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

Abstract

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space 𝔑 k + = {x = (x 1,…,xk ): xk > 0} is considered. It is assumed that the boundary values of a solution u = (u 1,…,u m) have the form ψ 1 ξ 1 + · · · + ψ n ξ n, 1 ≤ n ≤ m, where ξ 1,· · ·,ξ n is an orthogonal system of m-component normed vectors and ψ 1,· · ·,ψ n are continuous and bounded functions on ∂𝔑 k +. We study the mappings [C(∂𝔑 k +)] n ∋ (ψ 1,…,ψ n) → u(x) ∋ 𝔑 m and [C(∂𝔑 k +)] n ∋ (ψ 1,…,ψ n) → u(x) ∋ 𝔑 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 ∥u| xk =0∥, 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.