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

Symmetries in an overdetermined problem for the Green's function

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

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.
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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