An Inequality for the Pointwise Estimation of Solutions of Second Order Elliptic Partial Differential Equations

Abstract

An inequality is derived giving a pointwise estimate for any given twice continuously differentiable function of two variables. The inequality takes one of two alternative forms, both of which involve space integrals of the function and its first derivatives; one of the forms also includes a logarithmic term. The inequality is a variant of the classical Sobolev result, but appareantly easier to apply for cases of practical interest. It is shown that the inequality may be used to estimate the solution of a singular perturbation problem posed by an elliptic equation whose highest derivatives are multiplied by a small parameter. The inequality may be extended to higher dimensions only if a priori bounds on the function and its derivatives are known.