Exceptional boundary sets for solutions of parabolic partial differential inequalities

Abstract:

Let $\mathcal {M}$ be a second order, linear, parabolic partial differential operator with coefficients defined in a domain $\mathcal {D} = \Omega \times (0, T)$ in ${{\mathbf {R}}^n} \times {\mathbf {R}}$, with $\Omega$ a domain in ${{\mathbf {R}}^n}$. Let $u$ be a suitably regular real function in $\mathcal {D}$ such that $u$ is bounded below and $\mathcal {M}u$ is bounded above in $\mathcal {D}$. If $u \geqslant 0$ on $\Omega \times \{ 0\}$ except on a set $\Gamma \times \{ 0\}$, with $\Gamma$ a subset of $\Omega$ of suitably restricted Hausdorff dimension, then necessarily $u \geqslant 0$ also on $\Gamma \times \{ 0\}$. The allowable Hausdorff dimension of $\Gamma$ depends on the coefficients of $\mathcal {M}$. For example, if $\mathcal {M}$ is the heat operator $\Delta - \partial /\partial t$, the Hausdorff dimension of $\Gamma$ needs to be smaller than the number of space dimensions $n$. Analogous results are valid for exceptional boundary sets on the lateral boundary, $\partial \Omega \times (0, T)$, of $\mathcal {D}$.