optimal Decay Rates, I. Lasiecka, R. Triggiani.- Uniform Asymptotics of Green's Kernels for Mixed and Neumann Problems in Domains with Small Holes and Inclusions, V. Maz'ya, A. Movchan.- Finsler Structures and Wave Propagation, M. Taylor.
