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Abstract

The role of exact fundamental solutions in the study of linear PDEs is illustrated by several examples among equations of mixed type, subelliptic and degenerate elliptic equations, and hyperbolic equations. In particular, we derive exact fundamental solutions for the degenerate hyperbolic operators \( \partial _t^2 - {\partial ^{2l}}\vartriangle \) in \( {{\mathbb{R}}^{{2p}}} \times \mathbb{R} \) for arbitrary1 p =1, 2,….

Keywords

Fundamental Solution Heat Kernel Heisenberg Group Pseudodifferential Operator Transport Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    J. Aaráo, A transport equation of mixed type, J. Diff. Equations 150 (1998), 188–202.zbMATHCrossRefGoogle Scholar
  2. [2]
    R. Bader and P. Germain, Solutions élémentaires de certaines équations aux dérivées partielles du type mixte, Bull. Soc. Math. France 81 (1953), 145–174.MathSciNetzbMATHGoogle Scholar
  3. [3]
    M. S. Baouendi and C. Goulaouic, Non-analytic hypoellipticity for some degenerate elliptic operators, Bull. Amer. Math. Soc. 78 (1972), 483–486.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    R. Beals, A note on fundamental solutions, Comm. P. D. E. 24 (1999), 369–376.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    R. Beals, B. Gaveau, and R C. Greiner, The Green function of model step two hypoelliptic operators and the analysis of certain tangential Cauchy Riemann complexes, Advances in Math. 121 (1996), 288–345.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    R. Beals, B. Gaveau, and R C. Greiner. On a geometric formula for the fundamental solutions of subelliptic laplacians, Math. Nach. 181 (1996), 81–163.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    R. Beals, B. Gaveau, and P.C. Greiner, Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians I, II, III, Bull. Sci. Math. 121 (1997), 1–36; 97–149; 195–259.Google Scholar
  8. [8]
    R. Beals, B. Gaveau, and R C. Greiner, Uniform hypoelliptic Green’s functions, J. Math. Pures Appl. 77 (1998), 209–248.MathSciNetzbMATHGoogle Scholar
  9. [9]
    R. Beals, B. Gaveau, and R C. Greiner, Green’s functions for some highly degenerate elliptic operators, J. Funct. Anal. 165 (1999), 407–429.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    R. Beals, B. Gaveau, R C. Greiner, and Y. Kannai, Exact fundamental solutions for a class of degenerate elliptic operators, Comm. R D. E. 24 (1999), 719–742.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    R. Beals and P. C. Greiner, “Calculus on Heisenberg Manifolds,” Annals of Math. Studies no. 119, Princeton Univ. Press, Princeton, NJ, 1988.Google Scholar
  12. [12]
    A. Bellaïche and J.-J. Risler, “Subriemannian Geometry,” Progress in Mathematics 144, Birkhäuser, Basel, 1996.Google Scholar
  13. [13]
    L. Boutet de Monvel, A. Grigis, and B. Helfer, Paramétrixes d’opérateurs pseudo-differentiels à caractéristiques multiples, Astérisque 34–35 (1976), 93–121.Google Scholar
  14. [14]
    L. Boutet de Monvel and F. Treves, On a class pseudodifferential operators with double characteristics, Inventiones Math. 24 (1974), 1–34.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    A. Bove and D. S. Tartakoff, Optimal non-isotropic Gevrey exponents for sums of squares of vector fields, Comm P D E 22 (1997), 1263–1282.MathSciNetzbMATHGoogle Scholar
  16. [16]
    S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), 1–89.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [[17]
    ] M. Christ, Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm P. D. E. 16 (1991), 1695–1707.CrossRefGoogle Scholar
  18. [18]
    M. Christ, Intermediate Gevrey exponents occur, Comm. P. D. E. 22 (1997), 225–235.MathSciNetCrossRefGoogle Scholar
  19. [19]
    J. Dadok and R. Harvey, The fundamental solution for the Kohn-Laplacian qb on the sphere in C“, Math Annalen 244 (1979), 89–104.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    R. M. Davis, On a regular Cauchy problem for the Euler-Poisson-Darboux equation, Ann. Mat. Pura Appl. 42 (1956), 205–226.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    A. Debiard and B. Gaveau, Analysis on root systems, Can. J. Math. 39 (1987), 1281–1404.MathSciNetzbMATHGoogle Scholar
  22. [22]
    S. Delache, Calcul des solutions élémentaires des opérateurs de TricomiClairaut auto-adjoints, strictement hyperboliques, Bull. Soc. Math. France 97 (1969), 5–79.MathSciNetzbMATHGoogle Scholar
  23. [23]
    S. Delache and J. Leray, Calcul de la solution élémentaire de l’opérateur d’Euler-Poisson-Darboux et de l’opérateur de Tricomi-Clairaut hyperbolique d’ordre 2, Bull. Soc. Math. France 99 (1971), 313–336.MathSciNetzbMATHGoogle Scholar
  24. [24]
    M. Derridj and D. S. Tartakoff, Local analyticity for qb and the 8-Neumann problem at certain weakly pseudo-convex boundary points, Comm P D E. 13 (1988), 1847–1868.MathSciNetCrossRefGoogle Scholar
  25. [25]
    M. Derridj and C. Zuily, Regularité analytique et Gevrey pour des classes d’opérateurs élliptiques paraboliques dégénérés du second ordre, Astérisque 2–3 (1973), 309–336.MathSciNetGoogle Scholar
  26. [26]
    G. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373–376.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    G. Folland and E. M. Stein, Estimates for the 8b-complex and analysis on the Heisenberg group, Comm Pure Appl. Math. 27 (1974), 429–522.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), 95153.MathSciNetCrossRefGoogle Scholar
  29. [29]
    M. Gevrey, Sur la nature analytique des solutions des équations aux dérivées partielles, Ann. Sci. École Norm. Sup. 35 (1918), 129–189.MathSciNetzbMATHGoogle Scholar
  30. [30]
    D. Gourdin, Systèmes faiblement hyperboliques à caracteristiques multiples, CRAS 278 (1974), 269–272.MathSciNetzbMATHGoogle Scholar
  31. [31]
    P. C. Greiner, A fundamental solution for a nonelliptic partial differential operator, Can. J. Math. 31 (1979), 1107–1120.MathSciNetzbMATHGoogle Scholar
  32. [32]
    P. C. Greiner, D. Holcman, and Y. Kannai, Wave kernels related to second order operators, preprint.Google Scholar
  33. [33]
    A. Grigis and J. Sjöstrand, Front d’ondes analytique et sommes de carrés de champs de vecteurs, Duke Math. J. 52 (1985), 35–51.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    J. Hadamard, “Le Problème de Cauchy et les Équations aux Dérivées Partielles Linéaires Hyperboliques,” Hermann, Paris 1932.Google Scholar
  35. [35]
    Y. Hamada, On the propagation of singularities of the solution of the Cauchy problem, Publ. RIMS Kyoto Univ. 6, no. 2 (1970), 357–384.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    Y. Hamada, J. Leray, and C. Wagschal, Systèmes d’équations aux dérivées partielles à caractéristiques multiples: problème de Cauchy ramifié: hyerbolicité partielle, J. Math. Pures Appl. 55 (1976), 297–352.MathSciNetGoogle Scholar
  37. [37]
    N. Hanges and A. Himonas, Singular solutions for some sums of squares of vector fields, Comm. R D. E. 16 (1991), 1503–1511.MathSciNetzbMATHGoogle Scholar
  38. [38]
    G. Herglotz, Über die Integration linearer partieller Dîfferentielgleichungen mit konstanten Koeffizienten, I, II, III, Bericht. Sächs. Akad. Wiss. zu Leipzig, Math. Phys. Kl. 78 (1926), 93–126, 2870–318; 80 (1928), 69–116.Google Scholar
  39. [39]
    L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math. 56 (1976), 165–173.MathSciNetzbMATHGoogle Scholar
  41. [41]
    F. John, “Plane Waves and Spherical Means Applied to Partial Differential Equations.” Interscience, New York, 1955.Google Scholar
  42. [42]
    A. Klingler, New derivation of the Heisenberg kernel, Comm. PDE 22 (1997), 2051–2060.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    A. N. Kolmogorov, Zufällige Bewegungen, Acta Math. 35 (1934), 116–117.MathSciNetzbMATHGoogle Scholar
  44. [44]
    J. Leray, Les solutions élémentaires d’une équation aux dérivées partielles à coefficients constants, C. R. Acad. Sci. Paris, Sér. I, 234 (1952), 1112–1114.MathSciNetzbMATHGoogle Scholar
  45. [45]
    J. Leray, Intégrales abéliennes et solutions élémentaires des équations hyperboliques, Colloque CBRM de Bruxelles ‘Equations aux Dérivées Partielles, Thorne and Gauthier-Villars, 1954, pp. 37–43.Google Scholar
  46. [46]
    J. Leray, Le problème de Cauchy pour une équation linéaire à coefficients polynomiaux, C. R. Acad. Sci. Paris, Sér. I, 242 (1956), 953–957.MathSciNetzbMATHGoogle Scholar
  47. [47]
    J. Leray and Y. Ohya, Systémes linéaires hyperboliques non stricts, Colloque CBRM de Liège d’Analyse fonctionelle, Thorne and Gauthier-Villars, 1965, pp. 105–144.Google Scholar
  48. [[48]
    ] A. I. Nachman, The wave equation on the Heisenberg group, Comm P. D. E. 6 (1982), 675–714.CrossRefGoogle Scholar
  49. [49]
    Y. Ohya, Le problème de Cauchy pour les équations hyperboliques à caractéristiques multiples, J, Math. Soc. Japan 16 (1964), 268–286.MathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    O. A. Oleinik and E. V. Radkevic, “Second Order Equations with Non-Negative Characteristic Form,” Moscow, 1971; English translation, Plenum, New York, London, 1973.CrossRefGoogle Scholar
  51. [51]
    M. Riesz, L’Intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math. 81, (1949), 1–223.MathSciNetzbMATHCrossRefGoogle Scholar
  52. [52]
    D. Schiltz, J. Vaillant, and C. Wagschal, Problème de Cauchy ramifié: racine caractéristique double ou triple en involution, J. Math. Pure Appl. 61 (1982), 423–443.MathSciNetzbMATHGoogle Scholar
  53. [53]
    J. Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics, Ark. för Mat. 12 (1974), 85–130.zbMATHCrossRefGoogle Scholar
  54. [54]
    D. S. Tartakoff, On the local Gevrey and quasi-analytic hypoellipticity for qb, Comm. Pure Appl. Math. 26 (1973), 699–712.MathSciNetCrossRefGoogle Scholar
  55. [55]
    D. S. Tartakoff, Local analytic hypoellipticity for qb on npn-degenerate Cauchy-Riemann manifolds, Proc. Nat. Acad. Sci. U. S. A. 75 (1978), 3027–3028.MathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    F. Treves, Analytic hypoellipticity of a class of pseudodifferential operators with double characteristics and applications to the 8-Neumann problem, Comm P. D. E. 3 (1978), 475–642.MathSciNetzbMATHCrossRefGoogle Scholar
  57. [57]
    C. Wagschal, Problème de Cauchy analytique à données méromorphes, J. Math. Pure Appl. 51 (1972), 373–397.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Richard Beals
    • 1
  1. 1.Department of MathematicsYale UniversityNew Haven

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