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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The Cauchy problem in $ {\bf C}\sp N$ for linear second order partial differential equations with data on a quadric surface


Author: Gunnar Johnsson
Journal: Trans. Amer. Math. Soc. 344 (1994), 1-48
MSC: Primary 35A20; Secondary 35B60, 35G10
MathSciNet review: 1242782
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Abstract: By means of a method developed essentially by Leray some global existence results are obtained for the problem referred to in the title. The partial differential equations are required to have constant principal part and the initial surface to be irreducible and not everywhere characteristic. The Cauchy data are assumed to be given by entire functions. Under these conditions the location of all possible singularities of solutions are determined. The sets of singularities can be divided into two types, K- and L-singularities. K, the set of K-singularities, is the global version of the characteristic tangent defined by Leray. The L-sets are here quadric surfaces which, in contrast to the Ksets, allow unbounded singularities. The L-sets are in turn divided into three types: initial, asymptotic, and latent singularities. The initial singularities appear when the characteristic points of the initial surface are exceptional according to Leray's local theory. These sets of singularity intersect the initial surface at characteristic points. The asymptotic case, where the set of singularities does not cut the initial surface, can be viewed as projectively equivalent to the initial case, the intersection taking place at infinite characteristic points. Finally the latent singularities are sets which intersect the initial surface, but where the solutions do not develop singularities initially. In the case of the Laplace equation with data on a real quadric surface it is shown that the K-singularities and the asymptotic singularities occur on the classical focal sets defined by Poncelet, Plücker, Darboux et al. There are also latent singularities appearing in coordinate subspaces of $ {\mathbb{R}^N}$. As a corollary a new proof is given of the fact that ellipsoids have the Pompeiu property.


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  • [B-S-T] Leon Brown, Bertram M. Schreiber, and B. Alan Taylor, Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 125–154 (English, with French summary). MR 0352492 (50 #4979)
  • [B-T] H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexer Veränderlichen, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 51. Zweite, erweiterte Auflage. Herausgegeben von R. Remmert. Unter Mitarbeit von W. Barth, O. Forster, H. Holmann, W. Kaup, H. Kerner, H.-J. Reiffen, G. Scheja und K. Spallek, Springer-Verlag, Berlin, 1970 (German). MR 0271391 (42 #6274)
  • [Be] Carlos Alberto Berenstein, An inverse spectral theorem and its relation to the Pompeiu problem, J. Analyse Math. 37 (1980), 128–144. MR 583635 (82b:35031), http://dx.doi.org/10.1007/BF02797683
  • [Da] Philip J. Davis, The Schwarz function and its applications, The Mathematical Association of America, Buffalo, N. Y., 1974. The Carus Mathematical Monographs, No. 17. MR 0407252 (53 #11031)
  • [Dar] G. Darboux, Principes de géométrie analytique, Gauthier-Villars, Paris, 1917.
  • [De] E. Delassus, Sur les équations linéaires aux dérivées partielles à caractéristiques réelles, Ann. Sci. Ecole Norm. Sup (3) 12 (1895).
  • [Du] Jacques Dunau, Un problème de Cauchy caractéristique, J. Math. Pures Appl. (9) 69 (1990), no. 3, 369–402 (French). MR 1070484 (91j:35004)
  • [E] Peter Ebenfelt, Singularities encountered by the analytic continuation of solutions to Dirichlet’s problem, Complex Variables Theory Appl. 20 (1992), no. 1-4, 75–91. MR 1284354 (95f:30059)
  • [G] L. Gårding, Partial differential equations : problems and uniformization in Cauchy's problem, Lectures on Modern Math., vol. II. (T. L. Sauty, ed.), Wiley, New York, 1964.
  • [G-K-L] Lars Gȧrding, Takeshi Kotake, and Jean Leray, Uniformisation et développement asymptotique de la solution du problème de Cauchy linéaire, à données holomorphes; analogie avec la théorie des ondes asymptotiques et approchées (Problème de Cauchy, I bis et VI), Bull. Soc. Math. France 92 (1964), 263–361 (French). MR 0196280 (33 #4472)
  • [Ga] F. R. Gantmacher, Applications of the theory of matrices, Translated by J. L. Brenner, with the assistance of D. W. Bushaw and S. Evanusa, Interscience Publishers, Inc., New York, 1959. MR 0107648 (21 #6372b)
  • [Gr-Fr] H. Grauert and K. Fritzsche, Several complex variables, Springer-Verlag, New York, 1976. Translated from the German; Graduate Texts in Mathematics, Vol. 38. MR 0414912 (54 #3004)
  • [Ha] Y. Hamada, Les singularités des solutions du problème de Cauchy à données holomorphes, Pitman Res. Notes Math. Ser., vol. 183, Longman, Harlow, 1988.
  • [Har] F. Hartogs, Über die aus den singulären Stellen einer analytischen Funktion mehrerer Veränderlichen bestehenden Gebilde, Acta Math. 32 (1909), no. 1, 57–79 (German). MR 1555046, http://dx.doi.org/10.1007/BF02403211
  • [He] M. Hervé, Several complex variables, Oxford, Bombay, 1963.
  • [Her] G. Herglotz, Über die analytische Fortsetzung des Potentials ins Innere der anziehenden Massen, Gekrönte Preisschr. Jablonowskischen Gesellsch. Leipzig, 1914, reproduced in Gustav Herglotz--Gesammelte Schriften, Vandenhoek and Ruprecht, Göttingen, 1979.
  • [Hö 1] L. Hörmander, Linear partial differential operators, Springer, Berlin, 1963.
  • [Hö 2] Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035 (85g:35002a)
  • [J] G. Johnsson, Global existence results for linear analytic partial differential equations, TRITA-MAT-1989-12.
  • [Ka] L. Karp, Construction of quadrature domains in $ {\mathbb{R}^n}$ from quadrature domains in $ {\mathbb{R}^2}$, preprint, 1990.
  • [Ke] Oliver Dimon Kellogg, Foundations of potential theory, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin, 1967. MR 0222317 (36 #5369)
  • [Kh] D. Khavinson, Singularities of harmonic functions in 𝐶ⁿ, Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 207–217. MR 1128595 (92m:35009)
  • [Kh-Sh 1] D. Khavinson and H. S. Shapiro, The Schwarz potential in $ {\mathbb{R}^n}$ and Cauchy's problem for the Laplace equation, TRITA-MAT-1989-36.
  • [Kh-Sh 2] -, Dirichlet's problem when the data is an entire function, TRITA-MAT-1991-19.
  • [Kr] H.-O. Kreiss, Numerical methods for hyperbolic partial differential equations, Numerical methods for partial differential equations (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978), Publ. Math. Res. Center Univ. Wisconsin, vol. 42, Academic Press, New York, 1979, pp. 213–254. MR 558220 (81b:65002)
  • [L] Jean Leray, Problème de Cauchy. I. Uniformisation de la solution du problème linéaire analytique de Cauchy près de la variété qui porte les données de Cauchy, Bull. Soc. Math. France 85 (1957), 389–429 (French). MR 0103328 (21 #2102)
  • [La] A. Wangerin, ed., Über die Anziehung homogener Ellipsoide, Abhandlungen von Laplace (1782), Ivory (1809), Gauss (1813), Chasles (1838) und Dirichlet (1839), Ostwalds Klassiker der Exakt. Wiss., Nr. 77, Leipzig.
  • [Mi] Masatake Miyake, Global and local Goursat problems in a class of holomorphic or partially holomorphic functions, J. Differential Equations 39 (1981), no. 3, 445–463. MR 612597 (82h:35003), http://dx.doi.org/10.1016/0022-0396(81)90068-1
  • [P] Jan Persson, On the local and global non-characteristic Cauchy problem when the solutions are holomorphic functions or analytic functionals or analytic functionals in the space variables, Ark. Mat. 9 (1971), 171–180 (1971). MR 0318702 (47 #7248)
  • [Pl] J. Plücker, Ueber solche Punkte, Kie bei Curven eine höhern Ordnung als den zweiten den Brennpunkten der Kegelschnitte entsprechen, Crelle J. 10 (1832), 84-91.
  • [S-S 1] B. Yu. Sternin and V. E. Shatalov, An integral transformation of complex analytic functions, Dokl. Akad. Nauk SSSR 280 (1985), no. 3, 553–556 (Russian). MR 775923 (86f:34018)
  • [S-S 2] B. Yu. Sternin and V. E. Shatalov, Differential equations on complex-analytic manifolds and the Maslov canonical operator, Uspekhi Mat. Nauk 43 (1988), no. 3(261), 99–124, 271, 272 (Russian, with English summary); English transl., Russian Math. Surveys 43 (1988), no. 3, 117–148. MR 955775 (90a:58172), http://dx.doi.org/10.1070/RM1988v043n03ABEH001749
  • [S-S 3] -, Notes on problem of balyage in $ {\mathbb{R}^n}$, preprint, 1990.
  • [S-S 4] B. Yu. Sternin and V. E. Shatalov, Continuation of solutions of elliptic equations and localization of singularities, Nonlinear operators in global analysis (Russian), Novoe Global. Anal., Voronezh. Gos. Univ., Voronezh, 1991, pp. 153–156, 165 (Russian). MR 1167026
  • [Sh 1] Harold S. Shapiro, The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, 9, John Wiley & Sons Inc., New York, 1992. A Wiley-Interscience Publication. MR 1160990 (93g:30059)
  • [Sh 2] -, Global aspects of Cauchy's problem for the Laplace operator, preprint, 1989.
  • [Sha] H. Shahgholian, On Newtonian potential of a heterogeneous ellipsoid, TRITA-MAT-1988-10.
  • [W] B. L. van der Waerden, Algebra I, Springer, Berlin, 1964.
  • [Wi] Stephen A. Williams, A partial solution of the Pompeiu problem, Math. Ann. 223 (1976), no. 2, 183–190. MR 0414904 (54 #2996)
  • [Z] Lawrence Zalcman, Analyticity and the Pompeiu problem, Arch. Rational Mech. Anal. 47 (1972), 237–254. MR 0348084 (50 #582)

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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1242782-7
PII: S 0002-9947(1994)1242782-7
Article copyright: © Copyright 1994 American Mathematical Society