×

A duality theorem for an overdetermined eigenvalue problem. (English) Zbl 0842.35066

The overdetermined eigenvalue problem for the Laplacian \[ \Delta u + \lambda u = 0 \quad \text{in } \Omega, \qquad u = b, \;{\partial u \over \partial n} = c \quad \text{on } \partial \Omega \tag{P} \] is studied in a bounded domain \(\Omega\), for \(\lambda\), \(b > 0\). Call \(w\) a Helmholtz function of wave number \(\lambda\) in \(\Omega\), if \(w\) satisfies the equation \(\Delta w + \lambda w = 0\). The main result of this paper is:
There exists a solution \(u \in C^2 (\Omega) \cap C^1 (\overline \Omega)\) of (P) if and only if \(\int_\Omega wdx = d \int_{\partial \Omega} wd \sigma\) for all Helmholtz functions \(w\) of wave number \(\lambda\) \((d\) is an arbitrary fixed constant).

MSC:

35P05 General topics in linear spectral theory for PDEs
35R25 Ill-posed problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. Alessandrini,A symmetry theorem for condensors, Math. Meth. Appl. Sci.,15(5), 315-320 (1992). · Zbl 0756.35118 · doi:10.1002/mma.1670150503
[2] G. Alessandrini,Symmetry and non-symmetry for the overdetermined Stekloff eigenvalue problem, J. Appl. Math. Phys. (ZAMP),45(1), 44-52 (1994). · Zbl 0801.35017 · doi:10.1007/BF00942845
[3] Patricio Aviles,Symmetry theorems related to Pompeiu’s problem, Amer. J. Math.,108, 1023-1036 (1986). · Zbl 0644.35075 · doi:10.2307/2374594
[4] Allan Bennett,Symmetry in an overdetermined fourth order elliptic boundary value problem, SIAM J. Math. Anal.,17(6), 1354-1358 (1986). · Zbl 0612.35039 · doi:10.1137/0517095
[5] C. Berenstein,An inverse spectral theorem and its relation to the Pompeiu problem, J. d’Analyse Math.,37, 128-144 (1980). · Zbl 0449.35024 · doi:10.1007/BF02797683
[6] Marc A. Chamberland,The Pompeiu Problem and Schiffer’s Conjecture, PhD thesis, University of Waterloo, Waterloo, Ontario, Canada, December 1994.
[7] Bennett Chow and Robert Gulliver,Aleksandrov reflection and nonlinear evolution equations, I: The n-sphere and n-ball. Preprint, January 1994. · Zbl 0851.58041
[8] R. Courant and D. Hilbert,Methods of Mathematical Physics, Vol. I, Interscience, New York 1953. · Zbl 0051.28802
[9] Nicola Garofalo and John L. Lewis,A symmetry result related to overdetermined boundary value problems, Amer. J. Math.,111(1), 9-33 (1989). · Zbl 0681.35016 · doi:10.2307/2374477
[10] Nicola Garofalo and Fausto Segala,New results on the Pompeiu problem. Trans. Amer. Math. Soc.,325(1), 273-286 (1991). · Zbl 0737.35147 · doi:10.2307/2001671
[11] Nicola Garofalo and Fausto Segala,Another step toward the solution of the Pompeiu problem in the plane, Comm. P.D.E.’s.18(3&4), 491-503 (1993). · Zbl 0818.35136 · doi:10.1080/03605309308820938
[12] B. Gidas, Wei-Ming Ni, and L. Nirenberg.Symmetry and related properties via the maximum principle, Comm. Math. Phys.,68, 209-243 (1979). · Zbl 0425.35020 · doi:10.1007/BF01221125
[13] Bernhard Kawohl,Rearrangements and Convexity of Level Sets in PDF, vol 1150 ofLect. Notes in Math., Springer-Verlag, Berlin, Heidelberg 1985. · Zbl 0593.35002
[14] L. E. Payne,Some remarks on overdetermined systems in linear elasticity.J. Elasticity, 18, 181-189 (1987). · Zbl 0631.73003 · doi:10.1007/BF00127557
[15] L. E. Payne and G. Philippin,On two free boundary problems in potential theory, J. Math. Anal. Appl.,161, 332-342 (1991). · Zbl 0760.31003 · doi:10.1016/0022-247X(91)90333-U
[16] L. E. Payne and G. Philippin,Some overdetermined boundary value problems for harmonic functions. J. Appl. Math. Phys. (ZAMP),42, 865-873 (1991). · Zbl 0767.35047
[17] L. E. Payne and P. W. Schaefer,Duality theorems in some overdetermined boundary value problems, Math. Meth. in the Appl. Sci.,11, 805-819 (1989). · Zbl 0698.35051 · doi:10.1002/mma.1670110606
[18] L. E. Payne and P. W. Schaefer,Some nonstandard problems for the Poisson equation, Quarterly of Applied Mathematics,51(1), 81-90 (1993). · Zbl 0808.35024
[19] L. E. Payne and P. W. Schaefer,On overdetermined boundary value problems for the biharmonic operator, J. Math. Anal. Appl.,187, 598-616 (1994). · Zbl 0810.35020 · doi:10.1006/jmaa.1994.1377
[20] G. A. Philippin,On a free boundary problem in electrostatics, Math. Meth. Appl. Sci.,12, 387-392 (1990). · Zbl 0712.35031 · doi:10.1002/mma.1670120503
[21] D. Pompeiu,Sur certains systémes d’équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables, C. R. Acad. Sci. Paris,188, 1138-1139 (1929).
[22] Giovanni Porru and Aklilu Zeleke,An overdetermined problem in nonlinear elliptic equations of second order, J. Appl. Math. Phys. (ZAMP),44, 923-928 (1993). · Zbl 0801.35018 · doi:10.1007/BF00942817
[23] Murray H. Protter and Hans F. Weinberger,Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey 1967. · Zbl 0549.35002
[24] James Serrin,A symmetry problem in potential theory. Arch. Rat. Mech.,43, 304-318 (1971). · Zbl 0222.31007 · doi:10.1007/BF00250468
[25] W. C. Troy,Symmetry properties in systems of semilinear elliptic equations, J. Diff. Eq.,42, 400-413 (1981). · Zbl 0486.35032 · doi:10.1016/0022-0396(81)90113-3
[26] H. F. Weinberger,Remark on the preceding paper of Serrin, Arch. Rat. Mech. Anal.,43, 319-320 (1971). · Zbl 0222.31008 · doi:10.1007/BF00250469
[27] S. A. Williams,A partial solution to the Pompeiu problem, Math. Ann.,223, 183-190 (1976). · Zbl 0329.35045 · doi:10.1007/BF01360881
[28] Stephen A. Williams,Analyticity of the boundary for Lipschitz domains without the Pompeiu property, Indiana University Math. J.,30, 357-369 (1981). · Zbl 0461.35061 · doi:10.1512/iumj.1981.30.30028
[29] N. B. Willms and G. M. L. Gladwell,Saddle points and overdetermined problems for the Helmholtz equation, J. Appl. Math. Phys. (ZAMP),45(1), 1-26 (1994). · Zbl 0793.35065 · doi:10.1007/BF00942843
[30] N. B. Willms and H. F. Weinberger,A symmetry theorem for the buckling of a clamped plate, Preprint, February 1995.
[31] N. Brad Willms,A condition sufficient to prove Schiffer’s conjecture, Applicable Analysis, September 1993 (Accepted).
[32] N. B. Willms, G. M. L. Gladwell and D. Siegel,Symmetry theorems for some overdetermined boundary value problems on ring domains, J. Appl. Math. Phys. (ZAMP),45(4), 556-579 (1994). · Zbl 0807.35099 · doi:10.1007/BF00991897
[33] S. T. Yau, editor,Seminar on Differential Geometry, chapt. 34 (Problem Section), Annals of Mathematics Studies. Princeton University Press, 1982.
[34] L. Zalcman,A bibliographic survey of the Pompeiu problem, In B. Fuglede et al., editor,Approximation by Solutions of Partial Differential Equations, pp. 185-194. Kluwer, Dordrecht 1992. · Zbl 0830.26005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.