Some Maximum Principles in Semilinear Elliptic Equations

Abstract

ABSTRACT.We develop maximum principles for functions defined on the solutions to a class of semilinear, second order, uniformly elliptic partial differential equations. These principles are related to recent theorems of Protter and Protter and Weinberger and to a technique initiated by Payne for the determination of gradient bounds on the solution of the equation.1. Introduction. In [3] Payne introduced a technique, which utilizes a maximum principle for a function defined on solutions to an elliptic partial differential equation, in order to obtain bounds for the gradient of the solution of the relèvent differential equation. Several authors have contributed to the growing literature developing this technique. In their work (see the references cited here, especially [7], and the references therein), the authors seek estimates on the solution, the gradient of the solution, or other quantities of physical importance and/or extend the method to more general elliptic or parabolic differential equations. Early in the development of this method, the results were obtained when the principal part of the elliptic equation was the Laplace operator.Sperb [6] was the first to extend the results of Payne to a second order uniformly elliptic equation of the form

On a Class of Singularly Perturbed Elliptic Systems with Asymptotic Phase Segregation

“…A standard maximum and nonnegativity principle for elliptic equations (cf. [14]) yields the following result. In sequel we use this result.…”