Vladimir Maz’ya

Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems

The main goal of this book is to present results pertaining to various versions of the maximum principle for elliptic and parabolic systems of arbitrary order. In particular, the authors present necessary and sufficient conditions for validity of the classical maximum modulus principles for systems of second order and obtain sharp constants in inequalities of Miranda-Agmon type and in many other inequalities of a similar nature. Somewhat related to this topic are explicit formulas for the norms and the essential norms of boundary integral operators. The proofs are based on a unified approach using, on one hand, representations of the norms of matrix-valued integral operators whose target spaces are linear and finite dimensional, and, on the other hand, on solving certain finite dimensional optimization problems. This book reflects results obtained by the authors, and can be useful to research mathematicians and graduate students interested in partial differential equations.

ON THE FAST COMPUTATION OF HIGH DIMENSIONAL VOLUME POTENTIALS

A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate ...

Pointwise estimates for the polyharmonic Green function in general domains

In the present paper we establish sharp estimates on the polyharmonic Green function and its derivatives in an arbitrary bounded open set.

Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds

<abstract abstract-type=TeX><p> We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds

Sharp pointwise estimates for solutions of strongly elliptic second order systems with boundary data from L p

M

Basic Properties of Sobolev Spaces

of finite volume. Sharp conditions ensuring

Approximate approximations with data on a perturbed uniform grid

L^q(M)

Invariant convex bodies for strongly elliptic systems

and

Generalizations for Functions on Manifolds and Topological Spaces

L^infty (M)

Inequalities for Functions Vanishing at the Boundary

bounds for eigenfunctions are exhibited in terms of either the isoperimetric function or the isocapacitary function of