Uniqueness in some higher order elliptic boundary value problems in n dimensional domains

C.-P Danet

doi:10.14232/ejqtde.2011.1.54

Abstract

We develop maximum principles for several P functions which are defined on solutions to equations of fourth and sixth order (including a equation which arises in plate theory and bending of cylindrical shells). As a consequence, we obtain uniqueness results for fourth and sixth order boundary value problems in arbitrary n dimensional domains.

Highlights

Our aim here is to remove via the P function method dimension and geometry conditions with, further conditions on the coefficients a, b, c and ρ
We deal with a equation that arises in plate theory and in bending of cylindrical shells
We prove the uniqueness result for the corresponding homogeneous boundary value problem without the hypothesis that the plate has a convex shape

Summary

Introduction

This paper represents the n dimensional analogue of Schaefer’s paper [9] and is concerned with uniqueness results for boundary value problems of fourth and sixth order. Schaefer [9] investigated the uniqueness of the solution for the boundary value problems. Where a, b, ≥ 0, c > 0 are constants, φ ≡ 0, ρ > 0 in the bounded domain Ω, n = 2 and the curvature of the boundary is strictly positive. We prove the uniqueness result for the corresponding homogeneous boundary value problem without the hypothesis that the plate has a convex shape. Throughout the paper Ω and diamΩ denote respectively a bounded domain in IRn, the diameter of Ω

Some useful results

Maximum principles and uniqueness results for sixth order equations

A maximum principle and an uniqueness result for a fourth order equation