Transactions of the American Mathematical Society

Author(s): Sun-Sig Byun
Journal: Trans. Amer. Math. Soc. 357 (2005), 1025-1046.
MSC (2000): Primary 35R05, 35R35; Secondary 35J15, 35J25
Posted: May 28, 2004
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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 $W^{1,p} (1<p<\infty)$ 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 $W^{1,p}$ 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 $W^{1,p}$ 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 $C^{1}$ 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: $L^{p}$ theory, Math. Ann., 320 (2001), 577-623. MR 2003e:47083
5.: M. Bramanti and L. Brandolini, $L^{p}$ 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, $W^{1,2}_{p}$ 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 $W^{1,p}$ 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, $W^{2,p}$ -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 $W^{2,p}$ 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.

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35R05, 35R35, 35J15, 35J25

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: 10.1090/S0002-9947-04-03624-4
PII: S 0002-9947(04)03624-4
Keywords: Elliptic equations, Lipschitz domains, BMO, maximal function, Vitali covering lemma, compactness method
Received by editor(s): July 23, 2003
Posted: May 28, 2004
Additional Notes: This work was supported in part by NSF Grant \#0100679
Copyright of article: Copyright 2004, American Mathematical Society

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