On the parabolic Lipschitz approximation of parabolic uniform rectifiable sets

Abstract

Abstract. We prove the existence of big pieces of regular parabolic Lipschitz graphs for a class of parabolic uniform rectifiable sets satisfying what we call a synchronized two cube condition. An application to the fine properties of parabolic measure is given.2000 Mathematics Subject Classification.

A Weak Reverse Hölder Inequality for Caloric Measure

“…A sketch of the proof is as follows. One first invokes the deep fact proved in [NS,Theorem 1.2] that under the hypotheses of Theorem 5.1, one obtains an interior "big pieces" approximation (see [NS] for the precise definition), analogous to that proved in the elliptic setting in [DJ], by domains of the sort considered in [LM]. By the result of [LM], plus a standard maximum principle argument, one obtains the (θ, β)-local ampleness condition 1.5.…”

Removable singularities for Lipschitz caloric functions in time varying domains

“…In the present paper we study removable singularities for regular .1; 1=2/-Lipschitz solutions of the heat equation in time varying domains. The parabolic theory in time varying domains is an area that has experienced a lot of activity in the last years, with fundamental contributions by Hofmann, Lewis, Murray, Nyström, Silver, and Strömqvist [6], [7], [8], [9], [10], [12], [13], [19].…”

“…/ < 1 and k@ 1=2 t f k ; ;p < 1 satisfying the heat equation in n E, also satisfies the heat equation in the whole . Functions satisfying (1.2) are called regular .1; 1=2/-Lipschitz in the literature (see [19], for example). So perhaps it would be more precise to talk about regular .1; 1=2/-Lipschitz removability.…”

Parabolic NTA domains in $\mathbb R^2$

“…The boundary behavior of caloric functions has been studied in parabolic Lipschitz domains (see [FGS84], [Bro89] and [LM95]) and parabolic Reifenberg flat domains (see [HLN04] and [Eng17]) but for arbitrary parabolic NTA domains it is unknown, for example, if caloric measure is always doubling (as the domain may fail to "separate" space in the sense of equation (1.1) in [HLN04]). Similarly, the relationship between the topological constraint of being parabolic NTA and uniform rectifiability (in the parabolic sense) is still being investigated (for some important recent work in this direction, see [NS15]). Our Theorem 2.2 is a step towards understanding the geometry of parabolic NTA domains.…”