Conormal problem of higher-order parabolic systems with time irregular coefficients
HTML articles powered by AMS MathViewer
- by Hongjie Dong and Hong Zhang PDF
- Trans. Amer. Math. Soc. 368 (2016), 7413-7460 Request permission
Abstract:
The paper is a comprehensive study of $L_p$ and Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions. For the $L_p$ estimates, we assume that the leading coefficients are only bounded and measurable in the $t$ variable and have vanishing mean oscillations (VMO$_x$) with respect to $x$. We also prove the Schauder estimates in two situations: the coefficients are Hölder continuous only in the $x$ variable; the coefficients are Hölder continuous in the $t$ variable as well on the lateral boundary.References
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
- Oleg V. Besov, Valentin P. Il′in, and Sergey M. Nikol′skiĭ, Integral representations of functions and imbedding theorems. Vol. I, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978. Translated from the Russian; Edited by Mitchell H. Taibleson. MR 519341
- Oleg V. Besov, Valentin P. Il′in, and Sergey M. Nikol′skiĭ, Integral representations of functions and imbedding theorems. Vol. II, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1979. Edited by Mitchell H. Taibleson. MR 521808
- 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
- M. Bramanti, M. C. Cerutti, and M. Manfredini, $\scr L^p$ estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. Appl. 200 (1996), no. 2, 332–354. MR 1391154, DOI 10.1006/jmaa.1996.0209
- Sergio Campanato, Equazioni paraboliche del secondo ordine e spazi ${\cal L}^{2,\,\theta }\,(\Omega ,\,\delta )$, Ann. Mat. Pura Appl. (4) 73 (1966), 55–102 (Italian). MR 213737, DOI 10.1007/BF02415082
- Filippo Chiarenza, Michele Frasca, and Placido Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), no. 1, 149–168. MR 1191890
- 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
- Hongjie Dong and Doyoon Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients, Calc. Var. Partial Differential Equations 40 (2011), no. 3-4, 357–389. MR 2764911, DOI 10.1007/s00526-010-0344-0
- Hongjie Dong and Doyoon Kim, Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains, J. Funct. Anal. 261 (2011), no. 11, 3279–3327. MR 2835999, DOI 10.1016/j.jfa.2011.08.001
- Hongjie Dong and Doyoon Kim, On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 889–941. MR 2771670, DOI 10.1007/s00205-010-0345-3
- Hongjie Dong and Doyoon Kim, The conormal derivative problem for higher order elliptic systems with irregular coefficients, Recent advances in harmonic analysis and partial differential equations, Contemp. Math., vol. 581, Amer. Math. Soc., Providence, RI, 2012, pp. 69–97. MR 3013054, DOI 10.1090/conm/581/11534
- Hongjie Dong and Hong Zhang, Schauder estimates for higher-order parabolic systems with time irregular coefficients, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 47–74. MR 3385152, DOI 10.1007/s00526-014-0777-y
- G. Di Fazio, $L^p$ estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7) 10 (1996), no. 2, 409–420 (English, with Italian summary). MR 1405255
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Mariano Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR 1239172
- Barry F. Knerr, Parabolic interior Schauder estimates by the maximum principle, Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 51–58. MR 592103, DOI 10.1007/BF00284620
- Gary M. Lieberman, The conormal derivative problem for elliptic equations of variational type, J. Differential Equations 49 (1983), no. 2, 218–257. MR 708644, DOI 10.1016/0022-0396(83)90013-X
- Gary M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4) 148 (1987), 77–99. MR 932759, DOI 10.1007/BF01774284
- Gary M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity, Differential Integral Equations 5 (1992), no. 6, 1219–1236. MR 1184023
- Gary M. Lieberman, The conormal derivative problem for equations of variational type in nonsmooth domains, Trans. Amer. Math. Soc. 330 (1992), no. 1, 41–67. MR 1116317, DOI 10.1090/S0002-9947-1992-1116317-1
- Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
- Gary M. Lieberman, Oblique derivative problems for elliptic equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. MR 3059278, DOI 10.1142/8679
- Luca Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM J. Math. Anal. 32 (2000), no. 3, 588–615. MR 1786159, DOI 10.1137/S0036141098342842
- N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996. MR 1406091, DOI 10.1090/gsm/012
- N. V. Krylov, On weak uniqueness for some diffusions with discontinuous coefficients, Stochastic Process. Appl. 113 (2004), no. 1, 37–64. MR 2078536, DOI 10.1016/j.spa.2004.03.012
- N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (2007), no. 1-3, 453–475. MR 2304157, DOI 10.1080/03605300600781626
- N. V. Krylov, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal. 250 (2007), no. 2, 521–558. MR 2352490, DOI 10.1016/j.jfa.2007.04.003
- N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, vol. 96, American Mathematical Society, Providence, RI, 2008. MR 2435520, DOI 10.1090/gsm/096
- N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations 35 (2010), no. 1, 1–22. MR 2748616, DOI 10.1080/03605300903424700
- Antonino Maugeri, Dian K. Palagachev, and Lubomira G. Softova, Elliptic and parabolic equations with discontinuous coefficients, Mathematical Research, vol. 109, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. MR 2260015, DOI 10.1002/3527600868
- Dian Palagachev and Lubomira Softova, A priori estimates and precise regularity for parabolic systems with discontinuous data, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 721–742. MR 2153140, DOI 10.3934/dcds.2005.13.721
- Serena Boccia, Schauder estimates for solutions of higher-order parabolic systems, Methods Appl. Anal. 20 (2013), no. 1, 47–67. MR 3085781, DOI 10.4310/MAA.2013.v20.n1.a3
- Wilhelm Schlag, Schauder and $L^p$ estimates for parabolic systems via Campanato spaces, Comm. Partial Differential Equations 21 (1996), no. 7-8, 1141–1175. MR 1399194, DOI 10.1080/03605309608821221
Additional Information
- Hongjie Dong
- Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
- MR Author ID: 761067
- ORCID: 0000-0003-2258-3537
- Email: Hongjie_Dong@brown.edu
- Hong Zhang
- Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
- Email: Hong_Zhang@brown.edu
- Received by editor(s): January 13, 2014
- Received by editor(s) in revised form: September 11, 2014, and October 15, 2014
- Published electronically: November 16, 2015
- Additional Notes: The first author was partially supported by the NSF under agreement DMS-1056737.
The second author was partially supported by the NSF under agreement DMS-1056737. - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7413-7460
- MSC (2010): Primary 35K52, 35J58, 35B45, 35R05
- DOI: https://doi.org/10.1090/tran/6605
- MathSciNet review: 3471096