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, p. 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 : https://doi.org/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
@article{AIHPC_2010__27_6_1471_0,
     author = {Juutinen, Petri and Lukkari, Teemu and Parviainen, Mikko},
     title = {Equivalence of viscosity and weak solutions for the $ p(x)$-Laplacian},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {6},
     year = {2010},
     pages = {1471-1487},
     doi = {10.1016/j.anihpc.2010.09.004},
     zbl = {1205.35136},
     mrnumber = {2738329},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_6_1471_0}
}
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://www.numdam.org/item/AIHPC_2010__27_6_1471_0/

[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 1814973 | Zbl 0984.49020

[2] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213-259 | MR 1930392 | Zbl 1038.76058

[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 2646346 | Zbl 1206.35134

[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 1669915

[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 | MR 2248821 | Zbl 1101.43003

[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 0920.49020

[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 1462699 | Zbl 0901.49026

[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 0755.35015

[9] L. Diening, Maximal function on generalized Lebesgue spaces L p(·) , Math. Inequal. Appl. 7 no. 2 (2004), 245-253 | MR 2057643 | Zbl 1071.42014

[10] D.E. Edmunds, J. Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 no. 3 (2000), 267-293 | MR 1815935 | Zbl 0974.46040

[11] X. Fan, D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal. 36 no. 3 (1999), 295-318 | MR 1688232 | Zbl 0927.46022

[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 2350898 | Zbl 1130.31004

[13] P. Harjulehto, J. Kinnunen, T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl. (2007), 1-20 | MR 2291928 | Zbl 1161.35020

[14] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge (1985) | MR 832183

[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 920674 | Zbl 0708.35019

[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 2131056 | Zbl 1115.35029

[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 1871417 | Zbl 0997.35022

[18] S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Mem. vol. 13, Mathematical Society of Japan, Tokyo (2004) | MR 2084272 | Zbl 1056.49027

[19] O. Kováčik, J. Rákosník, On spaces L p(x) and W 1,p(x) , Czechoslovak Math. J. 41 no. 116 (1991), 592-618 | MR 1134951 | Zbl 0784.46029

[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 2246061 | Zbl 1102.49010

[21] P. Lindqvist, On the definition and properties of p-superharmonic functions, J. Reine Angew. Math. 365 (1986), 67-79 | MR 826152 | Zbl 0572.31004

[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 | Numdam | MR 2569909 | Zbl 1180.35242

[23] O. Martio, Counterexamples for unique continuation, Manuscripta Math. 60 no. 1 (1988), 21-47 | MR 920758 | Zbl 0653.30013

[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 2449057 | Zbl 1206.91002

[25] M. Ružička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Math. vol. 1748, Springer-Verlag, Berlin (2000) | MR 1810360 | Zbl 0962.76001 | Zbl 0968.76531

[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 1766309 | Zbl 0985.46021

[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 864171 | Zbl 0599.49031