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An inverse spectral theorem and its relation to the Pompeiu problem. (English) Zbl 0449.35024

MSC:
35J25 Boundary value problems for second-order elliptic equations
35R30 Inverse problems for PDEs
35N99 Overdetermined problems for partial differential equations and systems of partial differential equations
35P99 Spectral theory and eigenvalue problems for partial differential equations
42B99 Harmonic analysis in several variables
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[1] S. Agmon, A. Douglis and L. Nirenberg,Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure. Appl. Math.12 (1959), 623–727. · Zbl 0093.10401 · doi:10.1002/cpa.3160120405
[2] C. A. Berenstein,On the converse to Pompeiu’s problem, Notas e CommunicacŌes de Mathemática, Univ. Fed. de Pernarabuco,73, 1976. · Zbl 0332.35033
[3] L. Brown, B. M. Schreiber, and B. A. Taylor,Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier23 (1973), 125–154. · Zbl 0265.46044
[4] N. de Bruijn,Asymptotic Methods in Analysis, North-Holland, 1961. · Zbl 0109.03502
[5] L. A. Cafarelli,The regularity of free boundaries in higher dimensions, Acta Math.139 (1977), 155–184. · Zbl 0386.35046 · doi:10.1007/BF02392236
[6] R. Courant and D. Hilbert,Methods of Mathematical Physics, Interscience, 1953. · Zbl 0051.28802
[7] C. Dafermos,Contraction Semigroups and Trend to Equilibrium in Continuum Mechanics, Springer Lecture Notes in Mathematics503 (1976), 295–306. · Zbl 0345.47032 · doi:10.1007/BFb0088765
[8] L. Ehrenpreis,Fourier Analysis in Several Complex Variables, Interscience, 1970. · Zbl 0195.10401
[9] D. Kinderlehrer and L. Nirenberg,Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa4 (1977), 373–391. · Zbl 0352.35023
[10] P. Nowosad, Operadores positivos e optimizacāo; aplicacŌes à energia nuclear, preprint, 1977.
[11] R. Osserman,The isoperimetric inequality, Bull. Amer. Math. Soc.84 (1974), 1182–1238. · Zbl 0411.52006 · doi:10.1090/S0002-9904-1978-14553-4
[12] L. E. Payne,Inequalities for eigenvalues of membranes and plates, J. Rat. Mech. Analysis4 (1955), 517–529. · Zbl 0064.34802
[13] L. E. Payne,Isoperimetric inequalities and their applications, SIAM Rev.9 (1967), 413–488. · Zbl 0154.12602 · doi:10.1137/1009070
[14] Lord Rayleigh,The Theory of Sound, MacMillan, 1877. · JFM 09.0656.03
[15] F. Rellich,Darstellung der Eigenwerte \(\delta\)u + \(\lambda\)u durch ein Randintegral, Math. Z.46 (1940), 635–646. · Zbl 0023.04204 · doi:10.1007/BF01181459
[16] J. Serrin,A symmetry problem in potential theory, Arch. Rational Mech. Anal.43 (1971), 304–318. · Zbl 0222.31007 · doi:10.1007/BF00250468
[17] L. A. Shepp and J. B. Kruskal,Computerized tomography: the new medical X-ray technology, Amer. Math. Monthly85 (1978), 420–439. · Zbl 0381.68079 · doi:10.2307/2320062
[18] K. T. Smith, D. C. Solmon and S. L. Wagner,Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc.83 (1977), 1227–1270. · Zbl 0521.65090 · doi:10.1090/S0002-9904-1977-14406-6
[19] R. Temam,A non-linear eigenvalue problem: The shape at equilibrium of a confined plasma. Arch. Rational Mech. Anal.60 (1975), 51–73. · Zbl 0328.35069 · doi:10.1007/BF00281469
[20] H. F. Weinberger,Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal.43 (1971), 319–320. · Zbl 0222.31008 · doi:10.1007/BF00250469
[21] S. A. Williams,A partial solution of the Pompeiu problem, Math. Ann.223 (1976), 183–190. · Zbl 0329.35045 · doi:10.1007/BF01360881
[22] L. Zalcman,Analyticity and the Pompeiu problem, Arch. Rational Mech. Anal.47 (1972), 237–254. · Zbl 0251.30047 · doi:10.1007/BF00250628
[23] L. Zalcman,Offbeat integral geometry, preprint, 1978. · Zbl 0433.53048
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