Several authors have found the characteristic mean value formula for temperatures over heat spheres. Those who derived a corresponding formula over heat balls have all chosen different mean values. In this paper we discuss an infinity of possible means over heat balls, and show that, in the wider context of subtemperatures, some are more desirable than others.
Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip . We prove a representation theorem for an arbitrary function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of...
We present simple elementary proofs of several theorems about temperatures and subtemperatures. Most of these are concerned with mean values over heat spheres, heat balls, and modified heat balls, with applications to proving Harnack theorems and the monotone approximation of subtemperatures by smooth subtemperatures.
We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set whose supports are compact polar subsets of . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set , we prove...
A generalization of Nevanlinna’s First Fundamental Theorem to superharmonic functions on Green balls is proved. This enables us to generalize many other theorems, on the behaviour of mean values of superharmonic functions over Green spheres, on the Hausdorff measures of certain sets, on the Riesz measures of superharmonic functions, and on differences of positive superharmonic functions.
Let be a -subharmonic function with associated measure , and let be a superharmonic function with associated measure , on an open set . For any closed ball , of centre and radius , contained in , let denote the mean value of over the surface of the ball. We prove that the upper and lower limits as with of the quotient , lie between the upper and lower limits as of the quotient . This enables us to use some well-known measure-theoretic results to prove new variants and generalizations...
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