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 & 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.

[2]

P. L. Duren, "Theory of $H^p$ Spaces,", Academic Press, (1970).

[3]

P. L. Duren, "Univalent Functions,", Springer-Verlag, (1983).

[4]

L. E. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,", Cambridge University Press, (2000).

[5]

G. M. Goluzin, "Geometric Theory of Functions of a Complex Variable,", American Mathematical Society, (1969).

[6]

B. Gustafsson and A. Vasil'ev, "Conformal and Potential Analysis in Hele-Shaw Cells,", Birkh\, (2006).

[7]

P. Koosis, "Introduction to $H_p$ Spaces,", Cambridge University Press, (1998).

[8]

J. L. Lewis and A. Vogel, On some almost everywhere symmetry theorems,, in, 7 (1992), 347.

[9]

A. I. Markushevich, "Theory of Functions of a Complex Variable,", Prentice-Hall, (1965).

[10]

L. E. Payne and P. W. Schaefer, Duality theorems in some overdetermined boundary value problems,, Math. Meth. Appl. Sci., 11 (1989), 805. doi: doi:10.1002/mma.1670110606.

[11]

J. Privalov, Sur les fonctions conjuguées,, Bulletin de la S. M. F., 44 (1916), 100.

[12]

M. Sakai, "Quadrature Domains,", Springer-Verlag, (1982).

[13]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. doi: doi:10.1007/BF00250468.

[14]

H. F. Weinberger, Remark on the preceding paper of Serrin,, Arch. Rational Mech. Anal., 43 (1971), 319. 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.

[2]

P. L. Duren, "Theory of $H^p$ Spaces,", Academic Press, (1970).

[3]

P. L. Duren, "Univalent Functions,", Springer-Verlag, (1983).

[4]

L. E. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,", Cambridge University Press, (2000).

[5]

G. M. Goluzin, "Geometric Theory of Functions of a Complex Variable,", American Mathematical Society, (1969).

[6]

B. Gustafsson and A. Vasil'ev, "Conformal and Potential Analysis in Hele-Shaw Cells,", Birkh\, (2006).

[7]

P. Koosis, "Introduction to $H_p$ Spaces,", Cambridge University Press, (1998).

[8]

J. L. Lewis and A. Vogel, On some almost everywhere symmetry theorems,, in, 7 (1992), 347.

[9]

A. I. Markushevich, "Theory of Functions of a Complex Variable,", Prentice-Hall, (1965).

[10]

L. E. Payne and P. W. Schaefer, Duality theorems in some overdetermined boundary value problems,, Math. Meth. Appl. Sci., 11 (1989), 805. doi: doi:10.1002/mma.1670110606.

[11]

J. Privalov, Sur les fonctions conjuguées,, Bulletin de la S. M. F., 44 (1916), 100.

[12]

M. Sakai, "Quadrature Domains,", Springer-Verlag, (1982).

[13]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. doi: doi:10.1007/BF00250468.

[14]

H. F. Weinberger, Remark on the preceding paper of Serrin,, Arch. Rational Mech. Anal., 43 (1971), 319. doi: doi:10.1007/BF00250469.

[1]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001

[2]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307

[3]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[4]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[5]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487

[6]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098

[7]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767

[8]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657

[9]

Panos K. Palamides, Alex P. Palamides. Singular boundary value problems, via Sperner's lemma. Conference Publications, 2007, 2007 (Special) : 814-823. doi: 10.3934/proc.2007.2007.814

[10]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167

[11]

Olga A. Brezhneva, Alexey A. Tret’yakov, Jerrold E. Marsden. Higher--order implicit function theorems and degenerate nonlinear boundary-value problems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 293-315. doi: 10.3934/cpaa.2008.7.293

[12]

J. Cruz-Sampedro. Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1061-1076. doi: 10.3934/dcds.2013.33.1061

[13]

X. X. Huang, D. Li, Xiaoqi Yang. Convergence of optimal values of quadratic penalty problems for mathematical programs with complementarity constraints. Journal of Industrial & Management Optimization, 2006, 2 (3) : 287-296. doi: 10.3934/jimo.2006.2.287

[14]

Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321

[15]

Alfonso Sorrentino. Computing Mather's $\beta$-function for Birkhoff billiards. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5055-5082. doi: 10.3934/dcds.2015.35.5055

[16]

Hongjie Dong, Seick Kim. Green's functions for parabolic systems of second order in time-varying domains. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1407-1433. doi: 10.3934/cpaa.2014.13.1407

[17]

Mourad Choulli. Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions. Evolution Equations & Control Theory, 2015, 4 (1) : 61-67. doi: 10.3934/eect.2015.4.61

[18]

Jiyoung Han, Seonhee Lim, Keivan Mallahi-Karai. Asymptotic distribution of values of isotropic here quadratic forms at S-integral points. Journal of Modern Dynamics, 2017, 11: 501-550. doi: 10.3934/jmd.2017020

[19]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091

[20]

Simon Hochgerner, Luis García-Naranjo. $G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball. Journal of Geometric Mechanics, 2009, 1 (1) : 35-53. doi: 10.3934/jgm.2009.1.35

2016 Impact Factor: 0.781

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]