Certain Estimates for Solutions of Nonlinear Elliptic Differential Equations Applicable to the Theory of Thin Plates

Abstract

This article derives certain estimates based on properties of the Lagrangian integral for (classical) solutions of equations of the form \[ ( 1 )\qquad \sum\limits_i {\left( {a_i ( {\bf x} )\phi ( w )w_{x_i } } \right)} _{x_i } + c( {\bf x} )f( w ) = 0, \] and \[ ( 2 )\qquad \sum\limits_j {\sum\limits_i {\left( {a_i ( {\bf x} )\phi ( w )w_{x_i x_i } } \right)} } _{x_j x_j } + c( {\bf x} )f( w ) = 0, \], ${\bf x} \in \Omega \subset \mathbb{R}^n $ where $a_i ( {\bf x} )$ are positive in $\Omega $. These estimates depend on the size and shape of $\Omega $, but do not depend on the boundary conditions imposed on $\partial \Omega $. Examples are given of applications to nonlinear membrane and both linear and nonlinear thin plate theory.