On non-local reflection for elliptic equations of the second order in R^2 (the Dirichlet condition)

Details On non-local reflection for elliptic equations of the second order in R^2 (the Dirichlet condition) On non-local reflection for elliptic equations of the second order in R^2 (the Dirichlet condition)

2010-09-07

arxiv.org/abs/1009.1262v1

Mathematics

Complex Variables

25 pages, 2 figures

Scientific paper

Point-to-point reflection holding for harmonic functions subject to the Dirichlet or Neumann conditions on an analytic curve in the plane almost always fails for solutions to more general elliptic equations. We develop a non-local, point-to-compact set, formula for reflecting a solution of an analytic elliptic partial differential equation across a real-analytic curve on which it satisfies the Dirichlet conditions. We also discuss the special cases when the formula reduces to the point-to-point forms.

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