Domains of uniqueness for parabolic equations

Abstract

as well as certain extensions of the well-known maximuim principle. In the usual type of boundary value problem, the domain of the solution lies in the upper half plane t >0 and part of its boundary lies on the hyperplane t = 0. The data prescribed on the latter part are referred to as initial data. The present paper, on the other hand, treats domains extending indefinitely in the t-* co direction. Thus instead of initial-value problems, we consider generalized steady-state (see [2]) or without initial conditions. A particular case of such a domain is the cylinder QX(oo, m), where Q is a bounded domain in x-space. Some existence and uniqueness questions for boundary value problems in such a domain were treated in a previous paper [2] by the author. Another special case is the complement of such a cylindrical domain; here some obvious results can be obtained by combining the maximum principle developed in our Theorem 4 with the technique of Meyers and Serrin [4]. (See also [3] for certain results in one space dimension when the solution is periodic in time.) In the present paper, however, the domain of the solution (which we hereafter denote by c) is generally allowed to be noncylindrical; the goal in fact is to obtain geometric properties of D which guarantee uniqueness within the class of bounded solutions. We shall usually require no regularity properties of the coefficients of L other than local boundedness, and for our general result (Theorem 3) we assume only that L is locally parabolic. We shall be concerned with functions satisfying Lu >0. The well-known standard weak maximum principle holds for such functions if the derivatives appearing in the operator are continuous; we shall designate these functions by the term subsolutions.