Function Space and Operator Theory for Nonlinear Analysis

  • Michael E. Taylor
Part of the Applied Mathematical Sciences book series (AMS, volume 117)

Abstract

This chapter examines a number of analytical techiques, which will be applied to diverse nonlinear problems in the remaining chapters. For example, we study Sobolev spaces based on L p , rather than just L 2. Sections 1 and 2 discuss the definition of Sobolev spaces H k, p , for k ∈ Z+, and inclusions of the form H k, p L q . Estimates based on such inclusions have refined forms, due to E. Gagliardo and L. Nirenberg. We discuss these in §3, together with results of J. Moser on estimates on nonlinear functions of an element of a Sobolev space, and on commutators of differential operators and multiplication operators. In §4 we establish some integral estimates of N. Trudinger, on functions in Sobolev spaces for which L-bounds just fail. In these sections we use such basic tools as Hölder’s inequality and integration by parts.

Keywords

Function Space Operator Theory Nonlinear Analysis Hardy Space Compact Manifold 
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. [Ad]
    R. Adams, Sobolev Spaces, Academic Press, New York, 1975.MATHGoogle Scholar
  2. [ADN]
    S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic differential equations satisfying general boundary conditions, CPAM 12(1959), 623–727.MathSciNetMATHGoogle Scholar
  3. [AG]
    S. Ahlinac and P. Gerard, Operateurs Pseudo-differentiels et Théorème de Nash-Moser, Editions du CNRS, Paris, 1991.Google Scholar
  4. [Au]
    T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampere Equations, Springer-Verlag, New York, 1982.MATHCrossRefGoogle Scholar
  5. [AT]
    P. Auscher and M. Taylor, Paradifferential operators and commutator estimates, Comm. PDE 20(1995), 1743–1775.MathSciNetMATHCrossRefGoogle Scholar
  6. [Ba]
    J. Ball (ed.), Systems of Nonlinear Partial Differential Equations, Reidel, Boston, 1983.MATHGoogle Scholar
  7. [BL]
    J. Bergh and J. Löfstrom, Interpolation Spaces, an Introduction, Springer-Verlag, New York, 1976.MATHCrossRefGoogle Scholar
  8. [BJS]
    L. Bers, F. John, and M. Sehechter, Partial Differential Equations, Wiley, New York, 1964.MATHGoogle Scholar
  9. [Bon]
    J. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées nonlinéaires, Ann. Sci. Ecole Norm. Sup. 14(1981), 209–246.MathSciNetMATHGoogle Scholar
  10. [Bou]
    G. Bourdaud, Une algèbre maximale d’operateurs pseudodifferentiels, Comm.PDE 13(1988), 1059–1083.MathSciNetMATHCrossRefGoogle Scholar
  11. [BrG]
    H. Brezis and T. Gallouet, Nonlinear Schrodinger evolutions, J. Nonlinear Anal. 4(1980), 677–681.MathSciNetMATHCrossRefGoogle Scholar
  12. [BrW]
    H. Brezis and S. Wainger, A note on limiting cases of Sobolev imbeddings, Comm. PDE 5(1980), 773–789.MathSciNetMATHCrossRefGoogle Scholar
  13. [Ca]
    A. P. Calderon, Intermediate spaces and interpolation, the complex method, Studia Math. 24(1964), 113–190.MathSciNetMATHGoogle Scholar
  14. [CLMS]
    R. Coifman, P. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures etAppl. 72(1993), 247–286.MathSciNetMATHGoogle Scholar
  15. [DST]
    E. B. Davies, B. Simon, and M. Taylor, L p spectral theory of Kleinian groups, J.Funct.Anal. 78(1988), 116–136.MathSciNetMATHCrossRefGoogle Scholar
  16. [DiP]
    R. DiPerna, Measure-valued solutions to conservation laws, Arch. Rat. Mech.Anal. 88(1985), 223–270.MathSciNetMATHGoogle Scholar
  17. [Ev]
    L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Reg. Conf. Ser. #74, Providence, R. I., 1990.MATHGoogle Scholar
  18. [Ev2]
    L. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rat. Mech. Anal. 116(1991), 101–113.MATHCrossRefGoogle Scholar
  19. [FS]
    C. Fefferman and E. Stein, H p spaces of several variables, Acta Math. 129(1972), 137–193.MathSciNetMATHCrossRefGoogle Scholar
  20. [Fo]
    G. Folland, Lectures on Partial Differential Equations, Tata Institute, Bombay, Springer, New York, 1983.MATHCrossRefGoogle Scholar
  21. [FJW]
    M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Reg. Conf. Ser. Math. #79, AMS, Providence, R. I., 1991.MATHGoogle Scholar
  22. [Frd]
    A. Friedman, Generalized Functions and Partial Differential Equations, Prentice-Hall, Englewood Cliffs, N. J., 1963.MATHGoogle Scholar
  23. [Gag]
    E. Gagliardo, Ulteriori proprieta de alcune classi de funzioni in piu variabili, Ricerche Mat. 8(1959), 24–51.MathSciNetMATHGoogle Scholar
  24. [H1]
    L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1, Springer-Verlag, New York, 1983.CrossRefGoogle Scholar
  25. [H2]
    L. Hörmander, Pseudo-differential operators of type 1,1. Comm. PDE 13(1988), 1085–1111.MATHCrossRefGoogle Scholar
  26. [H3]
    L. Hörmander, Non-linear Hyperbolic Differential Equations. Lecture Notes, Lund Univ., 1986–87.Google Scholar
  27. [Jos]
    J. Jost, Nonlinear Methods in Riemannian and Kahlerian Geometry, Birkhäuser, Boston, 1988.CrossRefGoogle Scholar
  28. [KS]
    D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, NY, 1980.MATHGoogle Scholar
  29. [Mai]
    J. Marschall, Pseudo-differential operators with non regular symbols, Inaugural-Dissertation, Freien Universität Berlin, 1985.MATHGoogle Scholar
  30. [Ma2]
    J. Marschall, Pseudo-differential operators with coefficients in Sobolev spaces, Trans. AMS 307(1988), 335–361.MathSciNetMATHGoogle Scholar
  31. [Mey]
    Y. Meyer, Regularité des solutions des équations aux derivées partielles non linéaires, Sem. Bourbaki 1979/80, 293–302, LNM #842, Springer-Verlag, New York, 1980.Google Scholar
  32. [Mik]
    S. Mikhlin, Multidimensional Singular Integral Equations, Pergammon Press, New York, 1965.MATHGoogle Scholar
  33. [Mor]
    C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966.MATHGoogle Scholar
  34. [Mos]
    J. Moser, A rapidly convergent iteration method and nonlinear partial differential equations, I, Ann. Sc. Norm. Sup. Pisa 20(1966), 265–315.MATHGoogle Scholar
  35. [Mos2]
    J. Moser, A sharp form of an inequality of N. Trudinger, Indiana Math. J. 20(1971), 1077–1092.CrossRefGoogle Scholar
  36. [Mur]
    F. Murat, Compacité par compensation, Ann. Sc. Norm. Sup. Pisa 5(1978), 485–507.MathSciNetGoogle Scholar
  37. [Mus]
    N. Muskheleshvili, Singular Integral Equations, P. Nordhoff, Groningen, 1953.Google Scholar
  38. [Ni]
    L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa, 13(1959), 116–162.MathSciNetGoogle Scholar
  39. [RS]
    M. Reed and B. Simon, Methods of Mathematical Physics, Academic Press, New York, Vols. 1, 2, 1975; Vols. 3, 4, 1978.MATHGoogle Scholar
  40. [RRT]
    J. Robbins, R. Rogers, and B. Temple, On weak continuity and the Hodge decomposition, Trans. AMS 303(1987), 609–618.CrossRefGoogle Scholar
  41. [Sch]
    L. Schwartz, Theorie des Distributions, Hermann, Paris, 1950.MATHGoogle Scholar
  42. [Sem]
    S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. PDE 19(1994), 277–319.MathSciNetMATHCrossRefGoogle Scholar
  43. [So]
    S. Sobolev, On a theorem of functional analysis, Mat. Sb. 4(1938), 471–497;MATHGoogle Scholar
  44. [Soa]
    S. Sobolev, On a theorem of functional analysis, AMS Transi. 34(1963), 39–68.MATHGoogle Scholar
  45. [So2]
    S. Sobolev, Partial Differential Equations of Mathematical Physics, Dover, New York, 1964.MATHGoogle Scholar
  46. [S1]
    E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N. J., 1970.MATHGoogle Scholar
  47. [S2]
    E. Stein, Singular Integrals and Pseudo-differential Operators, Graduate Lecture Notes, Princeton Univ., 1972.Google Scholar
  48. [S3]
    E. Stein, Harmonic Analysis, Princeton Univ. Press, Princeton, N. J., 1993.MATHGoogle Scholar
  49. [SW]
    E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Space, Princeton Univ. Press, Princeton, N. J., 1971.Google Scholar
  50. [Str]
    R. Strichartz, A note on Trudinger’s extension of Sobolev’s inequalities, Indiana Math. 7.21(1972), 841–842.CrossRefGoogle Scholar
  51. [Tar]
    L. Tartar, Compensated compactness and applications to partial differential equations, Heriot-Watt Symp. Vol. IV, Pitman, New York, 1979, pp. 136–212.Google Scholar
  52. [Tar2]
    L. Tartar, The compensated compactness method applied to systems of conservation laws, pp. 263–285 in [Ba].Google Scholar
  53. [T1]
    M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, N. J., 1981.MATHGoogle Scholar
  54. [T2]
    M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.MATHCrossRefGoogle Scholar
  55. [T3]
    M. Taylor, L p-estimates on functions of the Laplace operator, Duke Math. J. 58(1989), 773–793.MathSciNetMATHCrossRefGoogle Scholar
  56. [Tri]
    H. Triebel, Theory of Function Spaces, Birkhäuser, Boston, 1983.CrossRefGoogle Scholar
  57. [Tru]
    N. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math.Mech. 17(1967), 473–483.MathSciNetMATHGoogle Scholar
  58. [You]
    L. Young, Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders, Philadelphia, 1979.Google Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Michael E. Taylor
    • 1
  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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