The Green Function for Elliptic Systems in Two Dimensions

J. L. Taylor, S. Kim, R. M. Brown

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. Our main goal is construct the Green function for the operator with mixed boundary conditions in a Lipschitz domain. Thus we specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We require a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary conditions. Our proof proceeds by defining a variant of the space BMO(Ω) that is adapted to the boundary conditions and showing that the solution exists in this space. We also give a construction of the Green function with Neumann boundary conditions and the fundamental solution in the plane.

Original languageEnglish
Pages (from-to)1574-1600
Number of pages27
JournalCommunications in Partial Differential Equations
Volume38
Issue number9
DOIs
Publication statusPublished - 2013 Sep

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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