Wiener's criterion at $\infty$ for the heat equation

Abstract

This paper introduces a notion of regularity (or irregularity) of the point at infinity ($\infty$) for the unbounded open set $\Omega\subset \mathbb R ^{N+1}$ concerning the heat equation, according as whether the parabolic measure of $\infty$ is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the parabolic Dirichlet problem in arbitrary unbounded open subset of $ \mathbb R ^{N+1}$ is established. It is expressed in terms of the Wiener's criterion for the regularity of $\infty$. A geometric iterated logarithm test for the well-posedness of the parabolic Dirichlet problem in arbitrary open subset of $ \mathbb R ^{N+1}$ ($N \geq 2$) is proved. A domain is produced for which the parabolic Dirichlet problem always has a unique bounded solution for the heat equation $u_t=\Delta u$, and infinitely many for the equation $u_t=(1-\epsilon)\Delta u$ for all $0 < \epsilon < 1$.