Elliptic equations with BMO coefficients in Lipschitz domains

Author:
Sun-Sig Byun

Journal:
Trans. Amer. Math. Soc. **357** (2005), 1025-1046

MSC (2000):
Primary 35R05, 35R35; Secondary 35J15, 35J25

DOI:
https://doi.org/10.1090/S0002-9947-04-03624-4

Published electronically:
May 28, 2004

MathSciNet review:
2110431

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study inhomogeneous Dirichlet problems for elliptic equations in divergence form. Optimal regularity requirements on the coefficients and domains for the estimates are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO semi-norms. The domain is supposed to have Lipschitz boundary with small Lipschitz constant. These conditions for the theory do not just weaken the requirements on the coefficients; they also lead to a more general geometric condition on the domain.

**1.**P. Acquistapace,*On BMO regularity for linear elliptic systems*, Ann. Mat. Pura Appl.,**161**(1992), 231-269. MR**93i:35027****2.**P. Auscher and M. Qafsaoui,*Observations on estimates for divergence elliptic equations with VMO coefficients*, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat.,**5**(2002), 487-509. MR**2003e:35052****3.**P. Auscher and P. Tchamitchian,*Gaussian estimates for second order elliptic divergence operators on Lipschitz and domains*, Lecture Notes in Pure and Appl. Math.,**215**, Dekker, New York, 2001, 15-32. MR**2001m:35074****4.**P. Auscher and P. Tchamitchian,*Square roots of elliptic second order divergence operators on strongly Lipschitz domains: theory*, Math. Ann.,**320**(2001), 577-623. MR**2003e:47083****5.**M. Bramanti and L. Brandolini,*estimates for nonvariational hypoelliptic operators with VMO coefficients*, Trans. Amer. Math. Soc.,**352**(2000), 781-822. MR**2000c:35026****6.**M. Bramanti and M.C. Cerutti,*solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients*, Comm. Partial Differential Equations,**18**(1993), 1735-1763. MR**94j:35180****7.**S. Byun,*Parabolic equations with BMO coefficients in Lipschitz domains*, In preparation.**8.**S. Byun and L. Wang,*Elliptic equations with BMO coefficients in Reifenberg domains*, to appear in Comm. Pure Appl. Math.**9.**L.A. Caffarelli and X. Cabré,*Fully nonlinear elliptic equations*, Colloq. Publ., vol. 43, Amer. Math. Soc., Providence, RI, 1995. MR**96h:35046****10.**L.A. Caffarelli and I. Peral,*On estimates for elliptic equations in divergence form*, Comm. Pure Appl. Math.,**51**, (1998), 1-21. MR**95c:35053****11.**A.P. Calderón and A. Zygmmund,*On the existence of certain singular integrals*, Acta Math.,**88**, (1952), 85-139. MR**14:637f****12.**S. Campanato,*Sistemi elliptic in forma divergence. Regolarita all'interno*, Elliptic systems in divergence form. Interior regularity, Quaderni, Scuola Normale Superiore Pisa, Pisa, 1980. MR**83i:35067****13.**F. Chiarenza and M. Frasca and P. Longo,*-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients*, Trans. Amer. Math. Soc.,**336**, (1993), 841-853. MR**93f:35232****14.**F. Chiarenza and M. Frasca and P. Longo,*Interior estimates for nondivergence elliptic equations with discontinuous coefficients*, Ricerche Mat.,**40**, (1991), 149-168. MR**93k:35051****15.**L.C. Evans,*Partial differential equations*, Graduate Studies in Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998. MR**99e:35001****16.**G. Di Fazio,*estimates for divergence form elliptic equations with discontinuous coefficients*, Boll. Un. Mat. Ital A(7),**10**, (1996), 409-420. MR**97e:35034****17.**G. Di Fazio and D.K. Palagachev and M.A. Ragusa,*Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients*, J. Funct. Anal.,**166**, (1999), 179-196. MR**2000d:35037****18.**D. Jerison and C. Kenig,*The inhomogeneous Dirichlet problem in Lipschitz domains*, J. Funct. Anal.,**130**, (2001), 161-219. MR**96b:35042****19.**F. John and L. Nirenberg,*On functions of bounded mean oscillation*, Comm. Pure Appl. Math.,**14**, (1961), 415-426. MR**24:A1348****20.**P.W. Jones,*Extension theorems for BMO*, Indiana Univ. Math. J.,**29**, (1980), 41-66. MR**81b:42047****21.**C. Kenig and T. Toro,*Free boundary regularity for harmonic measures and the Poisson kernel*, Ann. of Math.,**150**, (1999), 369-454. MR**2001d:31004****22.**C. Kenig and T. Toro,*Free boundary regularity for the Poisson kernel below the continuous threshold*, Math. Res. Lett.,**54**, (2002), 247-253. MR**2003c:31005****23.**A. Maugeri and D.K. Palagachev and C. Vitanza,*Oblique derivative problem for uniformly elliptic operators with VMO coefficients and applications*, C. R. Acad. Sci. Paris Sér. I Math.,**327**, (1998), 53-58. MR**2000a:35039****24.**N.G. Meyers,*An -estimate for the gradient of solutions of second order elliptic divergence equations*, Ann. Scuola Norm. Sup. Pisa(3),**17**, (1963), 189-206. MR**28:2328****25.**C.B. Morrey,*Multiple integrals in the calculus of variations*, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR**34:2380****26.**M.A. Ragusa,*Local Hölder regularity for solutions of elliptic systems*, Duke Math. J.,**113**, (2002), 385-397. MR**2003f:35070****27.**D. Sarason,*Functions of vanishing mean oscillation*, Trans. Amer. Math. Soc.,**207**, (1975), 391-405. MR**51:13690****28.**C.G. Simader,*On Dirichlet's boundary value problem. An theory based on a generalization of Gagliardo's inequality*, Lecture Notes in Mathematics, Vol. 268, Springer-Verlag, Berlin-New York, 1972. MR**57:13169****29.**L.G. Softova,*Oblique derivative problem for parabolic operators with VMO coefficients*, Manuscripte Math.,**103**, (2000), 203-220. MR**2001k:35180****30.**L.G. Softova,*Quasilinear parabolic equations with VMO coefficients*, C. R. Acad. Bulgare Sci.,**53**, (2000), 17-20.**31.**L.G. Softova,*Parabolic equations with VMO coefficients in Morrey spaces*, Electron. J. Differential Equations,**51**, (2001), 1-25. MR**2002e:35108****32.**E. M. Stein,*Harmonic Analysis*, Princeton University Press, Princeton, NJ, 1993. MR**95c:42002****33.**L. Wang,*A geometric approach to the Calderón-Zygmmund estimates*, Acta Mathematica Sinica,**19**, (2003), 381-396.

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Additional Information

**Sun-Sig Byun**

Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Address at time of publication:
Department of Mathematics, University of California, Irvine, California 92697

Email:
byun@math.uci.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03624-4

Keywords:
Elliptic equations,
Lipschitz domains,
BMO,
maximal function,
Vitali covering lemma,
compactness method

Received by editor(s):
July 23, 2003

Published electronically:
May 28, 2004

Additional Notes:
This work was supported in part by NSF Grant #0100679

Article copyright:
© Copyright 2004
American Mathematical Society