On non-local reflection for elliptic equations of the second order in $\mathbb {R}^2$ (the Dirichlet condition)
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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.References
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Additional Information
- Tatiana Savina
- Affiliation: Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701
- Received by editor(s): April 21, 2009
- Received by editor(s) in revised form: April 14, 2010
- Published electronically: January 20, 2012
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 364 (2012), 2443-2460
- MSC (2010): Primary 35J15; Secondary 32D15
- DOI: https://doi.org/10.1090/S0002-9947-2012-05462-6
- MathSciNet review: 2888214