IngentaConnect Best Constants in the Miranda-Agmon Inequalities for Solutions of...

Abstract:

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space $ shadRˆk_{+} = {x =(x_1 , ldots , x_k){:} x_k > 0 } $ is considered. It is assumed that the boundary values of a solution u = (u₁,···,u_m) have the form psi

₁

₁ + ··· +

_n, 1

m, where

₁,···,

_n is an orthogonal system of m-component normed vectors and psi

₁,···,

_n are continuous and bounded functions on $ partial shadR ˆk_ {+} $. We study the mappings $$ eqalign {[{rm C} (partial shadR ˆ k_{ + })] ˆ n i ( psi_1, ldots,psi_n) rightarrow u(x) in shadR ˆ m quad mbox {and} quad [{bf C} (partial shadR ˆ k_{ + })] ˆ n i (psi_1 , ldots,psi_n) rightarrow u(x) in shadC ˆ m} $$ generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and Verbar

u|_x_k =0} Verbar

, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·, ·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.

Keywords: Miranda-Agmon inequalities; Elliptic systems; Maximum modulus principle; Lamé and Stokes systems in a half-space; Planar deformed state; Biharmonic equation

Document Type: Research article

DOI: 10.1080/000368103763297456

Affiliations: 1: The Research Institute, The College of Judea and Samaria, 44837 Ariel, Israel 2: Department of Mathematics, Linköping University, S-58183 Linköping, Sweden

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Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces