The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition

Abstract

We establish L p , 2 ≤ p ≤ ∞ solvability of the Dirichlet boundary value problem for a parabolic equation ut − div(A∇u) − B · ∇u = 0 on time-varying domains with coefficient matrices A = [a ij ] and B = [b i ] that satisfy a small Carleson condition. The results are sharp in the following sense. For a given value of 1 < p < ∞ there exists operators that satisfy Carleson condition but fail to have L p solvability of the Dirichlet problem. Thus the assumption of smallness is sharp. Our results complements results of [18,31,32] where solvability of parabolic L p (for some large p) Dirichlet boundary value problem for coefficients that satisfy large Carleson condition was established. We also give a new (substantially shorter) proof of these results.In this metric, we consider the distance function δ of a point (X, t) to the boundary ∂Ω δ(X, t) = inf (Y,τ )∈∂Ω d[(X, t), (Y, τ )].

Parabolic Regularity and Dirichlet boundary value problems

“…We state a version of the maximum principle from [DH16] that is a modification of Lemma 3.38 from [HL01].…”

“…In particular in this paper the authors has solved the L 2 Dirichlet problem for the heat equation in graph domains of Lewis-Murray type. A related class of localised domains in which parabolic boundary value problems are solvable was considered in [Riv14] as well as in [DH16,DPP16]. The paper [DH16] has established L p solvability for parabolic Dirichlet problem under assumption that the coefficients satisfy certain natural small Carleson condition which also appears for elliptic PDEs.…”

“…A related class of localised domains in which parabolic boundary value problems are solvable was considered in [Riv14] as well as in [DH16,DPP16]. The paper [DH16] has established L p solvability for parabolic Dirichlet problem under assumption that the coefficients satisfy certain natural small Carleson condition which also appears for elliptic PDEs. The second paper [DPP16] finds sufficient and necessary condition for the parabolic measure to be A ∞ with respect to the parabolic analogue of the surface measure.…”

BMO solvability and the $A_∞$ condition for second order parabolic operators

“…As in [9], the results here are formulated for the class of admissible parabolic domains, which are, in effect, bounded time-varying domains that are "locally" of Lewis-Murray type. A related, but smaller, class of localized domains in which parabolic boundary value problems are solvable was considered in [31].…”

“…It is a fact that the parabolic PDE (2.12) with continuous boundary data is uniquely solvable (c.f. discussion under Definition 2.7 in [9]) and that there exists a a measure ω (X,t) such that (1.3) u(X, t) = ∂Ω f (y, s)dω (X,t) (y, s) for all continuous data, called the parabolic measure. Under the assumption of Definition 2.2, this measure is doubling (c.f.…”