Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Nondivergence parabolic equations in weighted variable exponent spaces


Authors: Sun-Sig Byun, Mikyoung Lee and Jihoon Ok
Journal: Trans. Amer. Math. Soc. 370 (2018), 2263-2298
MSC (2010): Primary 35K20; Secondary 46E30, 46E35
DOI: https://doi.org/10.1090/tran/7352
Published electronically: November 30, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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, https://doi.org/10.1007/s002050100117
  • [2] Emilio Acerbi and Giuseppe Mingione, Gradient estimates for the $ p(x)$-Laplacean system, J. Reine Angew. Math. 584 (2005), 117-148. MR 2155087, https://doi.org/10.1515/crll.2005.2005.584.117
  • [3] Emilio Acerbi and Giuseppe Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), no. 2, 285-320. MR 2286632, https://doi.org/10.1215/S0012-7094-07-13623-8
  • [4] 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
  • [5] 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, https://doi.org/10.4171/RMI/817
  • [6] 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, https://doi.org/10.1080/03605309308820991
  • [7] 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, https://doi.org/10.1007/s00208-015-1194-z
  • [8] 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, https://doi.org/10.1515/crelle-2014-0004
  • [9] 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, https://doi.org/10.1007/s00220-014-1962-8
  • [10] 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, https://doi.org/10.5186/aasfm.2016.4102
  • [11] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139. MR 0052553, https://doi.org/10.1007/BF02392130
  • [12] 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, https://doi.org/10.1137/050624522
  • [13] 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, https://doi.org/10.2307/2154379
  • [14] 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
  • [15] 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, https://doi.org/10.2478/s13540-011-0023-7
  • [16] L. Diening, Maximal function on generalized Lebesgue spaces $ L^{p(\cdot)}$, Math. Inequal. Appl. 7 (2004), no. 2, 245-253. MR 2057643, https://doi.org/10.7153/mia-07-27
  • [17] 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
  • [18] L. Diening and P. Hästö, Muckenhoupt weights in variable exponent spaces, preprint, 2011.
  • [19] 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, https://doi.org/10.1080/17476933.2010.504843
  • [20] 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, https://doi.org/10.1515/crll.2003.081
  • [21] Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943
  • [22] 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, https://doi.org/10.1016/j.jde.2007.01.008
  • [23] Loukas Grafakos, Modern Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. MR 2463316
  • [24] 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
  • [25] 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, https://doi.org/10.1080/17476930903276068
  • [26] 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, https://doi.org/10.4171/RMI/398
  • [27] 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, https://doi.org/10.1002/mana.200510542
  • [28] K. R. Rajagopal, and M. Růžička, Mathematical modeling of electrorheological materials, Contin. Mech. Thermodyn. 13 (2001), 59-78.
  • [29] Michael Růžička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. MR 1810360
  • [30] S. G. Samko, Density of $ C_0^\infty({\bf R}^n)$ in the generalized Sobolev spaces $ W^{m,p(x)}({\bf R}^n)$, Dokl. Akad. Nauk 369 (1999), no. 4, 451-454 (Russian). MR 1748959
  • [31] 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, https://doi.org/10.1016/j.jmaa.2005.02.002
  • [32] 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
  • [33] 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
  • [34] V. V. Zhikov, Solvability of the three-dimensional thermistor problem, Differ. Uravn. i Din. Sist. 261 (2008), 101-114 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 261 (2008), no. 1, 98-111. MR 2489700, https://doi.org/10.1134/S0081543808020090

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35K20, 46E30, 46E35

Retrieve articles in all journals with MSC (2010): 35K20, 46E30, 46E35


Additional Information

Sun-Sig Byun
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, 08826, Korea
Email: byun@snu.ac.kr

Mikyoung Lee
Affiliation: Department of Mathematical Sciences, KAIST, Daejeon 34141, Korea
Email: mikyounglee@kaist.ac.kr

Jihoon Ok
Affiliation: Department of Applied Mathematics and Institute of Natural Science, Kyung Hee University, Yongin 17104, Korea
Email: jihoonok@khu.ac.kr

DOI: https://doi.org/10.1090/tran/7352
Keywords: Parabolic equation, Calder\'on-Zygmund estimate, variable exponent, parabolic Muckenhoupt weight, BMO space
Received by editor(s): May 20, 2015
Published electronically: November 30, 2017
Additional Notes: The first author was supported by the National Research Foundation of Korea (NRF-2017R1A2B2003877). The second author was supported by the National Research Foundation of Korea (NRF-2015R1A4A1041675). The third author was supported by the National Research Foundation of Korea (NRF-2017R1C1B2010328)
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society