Caloric Measure Null Sets

We give a systematic treatment of caloric measure null sets on the essential boundary $\partial_eE$ of an arbitrary open set $E$ in ${\bf R}$. We discuss two characterisations of such sets and present some basic properties. We investigate the dependence of caloric measure null sets on the open set $E$. Thus, if $D$ is an open subset of $E$ and $Z\subseteq\partial_eE\cap\partial_eD$, we show that $Z$ is caloric measure null for $D$ if it is caloric measure null for $E$. We also give conditions on $E$ and $Z$ which imply that the reverse implication is true. We know from \cite{watson2011} that any polar subset of $\partial_eD$ is caloric measure null for $D$, but the reverse implication is not generally true. In our final result we show that, for subsets of a certain component of $\partial_eD$, caloric measure null sets are necessarily polar.

Absolute continuity of parabolic measure and the initial-dirichlet problem

This thesis is devoted to the study of parabolic measure corresponding to a divergence form parabolic operator. We first extend to the parabolic setting a number of basic results that are well known in the elliptic case. Then following a result of Bennewitz-Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Holder inequality on an open set [omega] R(n+1), assuming as a background hypothesis only that the essential boundary of [omega] satisfies an appropriate parabolic version of Ahlfors-David regularity (which entails some backwards in time thickness). We then show that the weak reverse Holder estimate is equivalent to solvability of the initial Dirichlet problem with "lateral" data in [Lp], for some p less than [infinity]. Finally, we prove that for the heat equation, BMO-solvability implies scale invariant quantitative absolute continuity of caloric measure with respect to surface measure, in an open set [omega] with time-backwards ADR boundary. Moreover, the same results apply to the parabolic measure associated to a uniformly parabolic divergence form operator (L), with estimates depending only on dimension, the ADR constants, and parabolicity, provided that the continuous Dirichlet problem is solvable for (L) in [omega]. By a result of Fabes, Garofalo and Lanconelli [FGL], this includes the case of [C1]-Dini coefficients.

CALORIC MEASURE FOR ARBITRARY OPEN SETS

We give a systematic treatment of caloric measure for arbitrary open sets. The caloric measure is defined only on the essential boundary of the set. Our main result gives criteria for the resolutivity of essential boundary functions, and their integral representation in terms of caloric measure. We also characterize the caloric measure null sets in terms of the boundary singularities of nonnegative supertemperatures.

Regularity below the continuous threshold in a two-phase parabolic free boundary problem

In this paper we study free boundary regularity in a parabolic two-phase problem below the continuous threshold. We consider unbounded domains $\Omega\subset\mathsf{R}^{n+1}$ assuming that $\partial\Omega$ separates $\mathsf{R}^{n+1}$ into two connected components $\Omega^1=\Omega$ and $\Omega^2=\mathsf{R}^{n+1}\setminus\overline\Omega$. We furthermore assume that both $\Omega^1$ and $\Omega^2$ are parabolic NTA-domains, that $\partial\Omega$ is Ahlfors regular and for $i\in\{1,2\}$ we define $\omega^i(\hat{X}^i,\hat{t}^i,\cdot)$ to be the caloric measure at $(\hat{X}^i,\hat{t}^i)\in \Omega^i$ defined with respect to $\Omega^i$. In the paper we make the additional assumption that $\omega^i(\hat{X}^i,\hat{t}^i,\cdot)$, for $i\in\{1,2\}$, is absolutely continuous with respect to an appropriate surface measure $\sigma$ on $\partial\Omega$ and that the Poisson kernels $k^i(\hat{X}^i,\hat{t}^i,\cdot)=d\omega^i(\hat{X}^i,\hat{t}^i,\cdot)/d\sigma$ are such that $\log k^i(\hat{X}^i,\hat{t}^i,\cdot)\in \mathrm{VMO}(d\sigma)$. Our main result (Theorem 1) states that, under these assumptions, $C_r(X,t)\cap\partial\Omega$ is Reifenberg flat with vanishing constant whenever $(X,t)\in\partial\Omega$ and $\min\{\hat{t}^1,\hat{t}^2\}>t+4r^2$. This result has an important consequence (Theorem 3) stating that if the two-phase condition on the Poisson kernels is fulfilled, $\Omega^1$ and $\Omega^2$ are parabolic NTA-domains and $\partial\Omega$ is Ahlfors regular then if $\Omega$ is close to being a chord arc domain with vanishing constant we can in fact conclude that $\Omega$ is a chord arc domain with vanishing constant.