The 𝐿^{𝑝} regularity problem on Lipschitz domains

Kilty, Joel; Shen, Zhongwei

doi:10.1090/S0002-9947-2010-05076-7

The $L^p$ regularity problem on Lipschitz domains
HTML articles powered by AMS MathViewer

by Joel Kilty and Zhongwei Shen PDF

Trans. Amer. Math. Soc. 363 (2011), 1241-1264 Request permission

Abstract:

This paper contains two results on the $L^p$ regularity problem on Lipschitz domains. For second order elliptic systems and $1<p<\infty$, we prove that the solvability of the $L^p$ regularity problem is equivalent to that of the $L^{p^\prime }$ Dirichlet problem. For higher order elliptic equations and systems, we show that if $p>2$, the solvability of the $L^p$ regularity problem is equivalent to a weak reverse Hölder condition with exponent $p$.

References

Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445
Björn E. J. Dahlberg, On the Poisson integral for Lipschitz and $C^{1}$-domains, Studia Math. 66 (1979), no. 1, 13–24. MR 562447, DOI 10.4064/sm-66-1-13-24
Björn E. J. Dahlberg and Carlos E. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace’s equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), no. 3, 437–465. MR 890159, DOI 10.2307/1971407
B. E. J. Dahlberg and C. E. Kenig, $L^p$ estimates for the three-dimensional systems of elastostatics on Lipschitz domains, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 621–634. MR 1044810
B. E. J. Dahlberg, C. E. Kenig, J. Pipher, and G. C. Verchota, Area integral estimates for higher order elliptic equations and systems, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 5, 1425–1461 (English, with English and French summaries). MR 1600375, DOI 10.5802/aif.1605
B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 109–135 (English, with French summary). MR 865663, DOI 10.5802/aif.1062
B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J. 57 (1988), no. 3, 795–818. MR 975122, DOI 10.1215/S0012-7094-88-05735-3
Eugene Fabes, Layer potential methods for boundary value problems on Lipschitz domains, Potential theory—surveys and problems (Prague, 1987) Lecture Notes in Math., vol. 1344, Springer, Berlin, 1988, pp. 55–80. MR 973881, DOI 10.1007/BFb0103344
E. B. Fabes, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), no. 3, 769–793. MR 975121, DOI 10.1215/S0012-7094-88-05734-1
Wen Jie Gao, Layer potentials and boundary value problems for elliptic systems in Lipschitz domains, J. Funct. Anal. 95 (1991), no. 2, 377–399. MR 1092132, DOI 10.1016/0022-1236(91)90035-4
David S. Jerison and Carlos E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203–207. MR 598688, DOI 10.1090/S0273-0979-1981-14884-9
Joel Kilty, The $L^p$ Dirichlet problem for the Stokes system on Lipschitz domains, Indiana Univ. Math. J. 58 (2009), no. 3, 1219–1234. MR 2541365, DOI 10.1512/iumj.2009.58.3568
Jill Pipher and Gregory Verchota, The Dirichlet problem in $L^p$ for the biharmonic equation on Lipschitz domains, Amer. J. Math. 114 (1992), no. 5, 923–972. MR 1183527, DOI 10.2307/2374885
J. Pipher and G. Verchota, A maximum principle for biharmonic functions in Lipschitz and $C^1$ domains, Comment. Math. Helv. 68 (1993), no. 3, 385–414. MR 1236761, DOI 10.1007/BF02565827
Jill Pipher and Gregory C. Verchota, Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators, Ann. of Math. (2) 142 (1995), no. 1, 1–38. MR 1338674, DOI 10.2307/2118610
Zhong Wei Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains, Proc. Amer. Math. Soc. 123 (1995), no. 3, 801–811. MR 1223521, DOI 10.1090/S0002-9939-1995-1223521-9
Zhongwei Shen, The $L^p$ Dirichlet problem for elliptic systems on Lipschitz domains, Math. Res. Lett. 13 (2006), no. 1, 143–159. MR 2200052, DOI 10.4310/MRL.2006.v13.n1.a11
Zhongwei Shen, Necessary and sufficient conditions for the solvability of the $L^p$ Dirichlet problem on Lipschitz domains, Math. Ann. 336 (2006), no. 3, 697–725. MR 2249765, DOI 10.1007/s00208-006-0022-x
Zhongwei Shen, On estimates of biharmonic functions on Lipschitz and convex domains, J. Geom. Anal. 16 (2006), no. 4, 721–734. MR 2271951, DOI 10.1007/BF02922138
Zhongwei Shen, The $L^p$ boundary value problems on Lipschitz domains, Adv. Math. 216 (2007), no. 1, 212–254. MR 2353255, DOI 10.1016/j.aim.2007.05.017
Zhongwei Shen, A relationship between the Dirichlet and regularity problems for elliptic equations, Math. Res. Lett. 14 (2007), no. 2, 205–213. MR 2318619, DOI 10.4310/MRL.2007.v14.n2.a4
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382, DOI 10.1016/0022-1236(84)90066-1
Gregory Verchota, The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Indiana Univ. Math. J. 39 (1990), no. 3, 671–702. MR 1078734, DOI 10.1512/iumj.1990.39.39034
Gregory C. Verchota, Potentials for the Dirichlet problem in Lipschitz domains, Potential theory—ICPT 94 (Kouty, 1994) de Gruyter, Berlin, 1996, pp. 167–187. MR 1404706
M. Wright and M. Mitrea, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, Preprint.