Symmetries in an overdetermined problem for the Green's function

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August 2011, 4(4): 791-800. doi: 10.3934/dcdss.2011.4.791

Symmetries in an overdetermined problem for the Green's function

Virginia Agostiniani ^1, and Rolando Magnanini ^2,

1.	SISSA, via Bonomea 265, 34136 Trieste, Italy
2.	Dipartimento di Matematica "U. Dini", Università degli Studi di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy

Received September 2009 Revised January 2010 Published November 2010

Abstract
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We consider in the plane the problem of reconstructing a domain from the normal derivative of its Green's function with pole at a fixed point in the domain. By means of the theory of conformal mappings, we obtain existence, uniqueness, (non-spherical) symmetry results, and a formula relating the curvature of the boundary of the domain to the normal derivative of its Green's function.

Keywords: Overdetermined boundary values problems, Green's function, symmetries..

Mathematics Subject Classification: Primary: 35N25; Secondary: 35J08, 35B0.

Citation: Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

References:

[1]	G. Alessandrini and E. Rosset, Symmetry of singular solutions of degenerate quasilinear elliptic equations, Rend. Sem. Mat. Univ. Trieste, 39 (2007), 1-8.
[2]	P. L. Duren, "Theory of $H^p$ Spaces," Academic Press, New York, 1970.
[3]	P. L. Duren, "Univalent Functions," Springer-Verlag, New York, 1983.
[4]	L. E. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems," Cambridge University Press, Cambridge, 2000.
[5]	G. M. Goluzin, "Geometric Theory of Functions of a Complex Variable," American Mathematical Society, Providence, 1969.
[6]	B. Gustafsson and A. Vasil'ev, "Conformal and Potential Analysis in Hele-Shaw Cells," Birkhäuser Verlag, Basel, 2006.
[7]	P. Koosis, "Introduction to $H_p$ Spaces," Cambridge University Press, Cambridge, 1998.
[8]	J. L. Lewis and A. Vogel, On some almost everywhere symmetry theorems, in "Progr. Nonlinear Differential Equations Appl.," 7, Birkhäuser Boston, Massachusetts, (1992), 347-374.
[9]	A. I. Markushevich, "Theory of Functions of a Complex Variable," Prentice-Hall, Englewood Cliffs, 1965.
[10]	L. E. Payne and P. W. Schaefer, Duality theorems in some overdetermined boundary value problems, Math. Meth. Appl. Sci., 11 (1989), 805-819. doi: doi:10.1002/mma.1670110606.
[11]	J. Privalov, Sur les fonctions conjuguées, Bulletin de la S. M. F., 44 (1916), 100-103.
[12]	M. Sakai, "Quadrature Domains," Springer-Verlag, Berlin, 1982.
[13]	J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. doi: doi:10.1007/BF00250468.
[14]	H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319-320. doi: doi:10.1007/BF00250469.

show all references

References:

[1]	G. Alessandrini and E. Rosset, Symmetry of singular solutions of degenerate quasilinear elliptic equations, Rend. Sem. Mat. Univ. Trieste, 39 (2007), 1-8.
[2]	P. L. Duren, "Theory of $H^p$ Spaces," Academic Press, New York, 1970.
[3]	P. L. Duren, "Univalent Functions," Springer-Verlag, New York, 1983.
[4]	L. E. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems," Cambridge University Press, Cambridge, 2000.
[5]	G. M. Goluzin, "Geometric Theory of Functions of a Complex Variable," American Mathematical Society, Providence, 1969.
[6]	B. Gustafsson and A. Vasil'ev, "Conformal and Potential Analysis in Hele-Shaw Cells," Birkhäuser Verlag, Basel, 2006.
[7]	P. Koosis, "Introduction to $H_p$ Spaces," Cambridge University Press, Cambridge, 1998.
[8]	J. L. Lewis and A. Vogel, On some almost everywhere symmetry theorems, in "Progr. Nonlinear Differential Equations Appl.," 7, Birkhäuser Boston, Massachusetts, (1992), 347-374.
[9]	A. I. Markushevich, "Theory of Functions of a Complex Variable," Prentice-Hall, Englewood Cliffs, 1965.
[10]	L. E. Payne and P. W. Schaefer, Duality theorems in some overdetermined boundary value problems, Math. Meth. Appl. Sci., 11 (1989), 805-819. doi: doi:10.1002/mma.1670110606.
[11]	J. Privalov, Sur les fonctions conjuguées, Bulletin de la S. M. F., 44 (1916), 100-103.
[12]	M. Sakai, "Quadrature Domains," Springer-Verlag, Berlin, 1982.
[13]	J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. doi: doi:10.1007/BF00250468.
[14]	H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319-320. doi: doi:10.1007/BF00250469.

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