Asymmetric Case; Comparison Principles and Applications

Abstract

We wish to estimate the behavior of capillary surfaces in tubes of general section. We approach the problem in the spirit of the preceding chapter, and attempt to gain information by comparison of a given solution with an explicitly known “near solution” of the equation. Since symmetry will no longer be assumed, we need a more sophisticated tool than Lemma 4.11, and we have recourse to a form of the maximum principle for elliptic equations. The statement we give is formally analogous to the familiar maximum principle for the Laplace operator; however there is an important new feature, which reflects itself in some striking distinctions, between the behavior of capillary surfaces and that of surfaces determined by the solutions of linear elliptic equations.