Gershon Kresin

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.

Criteria for validity of the maximum modulus principle for solutions of linear parabolic systems

We consider systems of partial differential equations of the first order int and of order 2s in thex variables, which are uniformly parabolic in the sense of Petrovskii. We show that the classical maximum modulus principle is not valid inRn×(0,T] fors≥2.For second order systems we obtain necessary and, separately, sufficient conditions for the classical maximum modulus principle, to hold in the layerRn×(0,T] and in the cylinder μ×(0,T], where μ is a bounded subdomain ofRn. If the coefficients of the system do not depend ont, these conditions coincide. The necessary and sufficient condition in this case is that the principal part of the system is scalar and that the coefficients of the system satisfy a certain algebraic inequality. We show by an example that the scalar character of the principal part of the system everywhere in the domain is not necessary for validity of the classical maximum modulus principle when the coefficients depend both onx andt.

Sharp Bohr Type Real Part Estimates

AbstractWe consider analytic functions ƒ in the unit disk

Criteria for Validity of the Maximum Norm Principle for Parabolic Systems

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Sharp constants and maximum principles for elliptic and parabolic systems with continuous boundary data

with Taylor coefficients c0, c1, … and derive estimates with sharp constants for the lq− norm (quasi-norm for 0 < q < 1) of the remainder of their Taylor series, where q ∈ (0, ∞]. As the main result, we show that given a function ƒ with Re ƒ in the Hardy space

Sharp estimates for the gradient of the generalized Poisson integral for a half-space

h_1 \left( \mathbb{D} \right)