Introduction to Boundary Value Problems of Nonlinear Elastostatics

Abstract

This paper provides a careful and accessible exposition of an $L^p$ approach to boundary value problems of nonlinear elastostatics in the case where solutions of the linearized problem correspond faithfully to those of the nonlinear problem, that is, in the case where there is no bifurcation. We prove that if the linearized problem has unique solutions, then so does the nonlinear one, nearby. This is done by using the linear $L^p$ theory and the inverse mapping theorem. The main theorem can be applied to the Saint Venant-Kirchhoff elastic material and the Hencky-Nadai elastoplastic material in a unified theory. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations.