The Green Function for Elliptic Systems in Two Dimensions

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

Research output: Contribution to journalArticle

7 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 1

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Elliptic Systems
Green's function
Two Dimensions
Boundary conditions
Fundamental Solution
BMO Space
Lipschitz Domains
Mixed Boundary Conditions
Remainder
Neumann Boundary Conditions
Accessibility
Dirichlet Boundary Conditions
Dirichlet
Divergence
Mathematical operators
Coefficient
Operator

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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The Green Function for Elliptic Systems in Two Dimensions. / Taylor, J. L.; Kim, S.; Brown, R. M.

In: Communications in Partial Differential Equations, Vol. 38, No. 9, 01.09.2013, p. 1574-1600.

Research output: Contribution to journalArticle

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