The Mazya Anniversary Collection: On Maz'Ya's Work in Functional Analysis, Partial Differential Equations and Applications

by Rossmann, Jurgen

• ISBN13: 9783764362010
• ISBN10: 3764362014
• Format: Hardcover
• Copyright: 1999-10-01
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Introduction	ix
G. Wildenhain
Vladimir Maz'ya: Friend and mathematician. Recollections	1	(6)
Gohberg
On Maz'ya's work in potential theory and the theory of function spaces	7	(10)
L. I. Hedberg
Introduction	7	(1)
Embeddings and isoperimetric inequalities	7	(3)
Regularity of solutions	10	(1)
Boundary regularity	11	(1)
Nonlinear potential theory	12	(5)
Maz'ya's works in the linear theory of water waves	17	(18)
N. G. Kuznetsov
B. R. Vainberg
Introduction	17	(1)
The unique solvability of the water wave problem	18	(7)
The Neumann-Kelvin problem	25	(3)
Asymptotic expansions for transient water waves due to brief and high-frequency disturbances	28	(7)
Maz'ya's work on integral and pseudodifferential operators	35	(18)
J. Elschner
Non-elliptic operators	35	(2)
Oblique derivative problem: breakthrough in the generic case of degeneration	37	(2)
Estimates for differential operators in the half-space	39	(1)
The characteristic Cauchy problem for hyperbolic equations	40	(2)
New methods for solving ill-posed boundary value problems	42	(1)
Applications of multiplier theory to integral operators	42	(1)
Integral equations of harmonic potential theory on general non-regular surfaces	43	(2)
Boundary integral equations on piecewise smooth surfaces	45	(8)
Contributions of V. Maz'ya to the theory of boundary value problems in nonsmooth domains	53	(46)
J. Rossmann
Maz'ya's early work on boundary value problems in nonsmooth domains	54	(11)
General elliptic boundary value problems in domains with point singularities	65	(12)
Boundary value problems in domains with edges	77	(6)
Spectral properties of operator pencils generated by elliptic boundary value problems in a cone	83	(3)
Applications to elastostatics and hydrodynamics	86	(5)
Singularities of solutions to nonlinear elliptic equations at a cone vertex	91	(8)
On some potential theoretic themes in function theory	99	(12)
V. Havin
Approximation theory	99	(4)
Uniqueness properties of analytic functions	103	(2)
The Cauchy problem for the Laplace equation	105	(6)
Approximate approximations and their applications	111	(26)
G. Schmidt
Introduction	111	(2)
Quasi-interpolation	113	(4)
Generating functions for quasi-interpolation of high order	117	(3)
Semi-analytic cubature formulas	120	(2)
Cubature of integral operators over bounded domains	122	(4)
Approximate waveletes	126	(4)
Numerical algorithms based upon approximate approximations	130	(7)
Maz'ya's work on the biography of Hadamard	137	(2)
R. Cooke
Isoperimetric inequalities and capacities on Riemannian manifolds	139	(16)
A. Grigoryan
Introduction	139	(2)
Capacity of balls	141	(2)
Parabolicity of manifolds	143	(2)
Isoperimetric inequality and Sobolev inequality	145	(1)
Capacity and the principal frequency	145	(2)
Cheeger's inequality	147	(1)
Eigenvalues of balls on spherically symmetric manifolds	148	(1)
Heat kernel on spherically symmetric manifolds	149	(6)
Multipliers of differentiable functions and their traces	155	(16)
T. Shaposhnikova
Introduction	155	(1)
Description and properties of multipliers	156	(6)
Multipliers in the space of Bessel potentials as traces of multipliers	162	(9)
An asymptotic theory of nonlinear abstract higher order ordinary differential equations	171	(4)
V. A. Kozlov
Sobolev spaces for domains with cusps	175	(12)
S. Poborchi
Introduction	175	(1)
Extension theorems	176	(3)
Embedding theorems	179	(2)
Boundary values of Sobolev functions	181	(6)
Extension theorems for Sobolev spaces	187	(14)
V. Burenkov
Introduction	187	(1)
Extensions with preservation of class	188	(2)
Estimates for the minimal norm of an extension operator	190	(3)
Extensions with deterioration of class	193	(8)
Contributions of V.G. Maz'ya to analysis of singularly perturbed boundary value problems	201	(12)
A. Movchan
Introduction	201	(1)
Domain with a small hole	202	(4)
General asymptotic theory by Maz'ya, Nazarov and Plamenevskii	206	(1)
Asymptotics of solutions of boundary integral equations under a small perturbation of a corner	207	(1)
Compound asymptotics for homogenization problems	208	(2)
Boundary value problems in 3D-1D multi-structures	210	(3)
Asymptotic analysis of a mixed boundary value problem in a singularly degenerating domain	213	(10)
J. Aslund
Introduction	213	(1)
Formulation of the problem	213	(2)
The leading order approximation	215	(8)
A history of the Cosserat spectrum	223	(12)
A. Kozhevnikov
Introduction	223	(1)
The first boundary value problem of elastostatics	224	(3)
The second and other boundary-value problems	227	(3)
Applications and other related results	230	(5)
Boundary integral equations for plane domains with cusps	235	(8)
A. Soloviev
Introduction	235	(2)
Integral equations in weighted Sobolev spaces	237	(6)
On Maz'ya type inequalities for convolution operators	243	(6)
S. Eilertsen
Introduction	243	(1)
One-dimensional polynomials	244	(2)
The functions |x|2λ in Rn	246	(3)
Sharp constants and maximum principles for elliptic and parabolic systems with continuous boundary data	249	(58)
G. Kresin
The norm and the essential norm of the double layer elastic and hydrodynamic potentials in the space of continuous functions	253	(11)
Exact constants in inequalities of maximum principle type for certain systems and equations of mathematical physics	264	(6)
Maximum modulus principle for elliptic systems	270	(8)
Maximum modulus principle for parabolic systems	278	(9)
Maximum norm principle for parabolic systems	287	(20)
Lp-contractivity of semigroups generated by parabolic matrix differential operators	307	(24)
M. Langer
Introduction	307	(3)
Preliminaries	310	(1)
Weakly coupled systems	311	(11)
Coupled systems	322	(9)
Curriculum vitae of Vladimir Maz'ya	331	(4)
Publications of Vladimir Maz'ya	335