Riv. Mat. Univ. Parma, Vol. 5, No. 1, 2014

Fatma Gamze Düzgün [1], Paolo Marcellini [2] and Vincenzo Vespri [2]

Space expansion for a solution of an anisotropic p-Laplacian equation by using a parabolic approach

Pages: 93-111
Received: 29 April 2013   
Accepted in revised form: 5 September 2013
Mathematics Subject Classification (2010): 35J70, 35J92, 35B65.

Keywords: Degenerate elliptic equations, anisotropic p-Laplacian, quantitative estimates.
Authors addresses:
[1] : Department of Mathematics, Hacettepe University, 06800, Beytepe, Ankara, Turkey
[2] : Dipartimento di Matematica e Informatica "U. Dini", UniversitÓ degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy

Abstract: In this study we show that a technique introduced in the parabolic setting works also in the elliptic context. More precisely we prove a space expansion of positivity for solutions of an elliptic equation with anisotropic growth.

References

[1] L. Boccardo, P. Marcellini and C. Sbordone, \(L^\infty\)-regularity for variational problems with sharp nonstandard growth conditions, Boll. Un. Mat. Ital. A (7) 4 (1990), 219-225.
[2] G. Cupini, P. Marcellini and E. Mascolo, Local boundedness of solutions to quasilinear elliptic systems, Manuscripta Math. 137 (2012), 287- 315.
[3] E. DiBenedetto, Degenerate parabolic equations, Springer-Verlag, New York 1993.
[4] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math. 200 (2008), 181-209.
[5] E. DiBenedetto, U. Gianazza and V. Vespri, Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), 385-422.
[6] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's inequality for degenerate and singular parabolic equations, Springer Monographs in Mathematics, Springer, New York 2012.
[7] U. Gianazza, M. Surnachev and V. Vespri, A new proof of the Hölder continuity of solutions to p-Laplace type parabolic equations, Adv. Calc. Var. 3 (2010), 263-278.
[8] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), 245-248.
[9] T. Kuusi, Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), 673-716.
[10] V. Liskevich and I. I. Skrypnik, Hölder continuity of solutions to an anisotropic elliptic equation, Nonlinear Anal. 71 (2009), 1699-1708.
[11] P. Marcellini, Un exemple de solution discontinue d'un probléme variationnel dans le cas scalaire, Preprint n. 11, Ist. Mat. "U. Dini", Firenze, 1987.
[12] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal. 105 (1989), 267-284.
[13] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations 90 (1991), 1-30.
[14] P. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl. 90 (1996), 161-181.
[15] S. M. Nikolskii, An imbedding theorem for functions with partial derivatives considered in different metrics, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 321-336. English translation: Amer. Math. Soc. Transl. 90 (1970), 27-44.
[16] M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat. 18 (1969), 3-24.
[17] N. S. Trudinger, An imbedding theorem for \(H_0(G,\Omega)\) spaces, Studia Math. 50 (1974), 17-30.
[18] J. M. Urbano, The method of intrinsic scaling. A systematic approach to regularity for degenerate and singular PDEs, Lecture Notes in Math., vol. 1930, Springer-Verlag, Berlin 2008.


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