Convexity of mean values of non-negative subtemperatures

Abstract

Convexity properties of hyperplane mean values of subharmonic functions have been studied over many years by several authors, including HARDY, INGHAM & POLYA [7], KURAN [10, 11, 12], FLETT [6], NUALTARANEE [13] and BRAWN [1, 3, 4]. Subtemperatures are the analogues of subharmonic functions for the heat equation, and their basic properties can be found in [18, 20]. A study of mean values of subtemperatures over hyperplanes orthogonal to the t-axis was made in [19], where it was shown that these mean values need not even be continuous, much less convex. However, in the present paper we show that if we take hyperplanes orthogonal to one of the other axes, then we do get convexity. Indeed, we still get convexity if, instead of integrating over the entire hyperplane, we integrate only up to some fixed final time t 0. As an application of our convexity theorem, we consider certain aspects of a pure boundary value problem for a half-space, that is, a problem with no initial conditions in which the solution exists for all time. Such a problem was studied by JONES [9] in connection with Lipschitz spaces of functions having mixed homogeneity. A special case of the problem also occurs in the study of temperatures on a semi-infinite strip, since if the boundary values vanish for t <0, then the solution can be viewed as a solution of the mixed problem for 0 < t < t o with initial values at t = 0 identically zero. Temperatures on a twodimensional semi-infinite strip were studied by WIDDER [21]. We use the convexity of the hyperplane means to show that the convergence of the solution to its boundary values cannot be too rapid unless those values are degenerate in some way. We use R" to denote n-dimensional real euclidean space. It is convenient to work in 1R "+ 2, and therefore to admit the possibility that n =0. For much of this paper the interpretation of the results in the case n = 0 is obvious, but where it is not so the appropriate forms are given. A typical point of IR "+2 is written (x, t, y), where t e ~ , y e ~ and x =(x 1 . . . . , x,)EIR". The heat equation takes the form ~ 02u 02u ~u