Elliptic equations with matrix weights and measurable nonlinearities on nonsmooth domains

Byun, Sun-Sig; Cho, Yumi; Lee, Ho-Sik

doi:10.1090/proc/16770

Elliptic equations with matrix weights and measurable nonlinearities on nonsmooth domains
HTML articles powered by AMS MathViewer

by Sun-Sig Byun, Yumi Cho and Ho-Sik Lee;

Proc. Amer. Math. Soc. 152 (2024), 2963-2982

DOI: https://doi.org/10.1090/proc/16770

Published electronically: May 21, 2024

HTML | PDF | Request permission

Abstract:

We study general elliptic equations with singular/degenerate matrix weights and measurable nonlinearities on nonsmooth bounded domains to obtain a global Calderón-Zygmund type estimate under possibly minimal assumptions that the logarithm of the matrix weight has a small bounded mean oscillation (BMO) norm, the nonlinearity is allowed to be merely measurable in one variable but has a small BMO norm in the other variables and that the boundary of the domain is sufficiently flat in Reifenberg sense.

References

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
Sumiya Baasandorj, Sun-Sig Byun, and Ho-Sik Lee, Global gradient estimates for a general class of quasilinear elliptic equations with Orlicz growth, Proc. Amer. Math. Soc. 149 (2021), no. 10, 4189–4206. MR 4305974, DOI 10.1090/proc/15585
Anna Kh. Balci, Sun-Sig Byun, Lars Diening, and Ho-Sik Lee, Global maximal regularity for equations with degenerate weights, J. Math. Pures Appl. (9) 177 (2023), 484–530. MR 4629762, DOI 10.1016/j.matpur.2023.07.010
Anna Kh. Balci, Lars Diening, Raffaella Giova, and Antonia Passarelli di Napoli, Elliptic equations with degenerate weights, SIAM J. Math. Anal. 54 (2022), no. 2, 2373–2412. MR 4410267, DOI 10.1137/21M1412529
Anna Kh. Balci, Lars Diening, and Markus Weimar, Higher order Calderón-Zygmund estimates for the $p$-Laplace equation, J. Differential Equations 268 (2020), no. 2, 590–635. MR 4021898, DOI 10.1016/j.jde.2019.08.009
Ariel Barton, Trace and extension theorems relating Besov spaces to weighted averaged Sobolev spaces, Math. Inequal. Appl. 21 (2018), no. 3, 817–870. MR 3844938, DOI 10.7153/mia-2018-21-58
Eric Bonnetier and Michael Vogelius, An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section, SIAM J. Math. Anal. 31 (2000), no. 3, 651–677. MR 1745481, DOI 10.1137/S0036141098333980
Sun-Sig Byun and Youchan Kim, Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math. 288 (2016), 152–200. MR 3436384, DOI 10.1016/j.aim.2015.10.015
Sun-Sig Byun and Youchan Kim, Riesz potential estimates for parabolic equations with measurable nonlinearities, Int. Math. Res. Not. IMRN 2018 (2018), no. 21, 6737–6779.
Sun-Sig Byun and Seungjin Ryu, Global weighted estimates for the gradient of solutions to nonlinear elliptic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), no. 2, 291–313. MR 3035978, DOI 10.1016/j.anihpc.2012.08.001
Sun-Sig Byun and Lihe Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math. 57 (2004), no. 10, 1283–1310. MR 2069724, DOI 10.1002/cpa.20037
Sun-Sig Byun and Lihe Wang, The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. London Math. Soc. (3) 90 (2005), no. 1, 245–272.
Sun-Sig Byun and Lihe Wang, Elliptic equations with measurable coefficients in Reifenberg domains, Adv. Math. 225 (2010), no. 5, 2648–2673. MR 2680179, DOI 10.1016/j.aim.2010.05.014
Sun-Sig Byun, Lihe Wang, and Shulin Zhou, Nonlinear elliptic equations with BMO coefficients in Reifenberg domains, J. Funct. Anal. 250 (2007), no. 1, 167–196. MR 2345911, DOI 10.1016/j.jfa.2007.04.021
Dat Cao, Tadele Mengesha, and Tuoc Phan, Weighted-$W^{1,p}$ estimates for weak solutions of degenerate and singular elliptic equations, Indiana Univ. Math. J. 67 (2018), no. 6, 2225–2277. MR 3900368, DOI 10.1512/iumj.2018.67.7533
Jongkeun Choi, Hongjie Dong, and Zongyuan Li, Optimal regularity for a Dirichlet-conormal problem in Reifenberg flat domain, Appl. Math. Optim. 83 (2021), no. 3, 1547–1583. MR 4261267, DOI 10.1007/s00245-019-09600-2
Guy David and Tatiana Toro, Reifenberg parameterizations for sets with holes, Mem. Amer. Math. Soc. 215 (2012), no. 1012, vi+102. MR 2907827, DOI 10.1090/S0065-9266-2011-00629-5
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
Hongjie Dong and Doyoon Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal. 196 (2010), no. 1, 25–70. MR 2601069, DOI 10.1007/s00205-009-0228-7
Hongjie Dong and Doyoon Kim, $L_q$-estimates for stationary Stokes system with coefficients measurable in one direction, Bull. Math. Sci. 9 (2019), no. 1, 1950004, 30. MR 3949432, DOI 10.1142/S1664360719500048
Hongjie Dong and Tuoc Phan, Parabolic and elliptic equations with singular or degenerate coefficients: the Dirichlet problem, Trans. Amer. Math. Soc. 374 (2021), no. 9, 6611–6647. MR 4302171, DOI 10.1090/tran/8397
Hongjie Dong and Tuoc Phan, Weighted mixed-norm $L_p$-estimates for elliptic and parabolic equations in non-divergence form with singular coefficients, Rev. Mat. Iberoam. 37 (2021), no. 4, 1413–1440. MR 4269403, DOI 10.4171/rmi/1233
Irene Drelichman and Ricardo G. Durán, Improved Poincaré inequalities with weights, J. Math. Anal. Appl. 347 (2008), no. 1, 286–293. MR 2433844, DOI 10.1016/j.jmaa.2008.06.005
Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Enrico Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1962933, DOI 10.1142/9789812795557
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
Antonio Giuseppe Grimaldi, Regularity results for solutions to a class of obstacle problems, Nonlinear Anal. Real World Appl. 62 (2021), Paper No. 103377, 26. MR 4273089, DOI 10.1016/j.nonrwa.2021.103377
Petteri Harjulehto and Peter Hästö, Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics, vol. 2236, Springer, Cham, 2019. MR 3931352, DOI 10.1007/978-3-030-15100-3
Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
Joshua Isralowitz and Kabe Moen, Matrix weighted Poincaré inequalities and applications to degenerate elliptic systems, Indiana Univ. Math. J. 68 (2019), no. 5, 1327–1377. MR 4030471, DOI 10.1512/iumj.2019.68.7686
Carlos E. Kenig and Tatiana Toro, Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math. (2) 150 (1999), no. 2, 369–454. MR 1726699, DOI 10.2307/121086
T. Kilpeläinen and P. Koskela, Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal. 23 (1994), no. 7, 899–909. MR 1302151, DOI 10.1016/0362-546X(94)90127-9
Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal. 26 (2007), no. 4, 345–361. MR 2300337, DOI 10.1007/s11118-007-9042-8
Doyoon Kim and N. V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal. 39 (2007), no. 2, 489–506. MR 2338417, DOI 10.1137/050646913
Youchan Kim, Gradient estimates for elliptic equations with measurable nonlinearities, J. Math. Pures Appl. (9) 114 (2018), 118–145 (English, with English and French summaries). MR 3801752, DOI 10.1016/j.matpur.2017.11.003
Youchan Kim and Seungjin Ryu, Global gradient estimates for parabolic equations with measurable nonlinearities, Nonlinear Anal. 164 (2017), 77–99. MR 3712021, DOI 10.1016/j.na.2017.08.009
Yan Yan Li and Michael Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal. 153 (2000), no. 2, 91–151. MR 1770682, DOI 10.1007/s002050000082
Tadele Mengesha and Nguyen Cong Phuc, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations 250 (2011), no. 5, 2485–2507. MR 2756073, DOI 10.1016/j.jde.2010.11.009
Tadele Mengesha and Nguyen Cong Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal. 203 (2012), no. 1, 189–216. MR 2864410, DOI 10.1007/s00205-011-0446-7
Giuseppe Mingione, Nonlinear aspects of Calderón-Zygmund theory, Jahresber. Dtsch. Math.-Ver. 112 (2010), no. 3, 159–191. MR 2722503, DOI 10.1365/s13291-010-0004-5
Giuseppe Mingione, Short tales from nonlinear Calderón-Zygmund theory, Nonlocal and nonlinear diffusions and interactions: new methods and directions, Lecture Notes in Math., vol. 2186, Springer, Cham, 2017, pp. 159–204. MR 3588124
Mikko Parviainen, Global higher integrability for parabolic quasiminimizers in nonsmooth domains, Calc. Var. Partial Differential Equations 31 (2008), no. 1, 75–98. MR 2342615, DOI 10.1007/s00526-007-0106-9
Tuoc Phan, Weighted Calderón-Zygmund estimates for weak solutions of quasi-linear degenerate elliptic equations, Potential Anal. 52 (2020), no. 3, 393–425. MR 4067298, DOI 10.1007/s11118-018-9737-z
Christoph Scheven, Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles, Manuscripta Math. 146 (2015), no. 1-2, 7–63. MR 3294416, DOI 10.1007/s00229-014-0684-8
R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. MR 1422252, DOI 10.1090/surv/049
Tatiana Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc. 44 (1997), no. 9, 1087–1094. MR 1470167