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...

The contributions in this volume are dedicated to Vladimir G. Maz'ya and are par tially based on talks given at the conference "Functional Analysis, Partial Differ ential Equations, and Applications," which took place at the University of Rostock from August 31 to September 4, 1998, to honour Prof. Maz'ya. This conference (a satellite meeting of the ICM) gave an opportunity to many friends and colleagues from all over the world to honour him. This academic community is very large. The scientific field of Prof. Maz'ya is impressively broad, which is reflected in the variety of contributions included in the volumes. Vladimir Maz'ya is the author and co-author of many publications (see the list of publications at the end of this volume), the topics of which extend from functional analysis, function theory and numerical analysis to partial differential equations and their broad applications. Vladimir G. Maz'ya provided significant contributions, among others to the the ory of Sobolev spaces, the capacity theory, boundary integral methods, qualitative and asymptotic methods of analysis of linear and nonlinear elliptic differential equations, the Cauchy problem for elliptic and hyperbolic equations, the theory of multipliers in spaces of differentiable functions, maximum principles for elliptic and parabolic systems, and boundary value problems in domains with piecewise smooth boundaries. Surveys on Maz'ya's work in different fields of mathematics and areas, where he made essential contributions, form a major part of the present first volume of The Maz'ya Anniversary Collection."

During the week of August 31 - September 4, 1998, a conference in honour of Vladimir Maz'ya was held in Rostock as a satellite meeting of the World Congress of Mathematicians. It was sponsored by the German Research Founda tion (Deutsche Forschungsgemeinschaft) and the Ministry of Education and Cul tural Affairs of the land Mecklenburg-Vorpommern. During his forty year career Maz'ya contributed to so many areas of mathematical analysis that such a broad topic of the conference as "Functional Analysis, Partial Differential Equations and Applications" sounds quite nat al. The conference was organized by the Depart ment of Mathematics of the University of Rostock and the Weierstrass Institute of Applied Analysis and Stochastics in Berlin on the occasion of his 60th birth day. For many years Maz'ya was connected with mathematicians from Berlin and Rostock through his work in potential theory, in differential and pseudodifferen tial equations and in approximation theory. In 1990 he was awarded an honorary doctorate by the University of Rostock. Shortly before the meeting, one of its organizers, an outstanding mathematician and Maz'ya's dear friend, Siegfried Pr6Bdorf died. This was a heavy loss for the and for the conference in particular. During the German mathematical community meeting the rector of the University of Rostock, Prof. Wildenhain, the director of the Weierstrass Institute, Prof. Sprekels, and Prof. Maz'ya remembered S. Pr6Bdorf very warmly. The conference was attended by 109 mathematicians from 21 countries, and the program included 24 invited lectures and 63 short communications."

Die Theorie des NEWToNschen Potentials von Massenverteilungen im Raum ist eines der altesten Beispiele einer Verbindung von physikalischer Anschauung und mathematischer Interpretation. Bedeutende Mathematiker vieler Generationen, wie C. F. GAUSS, H. POINCARE, D. lIILEERT, N. WIENER haben daran mitgearbeitet. Die Entwicklung der modernen Potentialtheorie ist auch wesentlich durch die Arbeiten von G. C. EVANS, M. RIEsz, O. FBOSTMAN, M. V. KELDYs, M. BRELoT, H. CARTAN, J. DENY, G. CHOQUET, J. L. DooE, H. BAUER, C. CONSTANTINESCU, V. G. MAz 'JA, B. FUGLEDE und anderen bestimmt worden. Historische Darstellungen wurden z. B. in [K6], [A30], [B40] gegeben. Obwohl einige Teile der Potentialtheorie heute als im wesentlichen abgeschlossen gelten, hat sich die Entwicklung in den letzten Jahren wieder erheblich verstarkt, seit sich viele ihrer leistungsfahigen Begriffe und Methoden durch den zunehmenden Einsatz funktionalanalytischer Methoden auf weite Klassen von Problemen aus der Theorie der partiellen Differentialgleichungen anwenden lassen. Daneben sind in der Analysis auch davon unabhangige Bestrebungen von potentialtheoretischem Charakter zu beobachten.