Behavior of solutions of the Dirichlet problem for quasilinear divergent higher-order elliptic equations in unbounded domains

Abstract

In this article we determine the energy apriori estimates of solutions of the Dirichlet problem in unbounded domains with noncompact boundaries. These estimates depend on the geometry of the boundary, and in the case of second-order equations, for a wide class of domains they coincide with those previously known [I, 2]. The method of obtaining these estimates is a new one, conceptually similar to the method used in [3, 4] to obtai n interior estimates of generalized solutions. On the basis of established energy estimates we prove alternative theorems of the type of the Phragmen--Lindel0f Theorem on the behavior of solutions at infinity. For linear equations, the first such theorems in terms of an interior diameter of the domain are given in [5]; in the same terms, for solutions of quasilinear divergent higher order elliptic equations, such a result is announced in [6]. We note further that for solutions of a polyharmonic equation in the case of some model domains, energy apriori estimates of the type of Saint-Venant's principle were established in [7].