Solvability of second-order equations with hierarchically partially BMO coefficients

Author:
Hongjie Dong

Journal:
Trans. Amer. Math. Soc. **364** (2012), 493-517

MSC (2010):
Primary 35J15, 35K15, 35R05

Published electronically:
August 25, 2011

MathSciNet review:
2833589

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By using some recent results for divergence-form equations, we study the -solvability of second-order elliptic and parabolic equations in nondivergence form for any . The leading coefficients are assumed to be in locally BMO spaces with suitably small BMO seminorms. We not only extend several previous results by Krylov and Kim to the full range of , but also deal with equations with more general coefficients.

**1.**Emilio Acerbi and Giuseppe Mingione,*Gradient estimates for a class of parabolic systems*, Duke Math. J.**136**(2007), no. 2, 285–320. MR**2286632**, 10.1215/S0012-7094-07-13623-8**2.**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**, 10.1080/03605309308820991**3.**Sun-Sig Byun and Lihe Wang,*Gradient estimates for elliptic systems in non-smooth domains*, Math. Ann.**341**(2008), no. 3, 629–650. MR**2399163**, 10.1007/s00208-008-0207-6**4.**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****5.**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**, 10.1090/S0002-9947-1993-1088476-1**6.**Giuseppe Chiti,*A 𝑊^{2,2} bound for a class of elliptic equations in nondivergence form with rough coefficients*, Invent. Math.**33**(1976), no. 1, 55–60. MR**0404848****7.**Hongjie Dong,*Solvability of parabolic equations in divergence form with partially BMO coefficients*, J. Funct. Anal.**258**(2010), no. 7, 2145–2172. MR**2584743**, 10.1016/j.jfa.2010.01.003**8.**H. Dong, D. Kim, Parabolic and elliptic systems with VMO coefficients,*Methods Appl. Anal.***16**(2009), no. 3, 365-388.**9.**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**, 10.1007/s00205-009-0228-7**10.**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**, 10.1007/s00526-010-0344-0**11.**Hongjie Dong and N. V. Krylov,*Second-order elliptic and parabolic equations with 𝐵(ℝ²,𝕍𝕄𝕆) coefficients*, Trans. Amer. Math. Soc.**362**(2010), no. 12, 6477–6494. MR**2678983**, 10.1090/S0002-9947-2010-05215-8**12.**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR**1814364****13.**Robert Haller-Dintelmann, Horst Heck, and Matthias Hieber,*𝐿^{𝑝}-𝐿^{𝑞} estimates for parabolic systems in non-divergence form with VMO coefficients*, J. London Math. Soc. (2)**74**(2006), no. 3, 717–736. MR**2286441**, 10.1112/S0024610706023192**14.**Doyoon Kim,*Parabolic equations with measurable coefficients. II*, J. Math. Anal. Appl.**334**(2007), no. 1, 534–548. MR**2332574**, 10.1016/j.jmaa.2006.12.077**15.**Doyoon Kim,*Elliptic and parabolic equations with measurable coefficients in 𝐿_{𝑝}-spaces with mixed norms*, Methods Appl. Anal.**15**(2008), no. 4, 437–467. MR**2550072**, 10.4310/MAA.2008.v15.n4.a3**16.**-, Parabolic equations with partially BMO coefficients and boundary value problems in Sobolev spaces with mixed norms,*Potential Anal.***33**(2010), no. 1, 17-46.**17.**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**, 10.1137/050646913**18.**Doyoon Kim and N. V. Krylov,*Parabolic equations with measurable coefficients*, Potential Anal.**26**(2007), no. 4, 345–361. MR**2300337**, 10.1007/s11118-007-9042-8**19.**Peer Christian Kunstmann,*On maximal regularity of type 𝐿^{𝑝}-𝐿^{𝑞} under minimal assumptions for elliptic non-divergence operators*, J. Funct. Anal.**255**(2008), no. 10, 2732–2759. MR**2464190**, 10.1016/j.jfa.2008.09.017**20.**N. V. Krylov,*Nonlinear elliptic and parabolic equations of the second order*, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR**901759****21.**N. V. Krylov,*Parabolic and elliptic equations with VMO coefficients*, Comm. Partial Differential Equations**32**(2007), no. 1-3, 453–475. MR**2304157**, 10.1080/03605300600781626**22.**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**, 10.1016/j.jfa.2007.04.003**23.**N. V. Krylov,*Second-order elliptic equations with variably partially VMO coefficients*, J. Funct. Anal.**257**(2009), no. 6, 1695–1712. MR**2540989**, 10.1016/j.jfa.2009.06.014**24.**Gary M. Lieberman,*Second order parabolic differential equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR**1465184****25.**Alfredo Lorenzi,*On elliptic equations with piecewise constant coefficients*, Applicable Anal.**2**(1972), no. 1, 79–96. MR**0296490****26.**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****27.**Dian K. Palagachev and Lubomira G. Softova,*Characterization of the interior for parabolic systems with discontinuous coefficients*, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.**16**(2005), no. 2, 125–132 (English, with English and Italian summaries). MR**2225506**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
35J15,
35K15,
35R05

Retrieve articles in all journals with MSC (2010): 35J15, 35K15, 35R05

Additional Information

**Hongjie Dong**

Affiliation:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912

Email:
Hongjie_Dong@brown.edu

DOI:
https://doi.org/10.1090/S0002-9947-2011-05453-X

Keywords:
Second-order equations,
vanishing mean oscillation,
partially BMO coefficients,
Sobolev spaces

Received by editor(s):
April 30, 2009

Received by editor(s) in revised form:
August 22, 2010

Published electronically:
August 25, 2011

Additional Notes:
The author was partially supported by NSF grant number DMS-0635607 from IAS, and NSF grant number DMS-0800129.

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.