We consider different notions of solutions to the -Laplace equation
Mots-clés : 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}, pages = {1471--1487}, publisher = {Elsevier}, volume = {27}, number = {6}, year = {2010}, doi = {10.1016/j.anihpc.2010.09.004}, mrnumber = {2738329}, zbl = {1205.35136}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.004/} }
TY - JOUR AU - Juutinen, Petri AU - Lukkari, Teemu AU - Parviainen, Mikko TI - Equivalence of viscosity and weak solutions for the $ p(x)$-Laplacian JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 1471 EP - 1487 VL - 27 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.004/ DO - 10.1016/j.anihpc.2010.09.004 LA - en ID - AIHPC_2010__27_6_1471_0 ER -
%0 Journal Article %A Juutinen, Petri %A Lukkari, Teemu %A Parviainen, Mikko %T Equivalence of viscosity and weak solutions for the $ p(x)$-Laplacian %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 1471-1487 %V 27 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.004/ %R 10.1016/j.anihpc.2010.09.004 %G en %F 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, Tome 27 (2010) no. 6, pp. 1471-1487. doi : 10.1016/j.anihpc.2010.09.004. http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.004/
[1] Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 no. 2 (2001), 121-140 | MR | Zbl
, ,[2] Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213-259 | MR | Zbl
, ,[3] Mappings of finite distortion and PDE with nonstandard growth, Int. Math. Res. Not. 2010 no. 10 (2010), 1940-1965 | MR | Zbl
, ,[4] 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] 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] Hölder continuity of the gradient of -harmonic mappings, C. R. Acad. Sci. Paris Sér. I Math. 328 no. 4 (1999), 363-368 | Zbl
, ,[7] 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] 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] Maximal function on generalized Lebesgue spaces , Math. Inequal. Appl. 7 no. 2 (2004), 245-253 | MR | Zbl
,[10] Sobolev embeddings with variable exponent, Studia Math. 143 no. 3 (2000), 267-293 | EuDML | MR | Zbl
, ,[11] A class of De Giorgi type and Hölder continuity, Nonlinear Anal. 36 no. 3 (1999), 295-318 | MR | Zbl
, ,[12] An obstacle problem and superharmonic functions with nonstandard growth, Nonlinear Anal. 67 no. 12 (2007), 3424-3440 | MR | Zbl
, , , , ,[13] Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl. (2007), 1-20 | EuDML | MR | Zbl
, , ,[14] Matrix Analysis, Cambridge University Press, Cambridge (1985) | MR
, ,[15] 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] Removability of a level set for solutions of quasilinear equations, Comm. Partial Differential Equations 30 no. 1–3 (2005), 305-321 | MR | Zbl
, ,[17] 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] A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Mem. vol. 13, Mathematical Society of Japan, Tokyo (2004) | MR | Zbl
,[19] On spaces and , Czechoslovak Math. J. 41 no. 116 (1991), 592-618 | EuDML | MR | Zbl
, ,[20] Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 no. 4 (2006), 1383-1406 | MR | Zbl
, , ,[21] On the definition and properties of p-superharmonic functions, J. Reine Angew. Math. 365 (1986), 67-79 | EuDML | MR | Zbl
,[22] -harmonic functions with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 2581-2595 | EuDML | Numdam | MR | Zbl
, , ,[23] Counterexamples for unique continuation, Manuscripta Math. 60 no. 1 (1988), 21-47 | EuDML | MR | Zbl
,[24] Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), 167-210 | MR | Zbl
, , , ,[25] Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Math. vol. 1748, Springer-Verlag, Berlin (2000) | MR | Zbl
,[26] Denseness of in the generalized Sobolev spaces , 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] 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
,Cité par Sources :