We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.
E.Acerbi and G.Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Rational Mech. Anal.156 (2001), 121–140. doi:10.1007/s002050100117.
2.
E.Acerbi and G.Mingione, Regularity results for a class of quasiconvex functionals with nonstandard growth, Ann. Scuola Norm. Sup Pisa30 (2001), 311–339.
3.
A.Bahrouni and V.D.Rădulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst., Ser. S11 (2018), 379–388.
4.
J.Bourgain, H.Brezis and P.Mironescu, Another look at Sobolev spaces, in: Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan’s 60th Birthday, J.L.Menaldi, E.Rofman and A.Sulem, eds, IOS Press, Amsterdam, 2001, pp. 439–455.
5.
J.Bourgain, H.Brezis and H.-M.Nguyen, A new estimate for the topological degree, C. R. Math. Acad. Sci. Paris340 (2005), 787–791. doi:10.1016/j.crma.2005.04.007.
6.
L.Diening, 2021, Private communication.
7.
L.Diening, P.Harjulehto, P.Hästö and M.Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 2011.
8.
L.Diening and P.Hästö, Muckenhoupt weights in variable exponent spaces, 2011, Preprint.
9.
M.Izuki, E.Nakai and Y.Sawano, The Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponent. Harmonic analysis and nonlinear partial differential equations, Res. Inst. Math. Sci. B42 (2013), 51–94.
10.
H.-M.Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal.237 (2006), 689–720. doi:10.1016/j.jfa.2006.04.001.
11.
H.-M.Nguyen, Further characterizations of Sobolev spaces, J. Eur. Math. Soc.10 (2008), 191–229.
12.
H.-M.Nguyen, Some inequalities related to Sobolev norms, Calc. Var. Partial Differential Equations41 (2011), 483–509. doi:10.1007/s00526-010-0373-8.
13.
H.-M.Nguyen, A.Pinamonti, M.Squassina and E.Vecchi, New characterizations of magnetic Sobolev spaces, Adv. Nonlinear Anal.7 (2018), 227–245. doi:10.1515/anona-2017-0239.
14.
M.M.Růžička, Electrorheological Fluids Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.
15.
E.Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
