Abstract
We develop a global Calderón–Zygmund theory for a quasilinear divergence form parabolic operator with discontinuous entries which exhibit nonlinearities both with respect to the weak solution u and its spatial gradient Du in a nonsmooth domain. The nonlinearity behaves as the parabolic p-Laplacian in Du, its discontinuity with respect to the independent variables is measured in terms of small-BMO, while only Hölder continuity is required with respect to u and the underlying domain is assumed to be \(\delta \)-Reifenberg flat. We introduce and employ essentially a new concept of the intrinsic parabolic maximal function in order to overcome the main difficulties stemming from both the parabolic scaling deficiency and the nonlinearity of u-variable of such a very general parabolic operator, obtaining optimal \(L^q\)-estimates for the spatial gradient under a minimal geometric condition on the domain.
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References
Acerbi, E., Mingione, G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136(2), 285–320 (2007)
Bögelein, V., Duzaar, F., Mingione, G.: The regularity of general parabolic systems with degenerate diffusion. Mem. Am. Math. Soc. 221(1041), vi+143 (2013)
Bögelein, V., Parviainen, M.: Self-improving property of nonlinear higher order parabolic systems near the boundary. NoDEA Nonlinear Differ. Equ. Appl. 17(1), 21–54 (2010)
Bögelein, V., Scheven, C.: Higher integrability in parabolic obstacle problems. Forum Math. 24(5), 931–972 (2012)
Byun, S.-S., Ok, J., Ryu, S.: Global gradient estimates for general nonlinear parabolic equations in nonsmooth domains. J. Differ. Equ. 254(11), 4290–4326 (2013)
Byun, S.-S., Ok, J., Palagachev, D.K., Softova, L.G.: Parabolic systems with measurable coefficients in weighted Orlicz spaces. Commun. Contemp. Math. 18(2), 1550018 (2016). 19 pp
Byun, S.-S., Palagachev, D.K., Softova, L.G.: Global gradient estimates in weighted Lebesgue spaces for parabolic operators. Ann. Acad. Sci. Fenn. Math. 41(1), 67–83 (2016)
Byun, S.-S., Palagachev, D.K., Shin, P.: Boundedness of solutions to quasilinear parabolic equations. J. Differ. Equ. 261(12), 6790–6805 (2016)
Byun, S.-S., Palagachev, D.K., Shin, P.: Global Sobolev regularity for general elliptic equations of \(p\)-Laplacian type. Calc. Var. Partial Differ. Equ. 57(5), 19 (2018). Art. 135
Byun, S.-S., Ryu, S.: Global weighted estimates for the gradient of solutions to nonlinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(2), 291–313 (2013)
Byun, S.-S., Wang, L.: \(L^p\) estimates for parabolic equations in Reifenberg domains. J. Funct. Anal. 223(1), 44–85 (2005)
Caffarelli, L.A., Peral, I.: On \(W^{1, p}\) estimates for elliptic equations in divergence form. Comm. Pure Appl. Math. 51(1), 1–21 (1998)
David, G., Toro, T.: Reifenberg parameterizations for sets with holes. Mem. Am. Math. Soc. 215(1012), vi+102 (2012)
DiBenedetto, E.: Degenerate Parabolic Equations. Universitext, Springer, New York (1993)
Diening, L., Schwarzacher, S.: Global gradient estimates for the \(p(\cdot )\)-Laplacian. Nonlinear Anal. 106, 70–85 (2014)
Duzaar, F., Mingione, G., Steffen, K.: Parabolic systems with polynomial growth and regularity. Mem. Am. Math. Soc. 214(1005), x+118 (2011)
Guliyev, V., Softova, L.G.: Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients. J. Differ. Equ. 259(6), 2368–2387 (2015)
Kinnunen, J., Lewis, J.L.: Higher integrability for parabolic systems of \(p\)-Laplacian type. Duke Math. J. 102(2), 253–271 (2000)
Milakis, E., Toro, T.: Divergence form operators in Reifenberg flat domains. Math. Z. 264(1), 15–41 (2010)
Nguyen, T.: Interior Calderón–Zygmund estimates for solutions to general parabolic equations of p-Laplacian type. Calc. Var. Partial Differ. Equ. 56, 42 (2017). Art. 173
Nguyen, T., Phan, T.: Interior gradient estimates for quasilinear elliptic equations. Calc. Var. Partial Differ. Equ. 55(3), 33 (2016). Art. 59
Parviainen, M.: Global gradient estimates for degenerate parabolic equations in nonsmooth domains. Ann. Mat. Pura Appl. 188(2), 333–358 (2009)
Parviainen, M.: Reverse Hölder inequalities for singular parabolic equations near the boundary. J. Differ. Equ. 246(2), 512–540 (2009)
Scheven, C.: Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles. Manuscripta Math. 146(1–2), 7–63 (2015)
Schwarzacher, S.: Hölder-Zygmund estimates for degenerate parabolic systems. J. Differ. Equ. 256(7), 2423–2448 (2014)
Toro, T.: Doubling and flatness: geometry of measures. Notices Amer. Math. Soc. 44(9), 1087–1094 (1997)
Acknowledgements
S.-S. Byun was supported by NRF-2017R1A2B003877. D.K. Palagachev is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). P. Shin was supported by NRF-2015R1A4A1041675.
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Byun, SS., Palagachev, D.K. & Shin, P. Optimal regularity estimates for general nonlinear parabolic equations. manuscripta math. 162, 67–98 (2020). https://doi.org/10.1007/s00229-019-01127-8
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DOI: https://doi.org/10.1007/s00229-019-01127-8