Global regularity and stability of solutions to elliptic equations with nonstandard growth

References

• Hästö , P . 2005 . Counter examples of regularity in variable exponent Sobolev spaces . Contemp. Math. , 367 : 133 – 143 .
   Google Scholar
• Zhikov , VV . 1997 . On some variational problems . Russ. J. Math. Phys. , 5 ( 1 ) : 105 – 116 .
   Google Scholar
• Zhikov , VV . 1995 . On Lavrentiev's phenomenon . Russ. J. Math. Phys. , 3 ( 2 ) : 249 – 269 .
   Google Scholar
• Acerbi , E and Mingione , G . 2001 . Regularity results for a class of functionals with nonstandard growth . Arch. Ration. Mech. Anal. , 156 : 121 – 140 .
   Google Scholar
• Acerbi , E and Mingione , G . 2005 . Gradient estimates for the p(x)-Laplacean system . J. Reine Angew. Math. , 584 : 117 – 148 .
   Google Scholar
• Diening , L . 2004 . Maximal function on generalized Lebesgue spaces L p(·) . Math. Inequal. Appl. , 7 ( 2 ) : 245 – 253 .
   Google Scholar
• Eleuteri , M . 2004 . Hölder continuity results for a class of functionals with non standard growth . Boll. Unione Mat. Ital. , 7-B ( 8 ) : 129 – 157 .
   Google Scholar
• M. Eleuteri and J. Habermann, Calderón-Zygmund type estimates for a class of obstacle problems with p(x) growth (submitted).
   Google Scholar
• Eleuteri , M and Habermann , J . A Hölder continuity result for a class of obstacle problems under non standard growth conditions . Math. Nachr. (to appear) ,
   Google Scholar
• Eleuteri , M and Habermann , J . 2008 . Regularity results for a class of obstacle problems under non standard growth conditions . J. Math. Anal. Appl. , 344 ( 2 ) : 1120 – 1142 .
   Google Scholar
• Fan , X and Zhao , D . 1999 . A class of De Giorgi type and Hölder continuity . Nonlinear Anal. , 36 ( 3 ) : 295 – 318 .
   Google Scholar
• Samko , S . 3 . Denseness of in the generalized Sobolev spaces , in Direct and Inverse Problems of Mathematical Physics , J. Kajiwara, and Y.S. Xa, eds., Proc. ISAAC Congress, Newark, DE : R.P. Gilbert . –7. June, 1997. ISAAC 5, Kluwer, Dordrecht, 2000, pp. 333–342
   Google Scholar
• Lewis , JL . 1988 . Uniformly fat sets . Trans. Amer. Math. Soc. , 308 ( 1 ) : 177 – 196 .
   Google Scholar
• Kilpeläinen , T and Koskela , P . 1994 . Global integrability of the gradients of solutions to partial differential equations . Nonlinear Anal. , 23 ( 7 ) : 899 – 909 .
   Google Scholar
• Lindqvist , P . 1993 . On nonlinear Rayleigh quotients . Potential Anal. , 2 ( 3 ) : 199 – 218 .
   Google Scholar
• Antontsev , S and Zhikov , VV . 2005 . Higher integrability for parabolic equations of p(x, t)-laplacian type . Adv. Differ. Equ. , 10 ( 9 ) : 1053 – 1080 .
   Google Scholar
• Pastukhova , SE and Zhikov , VV . 2008 . Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent . Sb. Math. , 199 ( 12 ) : 1751 – 1782 .
   Google Scholar
• Hedberg , LI . 1978 . Two approximation problems in function spaces . Ark. Mat. , 16 ( 1 ) : 51 – 81 .
   Google Scholar
• Mazya , VG . 1963 . The dirichlet problem for elliptic equations of arbitrary order in unbounded regions . Dokl. Akad. Nauk. SSS , 150 : 356 – 385 .
   Google Scholar
• Alkhutov , YuA and Krasheninnikova , OV . 2004 . Continuity at boundary points of solutions of quasilinear elliptic equations with a nonstandard growth condition . Izv. Ross. Akad. Nauk Ser. Mat. , 68 ( 6 ) : 3 – 60 .
   Google Scholar
• Li , G and Martio , O . 1998 . Stability of solutions of varying degenerate elliptic equations . Indiana Univ. Math. J. , 47 ( 3 ) : 873 – 891 .
   Google Scholar
• Lindqvist , P . 1987 . Stability for the solutions of div(|∇ u| p−2 ∇ u) = f with varying . p. J. Math. Anal. Appl. , 127 ( 1 ) : 93 – 102 .
   Google Scholar
• Maasalo , OE and Zatorska-Goldstein , A . 2008 . Stability of quasiminimizers of the p-Dirichlet integral with varying p on metric spaces . J. Lond. Math. Soc. (2) , 77 ( 3 ) : 771 – 788 .
   Google Scholar
• J. Kinnunen and M. Parviainen, Stability for degenerate parabolic equations, Adv. Calc. Var. 3(1) (2010), pp. 29–48.
   Google Scholar
• Kováčik , O and Rákosník , J . 1991 . On spaces L p(x) and W 1,p(x) . Czechoslovak Math. J. , 41 ( 116 ) : 592 – 618 .
   Google Scholar
• Harjulehto , P . 2007 . Variable exponent Sobolev spaces with zero boundary values . Math. Bohem. , 132 ( 2 ) : 125 – 136 .
   Google Scholar
• Hästö , P . 2007 . On the density of continuous functions in variable exponent Sobolev space . Rev. Mat. Iberoamerican. , 23 ( 1 ) : 215 – 237 .
   Google Scholar
• Showalter , RE . 1997 . “ Monotone operators in Banach Space and Nonlinear Partial Differential Equations ” . In Mathematical Surveys and Monographs , Vol. 49 , Providence, RI : American Mathematical Society .
   Google Scholar
• Harjulehto , P , Hästö , P , Koskenoja , M and Varonen , S . 2006 . The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values . Potential Anal. , 25 ( 3 ) : 205 – 222 .
   Google Scholar
• Harjulehto , P , Hästö , P and Koskenoja , M . 2007 . Properties of capacities in variable exponent Sobolev spaces . J. Anal. Appl. , 5 ( 2 ) : 71 – 92 .
   Google Scholar
• Harjulehto , P , Hästö , P , Koskenoja , M and Varonen , S . 2003 . Sobolev capacity on the space W 1,p(·)(ℝ n ) . J. Funct. Spaces Appl. , 1 : 17 – 33 .
   Google Scholar
• Diening , L . 2004 . Riesz potential and Sobolev embeddings on generalized Lebesgue and sobolev spaces L p(·) and W k,p(·) . Math. Nachr. , 268 : 31 – 43 .
   Google Scholar
• Harjulehto , P and Hästö , P . 2008 . Sobolev inequalities for variable exponents attaining the values 1 and n . Publ. Mat. , 52 ( 2 ) : 347 – 363 .
   Google Scholar
• Harjulehto , P and Latvala , V . 2008 . Fine topology of variable exponent energy superminimizers . Ann. Acad. Sci. Fenn. Math. , 33 : 491 – 510 .
   Google Scholar
• Mikkonen , P . 1996 . On the Wolff potential and quasilinear elliptic equations involving measures . Ann. Acad. Sci. Fenn. Math. Diss. , 104 : 1 – 71 .
   Google Scholar
• Edmunds , DE and Evans , WD . 1987 . “ Spectral Theory and Differential Operators ” . In Oxford Mathematical Monographs , New York : The Clarendon Press, Oxford University Press .
   Google Scholar
• Hajłasz , P . 1999 . Pointwise Hardy inequalities . Proc. Amer. Math. Soc. , 127 ( 2 ) : 417 – 423 .
   Google Scholar
• Kinnunen , J and Martio , O . 1997 . Hardy's inequalities for Sobolev functions . Math. Res. Lett. , 4 ( 4 ) : 489 – 500 .
   Google Scholar
• Giaquinta , M . 1983 . Multiple Integrals in the Calculus of Variations , Princeton, NJ : Princeton University Press .
   Google Scholar
• Giusti , E . 2003 . Direct Methods in the Calculus of Variations , Singapore : World Scientific .
   Google Scholar
• Alkhutov , YuA . 1997 . The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition . Differ. Equ. , 33 ( 12 ) : 1653 – 1663 .
   Google Scholar
• P. Harjulehto, J. Kinnunen, and T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl. 2007 (2007) ID48348, 20pp. Available at http://dx.doi.org/10.1155/2007/48348.
   Google Scholar
• Heinonen , J , Kilpeläinen , T and Martio , O . 2006 . Nonlinear Potential Theory of Degenerate Elliptic Equations , Mineola, NY : Dover . (Unabridged republication of the 1993 original)
   Google Scholar