Grand p-harmonic energy

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Abstract

To every nonlinear differential expression there corresponds the so-called natural domain of definition. Usually, such a domain consists of Sobolev functions, sometimes with additional geometric constraints. There are, however, special nonlinear differential expressions (Jacobian determinants, div-curl products, etc.) whose special properties (higher integrability, weak-continuity, etc.) cannot be detected within their natural domain. We must consider them in a slightly larger class of functions. The grand Lebesgue space, denoted by \(\mathscr {L}^{p})(\mathbb X)\), and the corresponding grand Sobolev space \(\mathscr {W}^{1,p})(\mathbb X)\), turn out to be most effective. They were studied by many authors, largely in analogy with the questions concerning \(\mathscr {L}^p (\mathbb X)\) and \(\mathscr {W}^{1,p}(\mathbb X)\) spaces. The present paper is a continuation of these studies. We take on stage the grand p-harmonic energy integrals. These variational functionals involve both one-parameter family of integral averages and supremum with respect to the parameter. It is for this reason that the existence and uniqueness of the grand p-harmonic minimal mappings becomes a new (rather challenging) problem.

Mathematics Subject Classification

Primary 35A30 Secondary 35B65 

References

  1. 1.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Mathematical Series, vol. 48. Princeton University Press, Princeton (2009)MATHGoogle Scholar
  2. 2.
    Capone, C., Formica, M.R., Giova, R.: Grand Lebesgue spaces with respect to measurable functions. Nonlinear Anal. 85, 125–131 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dacorogna, B.: Direct Methods in the calculus of variations. Applied Mathematical Sciences, vol. 78. Springer, Berlin (1989)Google Scholar
  4. 4.
    Fiorenza, A., Karadzhov, G.E.: Grand and small Lebesgue spaces and their analogs. Z. Anal. Anwendungen 23(4), 657–681 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977)CrossRefMATHGoogle Scholar
  6. 6.
    Greco, L.: A remark on the equality \(det\, Df= Det \, Df\). Differ. Int. Equ. 6(5), 1089–1100 (1993)MathSciNetMATHGoogle Scholar
  7. 7.
    Iwaniec, T., Martin, G.: Geometric Function Theory and Non-Linear Analysis. The Clarendon Press, Oxford University Press, New York, Oxford Math. Monogr (2001)Google Scholar
  8. 8.
    Iwaniec, T., Sbordone, C.: Weak minima of variational integrals. J. Reine Angew. Math. 454, 143–161 (1994)MathSciNetMATHGoogle Scholar
  9. 9.
    Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses Arch. Ration. Mech. Anal. 119, 129–143 (1992)CrossRefMATHGoogle Scholar
  10. 10.
    Radice, T., Zecca, G.: The maximum principle of Alexandrov for very weak solutions. J. Differ. Equ. 256, 1133–1150 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Umarkhadzhiev, S.M.: Generalization of a notion of grand Lebesgue space. Izv. Vyssh. Uchebn. Zaved. Mat. 4, 42–51 (2014)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Applicazioni, “R. Caccioppoli”Complesso Universitario “Monte S. Angelo”NaplesItaly

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