Decay Estimates for a Class of Nonlinear Boundary Value Problems in Two Dimensions

Abstract

This paper is concerned with second-order quasilinear partial differential equations in two independent variables of the form ${\operatorname {div}}[\rho ({\bf x},u,{\operatorname {grad }}u){\operatorname {grad }}u] = 0$. Previous work of the authors, establishing exponential decay estimates for Dirichlet problems on a semi-infinite strip subject to nonzero data on the finite end, is extended to include regions of arbitrary shape, and, in the case of unbounded regions, the a priori assumption that solutions must decay to zero as $| {\bf x} | \to \infty $ is removed. The results have application to Saint-Venant principles for nonlinear elasticity as well as to theorems of Phragmén–Lindelöf type.