A New Type of Generalized Solution of the Dirichlet Problem for the Heat Equation

Abstract

Let as consider in the Euclidean space R ⁿ harmonic functions as continuous solutions of the Laplace differential equation Δf = 0. Given a bounded open set U R ⁿ and a continuous function f on the boundary U* of U, we understand by the solution of the Dirichlet problem for f a continuous function F on the closure Ū of U which is harmonic in U and coincides with f on U*. A set U is termed regular if there exists a solution of the Dirichlet problem, for any continuous function f on U* and, besides that, it is non-negative if f is. Of course, not every open bounded set in R ⁿ is regular. There exist continuous functions on such sets for which, we cannot solve the Dirichlet problem. Nevertheless, we can assign to those functions something like a solution in a reasonable way. If we denote for a continuous function f on U* by H ^U_f . the infimum of all superharmonic functions on U whose limes inferior is at every boundary point z greater or equal to f(z), then H ^U_f is a harmonic function on U and it is called a generalized solution of the Dirichlet problem for f obtained by the Perron method. Briefly, we shall call H ^U_f the Perron solution of f. A point zed is called a regular boundary point of U if for any continuous function f on U*. The remaining points of U are termed irregular.

ArticleNote

This paper is an expanded version of a communication submitted for publication in Comment. Math. Univ. Carolinae 14, 773–778 (1973).