Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T02:55:56.998Z Has data issue: false hasContentIssue false
  • English
  • Français

ENTROPY AND RENORMALIZED SOLUTIONS FOR THE p(x)-LAPLACIAN EQUATION WITH MEASURE DATA

Part of: Elliptic equations and systems Linear function spaces and their duals

Published online by Cambridge University Press:  18 August 2010

CHAO ZHANG  and
SHULIN ZHOU
CHAO ZHANG*
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China (email: [email protected])
SHULIN ZHOU
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we prove the existence and uniqueness of both entropy solutions and renormalized solutions for the p(x)-Laplacian equation with variable exponents and a signed measure in L1(Ω)+W−1,p′(⋅)(Ω). Moreover, we obtain the equivalence of entropy solutions and renormalized solutions.

Keywords

variable exponentsentropy solutionsrenormalized solutionsexistenceuniqueness

MSC classification

Secondary: 35J70: Degenerate elliptic equations 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 3 , December 2010 , pp. 459 - 479
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This work was supported in part by the NBRPC under Grant 2006CB705700 and the NSFC under Grant 10990013.

References

[1]Acerbi, E. and Mingione, G., ‘Regularity results for a class of functionals with non-standard growth’, Arch. Ration. Mech. Anal. 156 (2001), 121–140.CrossRefGoogle Scholar
[2]Acerbi, E. and Mingione, G., ‘Regularity results for stationary electro-rheological fluids’, Arch. Ration. Mech. Anal. 164 (2002), 213–259.CrossRefGoogle Scholar
[3]Acerbi, E., Mingione, G. and Seregin, G. A., ‘Regularity results for parabolic systems related to a class of non Newtonian fluids’, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 25–60.CrossRefGoogle Scholar
[4]Alvino, A., Boccardo, L., Ferone, V., Orsina, L. and Trombetti, G., ‘Existence results for nonlinear elliptic equations with degenerate coercivity’, Ann. Mat. Pura Appl. 182 (2003), 53–79.CrossRefGoogle Scholar
[5]Antontsev, S. N. and Shmarev, S. I., ‘A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions’, Nonlinear Anal. 60 (2005), 515–545.CrossRefGoogle Scholar
[6]Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M. and Vazquez, J. L., ‘An L 1-theory of existence and uniqueness of solutions of nonlinear elliptic equations’, Ann. Sc. Norm. Super. Pisa Cl. Sci. 22 (1995), 241–273.Google Scholar
[7]Blanchard, D. and Murat, F., ‘Renormalised solutions of nonlinear parabolic problems with L 1 data: existence and uniqueness’, Proc. Roy. Soc. Edinburgh Sect. A 127(6) (1997), 1137–1152.CrossRefGoogle Scholar
[8]Blanchard, D., Murat, F. and Redwane, H., ‘Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems’, J. Differential Equations 177(2) (2001), 331–374.CrossRefGoogle Scholar
[9]Blanchard, D. and Redwane, H., ‘Renormalized solutions for a class of nonlinear evolution problems’, J. Math. Pure Appl. 77 (1998), 117–151.CrossRefGoogle Scholar
[10]Boccardo, L. and Cirmi, G. R., ‘Existence and uniqueness of solution of unilateral problems with L 1 data’, J. Convex. Anal. 6 (1999), 195–206.Google Scholar
[11]Boccardo, L., Gallouët, T. and Orsina, L., ‘Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data’, Ann. Inst. H. Poincaré Anal. Non Linéaire 13(5) (1996), 539–551.CrossRefGoogle Scholar
[12]Boccardo, L., Giachetti, D., Diaz, J. I. and Murat, F., ‘Existence and regularity of renormalized solutions for some elliptic problems involving derivations of nonlinear terms’, J. Differential Equations 106 (1993), 215–237.CrossRefGoogle Scholar
[13]Chen, Y., Levine, S. and Rao, M., ‘Variable exponent, linear growth functionals in image restoration’, SIAM J. Appl. Math. 66 (2006), 1383–1406.CrossRefGoogle Scholar
[14]Diening, L., ‘Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p(⋅) and W k,p(⋅) ’, Math. Nachr. 268 (2004), 31–43.CrossRefGoogle Scholar
[15]DiPerna, R. J. and Lions, P. L., ‘On the Cauchy problem for Boltzmann equations: global existence and weak stability’, Ann. of Math. (2) 130 (1989), 321–366.CrossRefGoogle Scholar
[16]Fan, X., ‘Global C 1,α regularity for variable exponent elliptic equations in divergence form’, J. Differential Equations 235 (2007), 397–417.CrossRefGoogle Scholar
[17]Fan, X., Shen, J. and Zhao, D., ‘Sobolev embedding theorems for spaces W k,p(x) (Ω)’, J. Math. Anal. Appl. 262 (2001), 749–760.CrossRefGoogle Scholar
[18]Fan, X. and Zhang, Q., ‘Existence of solutions for p(x)-Laplacian Dirichlet problem’, Nonlinear Anal. 52 (2003), 1843–1852.CrossRefGoogle Scholar
[19]Fan, X., Zhang, Q. and Zhao, D., ‘Eigenvalues of p(x)-Laplacian Dirichlet problem’, J. Math. Anal. Appl. 302 (2005), 306–317.CrossRefGoogle Scholar
[20]Fan, X. and Zhao, D., ‘On the spaces L p(x)(Ω) and W m,p(x) (Ω)’, J. Math. Anal. Appl. 263 (2001), 424–446.CrossRefGoogle Scholar
[21]Harjulehto, P., ‘Variable exponent Sobolev spaces with zero boundary values’, Math. Bohem. 132 (2007), 125–136.CrossRefGoogle Scholar
[22]Harjulehto, P., Hästö, P. and Koskenoja, M., ‘Properties of capacities in variable exponents Sobolev spaces’, J. Anal. Appl. 5(2) (2007), 71–92.Google Scholar
[23]Harjulehto, P., Hästö, P., Koskenoja, M. and Varonen, S., ‘The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values’, Potential Anal. 25(3) (2006), 205–222.CrossRefGoogle Scholar
[24]Kováčik, O. and Rákosník, J., ‘On spaces L p(x) and W k,p(x) ’, Czechoslovak Math. J. 41(116) (1991), 592–618.CrossRefGoogle Scholar
[25]Leone, C. and Porretta, A., ‘Entropy solutions for nonlinear elliptic equations in L 1’, Nonlinear Anal. 32(3) (1998), 325–334.CrossRefGoogle Scholar
[26]Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaire (Dunod et Gauthier Villars, Paris, 1969).Google Scholar
[27]Lions, P. L., Mathematical Topics in Fluid Mechanics, Incompressible Models, 1 (Oxford University Press, Oxford, 1996).Google Scholar
[28]Musielak, J., Orlicz Spaces and Modular Spaces (Springer, Berlin, 1983).CrossRefGoogle Scholar
[29]Palmeri, M. C., ‘Entropy subsolutions and supersolutions for nonlinear elliptic equations in L 1’, Ric. Mat. 53 (2004), 183–212.Google Scholar
[30]Rajagopal, K. and Ru̇žička, M., ‘Mathematical modelling of electro-rheological fluids’, Contin. Mech. Thermodyn. 13 (2001), 59–78.CrossRefGoogle Scholar
[31]Ru̇žička, M., Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748 (Springer, Berlin, 2000).CrossRefGoogle Scholar
[32]Sanchón, M. and Urbano, J. M., ‘Entropy solutions for the p(x)-Laplace equation’, Trans. Amer. Math. Soc. 361 (2009), 6387–6405.CrossRefGoogle Scholar
[33]Zhikov, V. V., ‘On some variational problems’, Russ. J. Math. Phys. 5 (1997), 105–116.Google Scholar
[34]Zhikov, V. V., ‘On the density of smooth functions in Sobolev–Orlicz spaces’, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 226 (2004), 67–81.Google Scholar