Equivalence of viscosity and weak solutions for the p(x)-Laplacian
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, pp. 1471-1487.

We consider different notions of solutions to the p(x)-Laplace equation

- div Du(x)| p(x)-2 Du(x))=0
with 1<p(x)<. We show by proving a comparison principle that viscosity supersolutions and p(x)-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Radó type removability theorem.

DOI: 10.1016/j.anihpc.2010.09.004
Classification: 35J92, 35D40, 31C45, 35B60
Keywords: Comparison principle, Viscosity solutions, Uniqueness, $ p(x)$-Superharmonic functions, Radó type theorem, Removability
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Juutinen, Petri; Lukkari, Teemu; Parviainen, Mikko. Equivalence of viscosity and weak solutions for the $ p(x)$-Laplacian. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, pp. 1471-1487. doi : 10.1016/j.anihpc.2010.09.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.004/

[1] E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 no. 2 (2001), 121-140 | MR | Zbl

[2] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213-259 | MR | Zbl

[3] T. Adamowicz, P. Hästö, Mappings of finite distortion and PDE with nonstandard growth, Int. Math. Res. Not. 2010 no. 10 (2010), 1940-1965 | MR | Zbl

[4] Y.A. Alkhutov, The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition, Differ. Uravn. 33 no. 12 (1997), 1651-1660 | MR

[5] T. Bieske, Equivalence of weak and viscosity solutions to the p-Laplace equation in the Heisenberg group, Ann. Acad. Sci. Fenn. Math. 31 no. 2 (2006), 363-379 | EuDML | MR | Zbl

[6] A. Coscia, G. Mingione, Hölder continuity of the gradient of p(x)-harmonic mappings, C. R. Acad. Sci. Paris Sér. I Math. 328 no. 4 (1999), 363-368 | Zbl

[7] M.G. Crandall, Viscosity solutions: a primer, Viscosity Solutions and Applications, Montecatini Terme, 1995, Lecture Notes in Math. vol. 1660, Springer, Berlin (1997), 1-67 | MR | Zbl

[8] M.G. Crandall, H. Ishii, P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 no. 1 (1992), 1-67 | Zbl

[9] L. Diening, Maximal function on generalized Lebesgue spaces L p(·) , Math. Inequal. Appl. 7 no. 2 (2004), 245-253 | MR | Zbl

[10] D.E. Edmunds, J. Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 no. 3 (2000), 267-293 | EuDML | MR | Zbl

[11] X. Fan, D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal. 36 no. 3 (1999), 295-318 | MR | Zbl

[12] P. Harjulehto, P. Hästö, M. Koskenoja, T. Lukkari, N. Marola, An obstacle problem and superharmonic functions with nonstandard growth, Nonlinear Anal. 67 no. 12 (2007), 3424-3440 | MR | Zbl

[13] P. Harjulehto, J. Kinnunen, T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl. (2007), 1-20 | EuDML | MR | Zbl

[14] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge (1985) | MR

[15] R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Ration. Mech. Anal. 101 no. 1 (1988), 1-27 | MR | Zbl

[16] P. Juutinen, P. Lindqvist, Removability of a level set for solutions of quasilinear equations, Comm. Partial Differential Equations 30 no. 1–3 (2005), 305-321 | MR | Zbl

[17] P. Juutinen, P. Lindqvist, J.J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal. 33 no. 3 (2001), 699-717 | MR | Zbl

[18] S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Mem. vol. 13, Mathematical Society of Japan, Tokyo (2004) | MR | Zbl

[19] O. Kováčik, J. Rákosník, On spaces L p(x) and W 1,p(x) , Czechoslovak Math. J. 41 no. 116 (1991), 592-618 | EuDML | MR | Zbl

[20] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 no. 4 (2006), 1383-1406 | MR | Zbl

[21] P. Lindqvist, On the definition and properties of p-superharmonic functions, J. Reine Angew. Math. 365 (1986), 67-79 | EuDML | MR | Zbl

[22] J.J. Manfredi, J.D. Rossi, J.M. Urbano, p(x)-harmonic functions with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 2581-2595 | EuDML | Numdam | MR | Zbl

[23] O. Martio, Counterexamples for unique continuation, Manuscripta Math. 60 no. 1 (1988), 21-47 | EuDML | MR | Zbl

[24] Y. Peres, O. Schramm, S. Sheffield, D.B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), 167-210 | MR | Zbl

[25] M. Ružička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Math. vol. 1748, Springer-Verlag, Berlin (2000) | MR | Zbl

[26] S. Samko, Denseness of C 0 (𝐑 N ) in the generalized Sobolev spaces W m,p(x) (𝐑 N ), Direct and Inverse Problems of Mathematical Physics, Newark, DE, 1997, Int. Soc. Anal. Appl. Comput. vol. 5, Kluwer Acad. Publ., Dordrecht (2000), 333-342 | MR | Zbl

[27] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 no. 4 (1986), 675-710, Math. USSR-Izv. 29 no. 1 (1987), 33-66 | MR | Zbl

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