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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Tatiana Savina
Journal: Trans. Amer. Math. Soc. 364 (2012), 2443-2460
MSC (2010): Primary 35J15; Secondary 32D15
Published electronically: January 20, 2012
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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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Additional Information

Tatiana Savina
Affiliation: Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05462-6
PII: S 0002-9947(2012)05462-6
Keywords: Elliptic equations, reflection principle, analytic continuation.
Received by editor(s): April 21, 2009
Received by editor(s) in revised form: April 14, 2010
Published electronically: January 20, 2012
Article copyright: © Copyright 2012 American Mathematical Society