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

Alejandro J. Castro, Kaj Nyström, Olow Sande

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

We consider parabolic operators of the form ∂ t+L, L:= -div A(X,t) ∇, Rn+2 ++ := {(X, t) = (x, xn+1,t) ∈ Rn × R × R : xn+1 > 0}, n≥ 1. We 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 xn + 1 as well as of the time coordinate t. We prove that the boundedness of associated single layer potentials, with data in L2, 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 ∂t+ L in R+n+2, with L2-data on Rn+1=∂R+n+2, and by way of layer potentials.

Original languageEnglish
Article number124
Number of pages49
JournalCalculus of Variations and Partial Differential Equations
Volume55
Issue number5
DOIs
Publication statusPublished - Oct 1 2016
Externally publishedYes

Keywords

  • 31B10
  • 35K20

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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