On the continuity of the Luxemburg norm of the gradient in 𝐿^{𝑝(⋅)} with respect to 𝑝(⋅)

Bocea, Marian; Mihăilescu, Mihai

doi:10.1090/S0002-9939-2013-12017-4

On the continuity of the Luxemburg norm of the gradient in $L^{p(\cdot )}$ with respect to $p(\cdot )$
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Proc. Amer. Math. Soc. 142 (2014), 507-517 Request permission

Abstract:

The asymptotic behavior of a sequence of functionals involving the Luxemburg norm of the gradient in variable exponent Lebesgue spaces is studied in the framework of $\Gamma$-convergence. As a consequence, we prove the convergence of minima for closely related functionals to a corresponding quantity associated to the $\Gamma$-limit.

References

F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, The limit as $p\to \infty$ in a nonlocal $p$-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles, Calc. Var. Partial Differential Equations 35 (2009), no. 3, 279–316. MR 2481827, DOI 10.1007/s00526-008-0205-2
Marian Bocea and Mihai Mihăilescu, $\Gamma$-convergence of power-law functionals with variable exponents, Nonlinear Anal. 73 (2010), no. 1, 110–121. MR 2645836, DOI 10.1016/j.na.2010.03.004
Marian Bocea, Mihai Mihăilescu, and Cristina Popovici, On the asymptotic behavior of variable exponent power-law functionals and applications, Ric. Mat. 59 (2010), no. 2, 207–238. MR 2738653, DOI 10.1007/s11587-010-0081-x
M. Bocea, M. Mihăilescu, M. Pérez-Llanos, and J. D. Rossi, Models for growth of heterogeneous sandpiles via Mosco convergence, Asymptot. Anal. 78 (2012), no. 1-2, 11–36. MR 2977657
Thierry Champion and Luigi De Pascale, Asymptotic behaviour of nonlinear eigenvalue problems involving $p$-Laplacian-type operators, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 6, 1179–1195. MR 2376876, DOI 10.1017/S0308210506000667
Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890, DOI 10.1007/978-3-642-51440-1
Yunmei Chen, Stacey Levine, and Murali Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406. MR 2246061, DOI 10.1137/050624522
Gianni Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152, DOI 10.1007/978-1-4612-0327-8
Ennio De Giorgi, Sulla convergenza di alcune successioni d’integrali del tipo dell’area, Rend. Mat. (6) 8 (1975), 277–294 (Italian, with English summary). MR 375037
Ennio De Giorgi and Tullio Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), no. 6, 842–850 (Italian). MR 448194
Lars Diening, Petteri Harjulehto, Peter Hästö, and Michael Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heidelberg, 2011. MR 2790542, DOI 10.1007/978-3-642-18363-8
L. Diening, Theoretical and numerical results for electrorheological fluids, Ph.D. thesis, University of Frieburg, Germany, 2002.
D. E. Edmunds, J. Lang, and A. Nekvinda, On $L^{p(x)}$ norms, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1981, 219–225. MR 1700499, DOI 10.1098/rspa.1999.0309
David E. Edmunds and Jiří Rákosník, Density of smooth functions in $W^{k,p(x)}(\Omega )$, Proc. Roy. Soc. London Ser. A 437 (1992), no. 1899, 229–236. MR 1177754, DOI 10.1098/rspa.1992.0059
David E. Edmunds and Jiří Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267–293. MR 1815935, DOI 10.4064/sm-143-3-267-293
Jürgen Jost and Xianqing Li-Jost, Calculus of variations, Cambridge Studies in Advanced Mathematics, vol. 64, Cambridge University Press, Cambridge, 1998. MR 1674720
Peter Lindqvist and Teemu Lukkari, A curious equation involving the $\infty$-Laplacian, Adv. Calc. Var. 3 (2010), no. 4, 409–421. MR 2734195, DOI 10.1515/ACV.2010.017
Ondrej Kováčik and Jiří Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. MR 1134951
J. J. Manfredi, J. D. Rossi, and J. M. Urbano, $p(x)$-harmonic functions with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 6, 2581–2595. MR 2569909, DOI 10.1016/j.anihpc.2009.09.008
J. J. Manfredi, J. D. Rossi, and J. M. Urbano, Limits as $p(x)\to \infty$ of $p(x)$-harmonic functions, Nonlinear Anal. 72 (2010), no. 1, 309–315. MR 2574940, DOI 10.1016/j.na.2009.06.054
Anna Mercaldo, Julio D. Rossi, Sergio Segura de León, and Cristina Trombetti, On the behaviour of solutions to the Dirichlet problem for the $p(x)$-Laplacian when $p(x)$ goes to 1 in a subdomain, Differential Integral Equations 25 (2012), no. 1-2, 53–74. MR 2906546
Mihai Mihăilescu and Vicenţiu Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2073, 2625–2641. MR 2253555, DOI 10.1098/rspa.2005.1633
Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. MR 724434, DOI 10.1007/BFb0072210
Enea Parini, Continuity of the variational eigenvalues of the $p$-Laplacian with respect to $p$, Bull. Aust. Math. Soc. 83 (2011), no. 3, 376–381. MR 2794523, DOI 10.1017/S000497271100205X
Mayte Perez-Llanos and Julio D. Rossi, Limits as $p(x)\to \infty$ of $p(x)$-harmonic functions with non-homogeneous Neumann boundary conditions, Nonlinear elliptic partial differential equations, Contemp. Math., vol. 540, Amer. Math. Soc., Providence, RI, 2011, pp. 187–201. MR 2807415, DOI 10.1090/conm/540/10665
K. R. Rajagopal and M. Ruzicka, Mathematical modelling of electrorheological fluids. Cont. Mech. Term. 13 (2001), 59-78.
Michael Růžička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. MR 1810360, DOI 10.1007/BFb0104029
S. Samko and B. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005), no. 1, 229–246. MR 2160685, DOI 10.1016/j.jmaa.2005.02.002
V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710, 877 (Russian). MR 864171