Nondivergence parabolic equations in weighted variable exponent spaces

Byun, Sun-Sig; Lee, Mikyoung; Ok, Jihoon

doi:10.1090/tran/7352

Nondivergence parabolic equations in weighted variable exponent spaces
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Trans. Amer. Math. Soc. 370 (2018), 2263-2298 Request permission

Abstract:

We prove the global Calderón-Zygmund estimates for second order parabolic equations in nondivergence form in weighted variable exponent Lebesgue spaces. We assume that the associated variable exponent is log-Hölder continuous, the weight is of a certain Muckenhoupt class with respect to the variable exponent, the coefficients of the equation are the functions of small bonded mean oscillation, and the underlying domain is a $C^{1,1}$-domain.

References

Emilio Acerbi and Giuseppe Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 (2001), no. 2, 121–140. MR 1814973, DOI 10.1007/s002050100117
Emilio Acerbi and Giuseppe Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math. 584 (2005), 117–148. MR 2155087, DOI 10.1515/crll.2005.2005.584.117
Emilio Acerbi and Giuseppe Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), no. 2, 285–320. MR 2286632, DOI 10.1215/S0012-7094-07-13623-8
Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
Paolo Baroni and Verena Bögelein, Calderón-Zygmund estimates for parabolic $p(x,t)$-Laplacian systems, Rev. Mat. Iberoam. 30 (2014), no. 4, 1355–1386. MR 3293436, DOI 10.4171/RMI/817
Marco Bramanti and M. Cristina Cerutti, $W_p^{1,2}$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1735–1763. MR 1239929, DOI 10.1080/03605309308820991
Sun-Sig Byun, Mikyoung Lee, and Jihoon Ok, $W^{2,p(\cdot )}$-regularity for elliptic equations in nondivergence form with BMO coefficients, Math. Ann. 363 (2015), no. 3-4, 1023–1052. MR 3412352, DOI 10.1007/s00208-015-1194-z
Sun-Sig Byun, Jihoon Ok, and Seungjin Ryu, Global gradient estimates for elliptic equations of $p(x)$-Laplacian type with BMO nonlinearity, J. Reine Angew. Math. 715 (2016), 1–38. MR 3507918, DOI 10.1515/crelle-2014-0004
Sun-Sig Byun, Jihoon Ok, and Lihe Wang, $W^{1,p(\cdot )}$-regularity for elliptic equations with measurable coefficients in nonsmooth domains, Comm. Math. Phys. 329 (2014), no. 3, 937–958. MR 3212875, DOI 10.1007/s00220-014-1962-8
Sun-Sig Byun, Dian K. Palagachev, and Lubomira G. Softova, Global gradient estimates in weighted Lebesgue spaces for parabolic operators, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 1, 67–83. MR 3467697, DOI 10.5186/aasfm.2016.4102
A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 52553, DOI 10.1007/BF02392130
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
Filippo Chiarenza, Michele Frasca, and Placido Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), no. 2, 841–853. MR 1088476, DOI 10.1090/S0002-9947-1993-1088476-1
D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239–264. MR 2210118
David Cruz-Uribe, Lars Diening, and Peter Hästö, The maximal operator on weighted variable Lebesgue spaces, Fract. Calc. Appl. Anal. 14 (2011), no. 3, 361–374. MR 2837636, DOI 10.2478/s13540-011-0023-7
L. Diening, Maximal function on generalized Lebesgue spaces $L^{p(\cdot )}$, Math. Inequal. Appl. 7 (2004), no. 2, 245–253. MR 2057643, DOI 10.7153/mia-07-27
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 and P. Hästö, Muckenhoupt weights in variable exponent spaces, preprint, 2011.
L. Diening, D. Lengeler, and M. Růžička, The Stokes and Poisson problem in variable exponent spaces, Complex Var. Elliptic Equ. 56 (2011), no. 7-9, 789–811. MR 2832214, DOI 10.1080/17476933.2010.504843
L. Diening and M. Růžička, Calderón-Zygmund operators on generalized Lebesgue spaces $L^{p(\cdot )}$ and problems related to fluid dynamics, J. Reine Angew. Math. 563 (2003), 197–220. MR 2009242, DOI 10.1515/crll.2003.081
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
Xianling Fan, Global $C^{1,\alpha }$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 (2007), no. 2, 397–417. MR 2317489, DOI 10.1016/j.jde.2007.01.008
Loukas Grafakos, Modern Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. MR 2463316, DOI 10.1007/978-0-387-09434-2
Qing Han and Fanghua Lin, Elliptic partial differential equations, Courant Lecture Notes in Mathematics, vol. 1, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997. MR 1669352
Vakhtang Kokilashvili and Alexander Meskhi, Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure, Complex Var. Elliptic Equ. 55 (2010), no. 8-10, 923–936. MR 2674873, DOI 10.1080/17476930903276068
Vakhtang Kokilashvili and Stefan Samko, Maximal and fractional operators in weighted $L^{p(x)}$ spaces, Rev. Mat. Iberoamericana 20 (2004), no. 2, 493–515. MR 2073129, DOI 10.4171/RMI/398
Vakhtang Kokilashvili, Natasha Samko, and Stefan Samko, Singular operators in variable spaces $L^{p(\cdot )}(\Omega ,\rho )$ with oscillating weights, Math. Nachr. 280 (2007), no. 9-10, 1145–1156. MR 2334666, DOI 10.1002/mana.200510542
K. R. Rajagopal, and M. Růžička, Mathematical modeling of electrorheological materials, Contin. Mech. Thermodyn. 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. G. Samko, Density of $C_0^\infty (\textbf {R}^n)$ in the generalized Sobolev spaces $W^{m,p(x)}(\textbf {R}^n)$, Dokl. Akad. Nauk 369 (1999), no. 4, 451–454 (Russian). MR 1748959
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
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
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
V. V. Zhikov, Solvability of the three-dimensional thermistor problem, Tr. Mat. Inst. Steklova 261 (2008), no. Differ. Uravn. i Din. Sist., 101–114 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 261 (2008), no. 1, 98–111. MR 2489700, DOI 10.1134/S0081543808020090