Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients

Abstract

We consider parabolic operators of the formWe assume that A is a (n + 1) × (n + 1)-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate x n+1 as well as of the time coordinate t. We prove that the boundedness of associated single layer potentials, with data in L 2 , can be reduced to two crucial estimates (Theorem 1.1), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two crucial estimates in the case of real, symmetric operators (Theorem 1.2). As part of this argument we establish a scale-invariant reverse Hölder inequality for the parabolic Poisson kernel (Theorem 1.3). Our results are important when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator+ , and by way of layer potentials. Mathematics Subject Classification 35K20 · 31B10Communicated by Y. Giga.

Homogenization of a parabolic Dirichlet problem by a method of Dahlberg

“…Hence, it is enough to show that in any unit neighborhood N := Q 1 (x, t) × (λ − 1, λ + 1) of each (x, t, λ) ∈ S, the Dirichlet problem for H 2 | N is solvable. When λ ≤ 1, A 2 (x, λ) does not depend on λ, and in this situation the solvability has been established previously in [3] and [16]. At unit distance from the remaining part of the boundary, A 2 = I, for which the solvability is well known.…”

The Dirichlet problem for second order parabolic operators in divergence form

“…Very recently, there have been advances in the theory of boundary value problems for second order parabolic equations (and systems) of the form Hu := ∂ t u − div λ,x A(x, t)∇ λ,x u = 0, (1.1) in the upper-half parabolic space R n+2 + := {(λ, x, t) ∈ R × R n × R : λ > 0}, n ≥ 1, with boundary determined by λ = 0, assuming only bounded, measurable, uniformly elliptic and complex coefficients. In [6,26,27], the solvability for Dirichlet, regularity and Neumann problems with L 2 -data were established for the class of parabolic equations (1.1) under the additional assumptions that the elliptic part is also independent of the time variable t and that it has either constant (complex) coefficients, real symmetric coefficients, or small perturbations thereof. Focusing on parabolic measure, a particular consequence of Theorem 1.3 in [6] is the generalization of [13] to equations of the form (1.1) but with A real, symmetric and time-independent.…”

“…In [6,26,27], the solvability for Dirichlet, regularity and Neumann problems with L 2 -data were established for the class of parabolic equations (1.1) under the additional assumptions that the elliptic part is also independent of the time variable t and that it has either constant (complex) coefficients, real symmetric coefficients, or small perturbations thereof. Focusing on parabolic measure, a particular consequence of Theorem 1.3 in [6] is the generalization of [13] to equations of the form (1.1) but with A real, symmetric and time-independent. This analysis was advanced further in [4], where a first order strategy to study boundary value problems of parabolic systems with second order elliptic part in the upper half-space was developed.…”

“…Note that in [18] the result of Dahlberg was proved for elliptic measure associated to the elliptic counterpart of (1.1) with symmetric A, that is, in this case the associated Poisson kernel exists and belongs to B 2 . In contrast, in the parabolic case it is not clear if such a result holds true if we allow for time-dependent coefficients (the case of time-independent coefficients was treated in [6] and does give B 2 ). Theorem 1.5 generalizes immediately to the setting of time-independent Lipschitz domains in the following sense.…”

“…. Let us recall that H and H * admit the following hidden coercivity used systematically in [4,6,26,27]. In fact, it appeared before in [17].…”