On non-local reflection for elliptic equations of the second order in ℝ² (the Dirichlet condition)

Savina, Tatiana

doi:10.1090/S0002-9947-2012-05462-6

On non-local reflection for elliptic equations of the second order in $\mathbb {R}^2$ (the Dirichlet condition)
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Trans. Amer. Math. Soc. 364 (2012), 2443-2460 Request permission

Abstract:

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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