On an initial-boundary value problem for a class of degenerate elliptic operators

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Necessary and sufficient conditions are established for the existence of solutions of some boundary value problems which are not well posed in the sense of Hadamard.


  1. [1]

    H.S. Carslaw,Theory of Fourier’s Series and Integrals, Dover Publications, New York, (1930).

  2. [2]

    F. Frankl,On the theory of the Laval nozzle, Izvestiya Akademii Nauk SSSR, Seriya Matematicéskaya, vol. 9 (1945), pp. 387–422.

  3. [3]

    —— ——,On the problems of Chaplygin for mixed sub-and supersonic flows, Izvestiya Akademii Nauk SSSR, Seriya Matematicéskaya, vol. 9 (1945), pp. 121–143. Also Tech. Memo. 1155, National Advisory Committee for Aeronautics, 1947.

  4. [4]

    P. Germain,Remarks on the theory of partial differential equations of mixed type and applications to the study of transonic flow, Comm. Pure Appl. Math, vol. 7 (1954), pp. 117–143.

  5. [5]

    J. Hadamard,Sur les problémes aux derivées partielles et leur signification physique, Bull. Univ. Princeton, vol. 13 (1902), pp. 49–52.

  6. [6]

    —— ——,Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover Publications, New York, (1952).

  7. [7]

    E. Holmgren,Sur un probléme aux limites pour l’equation \(y^m \frac{{\partial ^2 z}}{{\partial x^2 }} + \frac{{\partial ^2 z}}{{\partial y^2 }} = 0''\), Arkiv Mat. Astr. Fys., vol. 19B, No. 14 (1925), pp. 1–3.

  8. [8]

    F. John,Numerical solution of problems which are not well posed in the sense of Hadamard, Symposium on the Numerical Treatment of Partial Differential Equations with Real Characteristics, pp. 103–116, Provisional International Computation Centre Rome, (1959).

  9. [9]

    C. La Vallée Poussin,L’Approximation des fonctions d’une variable réelle, Gauthier-Villars, Paris, (1919).

  10. [10]

    M.M. Lavrentiev,Some Improperly Posed Problems of Mathematical Physics, Springer Verlag, New York, (1967).

  11. [11]

    S. Mandelbrojt,Analytic functions and classes of infinitely differentiable functions Rice Institute Pamphlet, vol. 29, No. 1 (1942).

  12. [12]

    L.E. Payne,On some non well posed problems for partial differential equations, Numerical Solutions of Nonlinear Differential Equations, pp. 239–263, Wiley and Sons, New York, (1966).

  13. [13]

    L.E. PayneD. Sather,On some improperly posed problems for the Chaplygin equation, J. Math. Anal. Appl., vol. 19 (1967), pp. 67–77.

  14. [14]

    ,On some improperly posed problems for quasilinear equations of mixed type, Trans. A.M.S., vol. 128 (1967), pp. 135–141.

  15. [15]

    C. Pucci,Discussione del problema di Cauchy per le equazioni di tipo ellittico, Ann. Mat. Pura Appl, vol. 46 (1958), pp. 131–153.

  16. [16]

    -- --,Some topics in parabolic and elliptic equations, Lecture Series No. 36, Institute for Fluid Mechanics and Applied Mathematics, University of Maryland, (1958).

  17. [17] a).

    F. Tricomi,Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto, Rend. Atti della Accademia Nazionale dei Lincei, Series 5, vol. 14 (1923), pp. 134–247.

  18. [17] b)

    On linear partial differential equations of the second order of mixed type, Translation No. A9-T-26, Brown University, 1948,

  19. [18]

    G.N. Watson,Theory of Bessel Functions, Cambridge University, Press, (1944).

  20. [19]

    W.H. Young,On a class of parametric integrals and their application in the theory of Fourier series, Proc. Royal Soc., Series A, vol. 85 (1911), pp. 401–414.

  21. [20]

    —— ——,On the convergence of certain series involving the Fourier constants of a function, Proc. Royal Soc., Series A, vol. 87 (1912), pp. 217–224.

  22. [21]

    A. Zygmund,Trignometrical Series, Chelsea Publishing Co., New York, (1952).

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This research was supported in part by National Science Foundation Grant No. GP 5882 and in part by Air Force Contract AF OSR 396–63.

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Payne, L.E., Sather, D. On an initial-boundary value problem for a class of degenerate elliptic operators. Annali di Matematica 78, 323–337 (1968).

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  • Elliptic Operator
  • Degenerate Elliptic Operator