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 H¨older inequality on an open set ? Rn+1, assuming as a background hypothesis only that the essential boundary of ? satisfies an appropriate parabolic version of Ahlfors-David regularity (which entails some backwards in time thickness). We then show that the weak reverse H "older estimate is equivalent to solvability of the initial Dirichlet problem with "lateral"data in [Lp], for some p< ?. 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 ? 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 ?. By a result of Fabes, Garofalo and Lanconelli [FGL], this includes the case of [C1]-Dini coefficients.