Synopses & Reviews
This is the first volume of a collection of articles dedicated to V.G Maz'ya on the occasion of his 60th birthday. It contains surveys on his work in different fields of mathematics or on areas to which he made essential contributions. Other articles of this book have their origin in the common work with Maz'ya. V.G Maz'ya is author or co-author of more than 300 scientific works on various fields of functional analysis, function theory, numerical analysis, partial differential equations and their application. The reviews in this book show his enormous productivity and the large variety of his work. The scond volume contains most of the invited lectures of the Conference on Functional Analysis, Partial Differential Equations and Applications held in Rostock in September 1998 in honor of V.G Maz'ya. Here different problems of functional analysis, potential theory, linear and nonlinear partial differential equations, theory of function spaces and numerical analysis are treated. The authors, who are outstanding experts in these fields, present surveys as well as new results.
Synopsis
Vladimir Maz'ya: Friend and mathematician. Recollections.- On Maz'ya's work in potential theory and the theory of function spaces.- 1. Introduction.- 2. Embeddings and isoperimetric inequalities.- 3. Regularity of solutions.- 4. Boundary regularity.- 5. Nonlinear potential theory.- Maz'ya's works in the linear theory of water waves.- 1. Introduction.- 2. The unique solvability of the water wave problem.- 3. The Neumann-Kelvin problem.- 4. Asymptotic expansions for transient water waves due to brief and high-frequency disturbances.- Maz'ya's work on integral and pseudodifferential operators.- 1. Non-elliptic operators.- 2. Oblique derivative problem: breakthrough in the generic case of degeneration.- 3. Estimates for differential operators in the half-space.- 4. The characteristic Cauchy problem for hyperbolic equations.- 5. New methods for solving ill-posed boundary value problems.- 6. Applications of multiplier theory to integral operators.- 7. Integral equations of harmonic potential theory on general non-regular surfaces.- 8. Boundary integral equations on piecewise smooth surfaces.- Contributions of V. Maz'ya to the theory of boundary value problems in nonsmooth domains.- 1. Maz'ya's early work on boundary value problems in nonsmooth domains.- 2. General elliptic boundary value problems in domains with point singularities.- 3. Boundary value problems in domains with edges.- 4. Spectral properties of operator pencils generated by elliptic boundary value problems in a cone.- 5. Applications to elastostatics and hydrodynamics.- 6. Singularities of solutions to nonlinear elliptic equations at a cone vertex.- On some potential theoretic themes in function theory.- 1. Approximation theory.- 2. Uniqueness properties of analytic functions.- 3. The Cauchy problem for the Laplace equation.- Approximate approximations and their applications.- 1. Introduction.- 2. Quasi-interpolation.- 3. Generating functions for quasi-interpolation of high order.- 4. Semi-analytic cubature formulas.- 5. Cubature of integral operators over bounded domains.- 6. Approximate wavelets.- 7. Numerical algorithms based upon approximate approximations.- Maz'ya's work on the biography of Hadamard.- Isoperimetric inequalities and capacities on Riemannian manifolds.- 1. Introduction.- 2. Capacity of balls.- 3. Parabolicity of manifolds.- 4. Isoperimetric inequality and Sobolev inequality.- 5. Capacity and the principal frequency.- 6. Cheeger's inequality.- 7. Eigenvalues of balls on spherically symmetric manifolds.- 8. Heat kernel on spherically symmetric manifolds.- Multipliers of differentiable functions and their traces.- 1. Introduction.- 2. Description and properties of multipliers.- 3. Multipliers in the space of Bessel potentials as traces of multipliers.- An asymptotic theory of nonlinear abstract higher order ordinary differential equations.- Sobolev spaces for domains with cusps.- 1. Introduction.- 2. Extension theorems.- 3. Embedding theorems.- 4. Boundary values of Sobolev functions.- Extension theorems for Sobolev spaces.- 1. Introduction.- 2. Extensions with preservation of class.- 3. Estimates for the minimal norm of an extension operator.- 4. Extensions with deterioration of class.- Contributions of V.G. Maz'ya to analysis of singularly perturbed boundary value problems.- 1. Introduction.- 2. Domain with a small hole.- 3. General asymptotic theory by Maz'ya, Nazarov and Plamenevskii.- 4. Asymptotics of solutions of boundary integral equations under a small perturbation of a corner.- 5. Compound asymptotics for homogenization problems.- 6. Boundary value problems in 3D-1D multi-structures.- Asymptotic analysis of a mixed boundary value problem in a singularly degenerating domain.- 1. Introduction.- 2. Formulation of the problem.- 3. The leading order approximation.- A history of the Cosserat spectrum.- 1. Introduction.- 2. The first boundary value problem of elastostatics.- 3. The second and other boundary-value problems.- 4. Applications and o...