Layer potentials and boundary value problems for elliptic equations with complex L coefficients satisfying the small Carleson measure norm condition

Steve Hofmann, Svitlana Mayboroda, Mihalis Mourgoglou

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Abstract

We consider divergence form elliptic equations Lu:=∇{dot operator}(A∇u)=0 in the half space R+n+1:={(x,t)∈Rn×(0,∞)}, whose coefficient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy A(x, t)-A(x, 0) satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in Lp, Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for L, with data in Lp(Rn), BMO(Rn), and Cα(Rn), and solvability of the Neumann and Regularity problems, with data in the spaces Lp(Rn)/Hp(Rn) and L1p(Rn)/H1,p(Rn) respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials for the t-independent operator L0:=-∇{dot operator}(A({dot operator}, 0)∇).

Original languageEnglish (US)
Pages (from-to)480-564
Number of pages85
JournalAdvances in Mathematics
Volume270
DOIs
StatePublished - Jan 2 2015

Bibliographical note

Publisher Copyright:
© 2014 Elsevier Inc.

Keywords

  • Boundary value problems
  • Carleson measures
  • Complex coefficients
  • Elliptic operator
  • Layer potentials

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