Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2012-05462-6
Published electronically: January 20, 2012
MathSciNet review: 2888214
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. D. Aberra and T. Savina, The Schwarz reflection principle for polyharmonic functions in $ \mathbb{R}^2$, Complex Var. Theory Appl., 41 (2000), no. 1, 27-44. MR 1758596 (2001a:31002)
  • 2. B.P. Belinskiy and T.V. Savina, The Schwarz reflection principle for harmonic functions in $ \mathbb{R}^2$ subject to the Robin condition, J. Math. Anal. Appl., 348 (2008), 685-691. MR 2445769 (2009h:31001)
  • 3. J. Bramble, Continuation of biharmonic functions across circular arcs, J. Math. Mech., 7 (1958), No. 6, 905-924. MR 0100180 (20:6614)
  • 4. L. Carroll, Through the Looking-glass, in: Alice in Wonderland, Wordsworth Edition, 1995.
  • 5. D. Colton and R.P. Gilbert, Singularities of solutions to elliptic partial differential equations with analytic coefficients, Q. J. Math. Oxford Ser. (2) 19 (1968), 391-396. MR 0237929 (38:6206)
  • 6. Ph. Davis, The Schwarz function and its applications, Carus Mathematical Monographs, No. 17, The Mathematical Association of America, 1974. MR 0407252 (53:11031)
  • 7. R.J. Duffin, Continuation of biharmonic functions by reflection, Duke Math. J., 22 (1955), No. 2, 313-324. MR 0079105 (18:29e)
  • 8. P. Ebenfelt and D. Khavinson, On point to point reflection of harmonic functions across real analytic hypersurfaces in $ \mathbb{R}^n$, J. d´Analyse Mathématique, 68 (1996), 145-182. MR 1403255 (97i:31001)
  • 9. P. Ebenfelt, Holomorphic extension of solutions of elliptic partial differential equations and a complex Huygens principle, J. London Math. Soc., 55 (1997), 87-104. MR 1423288 (98g:35029)
  • 10. R. Farwig, A note on a reflection principle for the biharmonic equation and the Stokes system, Acta Appl. Math., 37 (1994), 41-51. MR 1308744 (95k:35159)
  • 11. P.R. Garabedian, Partial differential equations with more than two independent variables in the complex domain, J. Math. Mech., 9 (1960), 241-271. MR 0120441 (22:11195)
  • 12. P.R. Garabedian, Partial differential equations, John Wiley and Sons, Inc., 1964. MR 0162045 (28:5247)
  • 13. J. Hadamard, Lectures on Cauchy's problem in linear partial differential equations, Yale University Press, New Haven, 1923.
  • 14. F. John, Continuation and reflection of solutions of partial differential equations, Bull. Amer. Math. Soc., 63 (1957), 327-344. MR 0089332 (19:653e)
  • 15. F. John, Plane waves and spherical means applied to partial differential equations, Springer-Verlag, New York-Berlin, 1981. MR 614918 (82e:35001)
  • 16. F. John, The fundamental solution of linear elliptic differential equations with analytic coefficients, Comm. Pure and Appl. Math., 2 (1950), 213-304. MR 0042030 (13:40h)
  • 17. D. Khavinson, Holomorphic partial differential equations and classical potential theory, Universidad de La Laguna, 1996. MR 1392698 (97i:35005)
  • 18. D. Khavinson and H.S. Shapiro, Remarks on the reflection principles for harmonic functions, J. d´Analyse Mathématique, 54 (1991), 60-76. MR 1041175 (91b:31006)
  • 19. H. Lewi, On the reflection laws of second order differential equations in two independent variables, Bull. Amer. Math. Soc., 65 (1959), 37-58. MR 0104048 (21:2810)
  • 20. R.R. López, On reflection principles supported on a final set, J. Math. Anal. Appl., 351 (2009), 556-566. MR 2473961 (2009m:35068)
  • 21. D. Ludwig, Exact and Asymptotic solutions of the Cauchy problem, Comm. Pure Appl. Math., 13 (1960), 473-508. MR 0115010 (22:5816)
  • 22. H. Poritsky, Application of analytic functions to two-dimensional biharmonic analysis, Trans. Amer. Math. Soc., 59 (1946), No. 2, 248-279. MR 0015630 (7:449b)
  • 23. T.V. Savina, B.Yu. Sternin and V.E. Shatalov, On a reflection formula for the Helmholtz equation, J. Comm. Techn. Electronics, 38 (1993), no. 7, 132-143.
  • 24. T.V. Savina, B.Yu. Sternin and V.E. Shatalov, On the reflection law for the Helmholtz equation, Dokl. Math., 45 (1992), no. 1, 42-45. MR 1158949 (93c:35026)
  • 25. T.V. Savina, A reflection formula for the Helmholtz equation with the Neumann condition, Comput. Math. Math. Phys. 39 (1999), no. 4, 652-660. MR 1691388 (2000c:35017)
  • 26. T.V. Savina, On splitting up singularities of fundamental solutions to elliptic equations in $ \mathbb{C}^2$, Cent. Eur. J. Math., 5 (2007), no. 4, 733-740. MR 2342283 (2008i:32054)
  • 27. T.V. Savina, On the dependence of the reflection operator on boundary conditions for biharmonic functions, J. Math. Anal. Appl., 370 (2010), 716-725. MR 2651690 (2011f:31007)
  • 28. H.S. Shapiro, The Schwarz function and its generalization to higher dimensions, John Wiley and Sons, Inc., 1992. MR 1160990 (93g:30059)
  • 29. I.N. Vekua, New methods for solving elliptic equations, North-Holland, 1967. MR 0212370 (35:3243)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35J15, 32D15

Retrieve articles in all journals with MSC (2010): 35J15, 32D15


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9947-2012-05462-6
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

American Mathematical Society