The problem of dirichlet for quasilinear elliptic differential equations with many independent variables

Abstract

This paper is concerned with the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order, the conclusions being in the form of necessary conditions and sufficient conditions for this problem to be solvable in a given domain with arbitrarily assigned smooth boundary data. A central position in the discussion is played by the concept of global barrier functions and by certain fundamental invariants of the equation. With the help of these invariants we are able to distinguish an important class of ‘ regularly elliptic5 equations which, as far as the Dirichlet problem is concerned, behave comparably to uniformly elliptic equations. For equations which are not regularly elliptic it is necessary to impose significant restrictions on the curvatures of the boundaries of the underlying domains in order for the Dirichlet problem to be generally solvable; the determination of the precise form of these restrictions constitutes a second primary aim of the paper. By maintaining a high level of generality throughout, we are able to treat as special examples the minimal surface equation, the equation for surfaces having prescribed mean curvature, and a number of other non-uniformly elliptic equations of classical interest.