Zalcman, Lawrence Mean values and differential equations. (English) Zbl 0263.35013 Isr. J. Math. 14, 339-352 (1973). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 49 Documents MSC: 35G05 Linear higher-order PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 31B99 Higher-dimensional potential theory × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Aczél, J.; Haruki, H.; McKiernan, M. A.; Sakovič, G. N., General and regular solutions of functional equations characterizing harmonic polynomials, Aequationes Math., 1, 37-53 (1968) · Zbl 0157.46102 · doi:10.1007/BF01817556 [2] Arsove, M. G., The Looman-Menchoff theorem and some subharmonic function analogues, Proc. Amer. Math. Soc., 6, 94-105 (1955) · Zbl 0065.33601 · doi:10.2307/2032660 [3] Baker, J. A., An analogue of the wave equation and certain related equations, Canad. Math. Bull., 12, 837-846 (1969) · Zbl 0187.09103 [4] Beckenbach, E. F.; Reade, M., Mean values and harmonic polynomials, Trans. Amer. Math. Soc., 53, 230-238 (1943) · Zbl 0063.00272 · doi:10.2307/1990344 [5] Beckenbach, E. F.; Reade, M., Regular solids and harmonic polynomials, Duke Math. J., 12, 629-644 (1945) · Zbl 0063.00270 · doi:10.1215/S0012-7094-45-01255-5 [6] C. A. Berenstein and M. A. Dostal,Analytically Uniform Spaces and their Applications to Convolution Equations, Springer-Verlag, 1972. · Zbl 0237.47025 [7] Bourget, J., Mémoire sur le mouvement vibratoire des membranes circulaires, Ann. Sci. Ecole Norm. Sup., 3, 55-95 (1866) [8] Bramble, J. H.; Payne, L. E., Mean value theorems for polyharmonic functions, Amer. Math. Monthly, 73, 124-127 (1966) · Zbl 0139.06303 · doi:10.2307/2313762 [9] Brödel, W., Funktionen mit Gaussischer Mittelwerteigenschaften für konvexe Kurven und Bereiche, Deutsche Mathematik, 4, 3-15 (1939) · Zbl 0020.21002 [10] Cheng, Min-Teh, On a theorem of Nicolesco and generalized Laplace operators, Proc. Amer. Math. Soc., 2, 77-86 (1951) · Zbl 0043.10901 · doi:10.2307/2032624 [11] Choquet, G.; Deny, J., Sur quelques propriétés de moyenne charactérstiques des fonctions harmoniques et polyharmoniques, Bull. Soc. Math. France, 72, 118-140 (1944) · Zbl 0063.00852 [12] R. Courant and D. Hilbert,Methods of Mathematical Physics, Vol. 2, Interscience, 1962. · Zbl 0099.29504 [13] L. Ehrenpreis,Fourier Analysis in Several Complex Variables, Interscience, 1970. · Zbl 0195.10401 [14] Fenyö, I., Remark on a paper of J. A. Baker, Aequationes Math., 8, 103-108 (1972) · Zbl 0254.39006 · doi:10.1007/BF01832740 [15] Flatto, L., Functions with a mean value property, J. Math. Mech., 10, 11-18 (1961) · Zbl 0097.30605 [16] Flatto, L., Functions with a mean value property II, Amer. J. Math., 85, 248-270 (1963) · Zbl 0145.37103 · doi:10.2307/2373214 [17] Flatto, L., Partial differential equations and difference equations, Proc. Amer. Math. Soc., 16, 858-863 (1965) · Zbl 0145.13301 · doi:10.2307/2035570 [18] Flatto, L., On polynomials characterized by a certain mean value property, Proc. Amer. Math. Soc., 17, 598-601 (1966) · Zbl 0145.28602 · doi:10.2307/2035374 [19] Friedman, A.; Littman, W., Functions satisfying the mean value property, Trans. Amer. Math. Soc., 102, 167-180 (1962) · Zbl 0103.32201 · doi:10.2307/1993885 [20] Garsia, A., A note on the mean value property, Trans. Amer. Math. Soc., 102, 181-186 (1962) · Zbl 0103.32202 · doi:10.2307/1993886 [21] Garsia, A. M.; Rodemich, E., On functions satisfying the mean value property with respect to a product measure, Proc. Amer. Math. Soc., 17, 592-594 (1966) · Zbl 0178.05004 · doi:10.2307/2035372 [22] L. Hörmander,Linear Partial Differential Operators, Springer-Verlag, 1964. · Zbl 0108.09301 [23] J. E. Littlewood,A Mathematican’s Miscellany, Methuen, 1953. · Zbl 0051.00101 [24] McKiernan, M. A., Boundedness on a set of positive measure and the mean value property characterizes polynomials on a space V^n, Aequations Math., 4, 31-36 (1970) · Zbl 0195.39602 · doi:10.1007/BF01817742 [25] M. Nicolesco,Les Fonctions Polyharmoniques, Hermann, 1936. · JFM 62.1302.01 [26] V. P. Palamodov,Linear Differential Operators with Constant Coefficients, Springer-Verlag, 1970. · Zbl 0191.43401 [27] Pizzetti, P., Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera, Rend. Lincei, ser. 5, 18, 182-185 (1909) · JFM 40.0455.01 [28] Pizzetti, P., Sul significato geometrico del secundo parametro differenziale di una funzione sopra una superficie qualunque, Rend. Lincei, ser. 5, 18, 309-316 (1909) · JFM 40.0664.01 [29] Poritsky, H., Generalizations of the Gauss law of the spherical mean, Trans. Amer. Math. Soc., 43, 199-225 (1938) · Zbl 0018.12604 · doi:10.2307/1990039 [30] L. Schwartz,Théorie des Distributions, Vols. 1 and 2, Hermann, 1957. · Zbl 0089.09601 [31] Shapiro, V. L., Sets of uniqueness for the vibrating string problem, Trans. Amer. Math. Soc., 141, 127-146 (1969) · Zbl 0216.12804 · doi:10.2307/1995094 [32] C. L. Siegel,Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys-math. Kl. 1929, no. 1. · JFM 56.0180.05 [33] Šwiatak, H., Criteria for the regularity of continuous and locally integrable solutions of a class of linear functional equations, Aequations Math, 6, 170-187 (1971) · Zbl 0231.39005 · doi:10.1007/BF01819749 [34] F. Trèves,Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, 1966. · Zbl 0164.40602 [35] B. L. van der Waerden,Modern Algebra, Vols. 1, and 2, Frederich Ungar, 1953, and 1950. [36] Volterra, V., Alcune osservazioni sopra proprietà atte ad individuare una funzione, Rend. Lincei, ser. 5, 18, 263-266 (1909) · JFM 40.0453.01 [37] Walsh, J. L., A mean value theorem for polynomials and harmonic polynomials, Bull. Amer. Math. Soc., 42, 923-930 (1936) · Zbl 0016.12302 · doi:10.1090/S0002-9904-1936-06468-2 [38] G. N. Watson,A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1962. [39] Weber, H., Ueber die Integration der partiellen Differentialgleichung: {ie352-1}, Math. Ann., 1, 1-36 (1869) · JFM 02.0217.01 · doi:10.1007/BF01447384 [40] Zalcman, L., Analyticity and the Pompeiu problem, Arch. Rat. Mech. Anal., 47, 237-254 (1972) · Zbl 0251.30047 · doi:10.1007/BF00250628 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.