Gradient estimate on convex domains and applications

Wang, Feng-Yu; Yan, Lixin

doi:10.1090/S0002-9939-2012-11480-7

Gradient estimate on convex domains and applications
HTML articles powered by AMS MathViewer

by Feng-Yu Wang and Lixin Yan PDF

Proc. Amer. Math. Soc. 141 (2013), 1067-1081 Request permission

Abstract:

By solving the Skorokhod equation for reflecting diffusion processes on a convex domain, gradient estimates for the associated Neumann semigroup are derived. As applications, functional/Harnack inequalities are established for the Neumann semigroup. When the domain is bounded, the gradient estimates are applied to the study of Riesz transforms and regularity of the inhomogeneous Neumann problems on convex domains.

References

Vilhelm Adolfsson and David Jerison, $L^p$-integrability of the second order derivatives for the Neumann problem in convex domains, Indiana Univ. Math. J. 43 (1994), no. 4, 1123–1138. MR 1322613, DOI 10.1512/iumj.1994.43.43049
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
Pascal Auscher, Thierry Coulhon, Xuan Thinh Duong, and Steve Hofmann, Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 911–957 (English, with English and French summaries). MR 2119242, DOI 10.1016/j.ansens.2004.10.003
Dominique Bakry and Michel Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 15, 775–778 (French, with English summary). MR 772092
Richard F. Bass and Pei Hsu, The semimartingale structure of reflecting Brownian motion, Proc. Amer. Math. Soc. 108 (1990), no. 4, 1007–1010. MR 1007487, DOI 10.1090/S0002-9939-1990-1007487-8
Richard F. Bass and Pei Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab. 19 (1991), no. 2, 486–508. MR 1106272
Richard F. Bass, Krzysztof Burdzy, and Zhen-Qing Chen, Uniqueness for reflecting Brownian motion in lip domains, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 2, 197–235 (English, with English and French summaries). MR 2124641, DOI 10.1016/j.anihpb.2004.06.001
Thierry Coulhon and Xuan Thinh Duong, Riesz transforms for $1\leq p\leq 2$, Trans. Amer. Math. Soc. 351 (1999), no. 3, 1151–1169. MR 1458299, DOI 10.1090/S0002-9947-99-02090-5
Thierry Coulhon and Adam Sikora, Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem, Proc. Lond. Math. Soc. (3) 96 (2008), no. 2, 507–544. MR 2396848, DOI 10.1112/plms/pdm050
Giuseppe Da Prato, Michael Röckner, and Feng-Yu Wang, Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups, J. Funct. Anal. 257 (2009), no. 4, 992–1017. MR 2535460, DOI 10.1016/j.jfa.2009.01.007
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239, DOI 10.1017/CBO9780511566158
X.T. Duong, S. Hofmann, D. Mitrea, M. Mitrea, L.X. Yan, Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems, to appear in Rev. Mat. Iberoamericana (2011).
Xuan Thinh Duong and Alan MacIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999), no. 2, 233–265. MR 1715407, DOI 10.4171/RMI/255
Fu-Zhou Gong and Feng-Yu Wang, Heat kernel estimates with application to compactness of manifolds, Q. J. Math. 52 (2001), no. 2, 171–180. MR 1838361, DOI 10.1093/qjmath/52.2.171
Alexander Grigor′yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom. 45 (1997), no. 1, 33–52. MR 1443330
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
P. Grisvard, G. Iooss, Problèmes aux limites unilatéraux dans des domaines non réguliers, Publications des Séminaires de Mathématiques, Université de Rennes, 9 (1975), 1–26.
Elton P. Hsu, Multiplicative functional for the heat equation on manifolds with boundary, Michigan Math. J. 50 (2002), no. 2, 351–367. MR 1914069, DOI 10.1307/mmj/1028575738
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
Svitlana Mayboroda and Marius Mitrea, Sharp estimates for Green potentials on non-smooth domains, Math. Res. Lett. 11 (2004), no. 4, 481–492. MR 2092902, DOI 10.4310/MRL.2004.v11.n4.a7
V. Maz′ya, B. Plamenevskiĭ, Estimates in $L_p$ and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary, Amer. Math. Soc. Transl. 123 (1984), 1–56.
Dorina Mitrea, Marius Mitrea, and Lixin Yan, Boundary value problems for the Laplacian in convex and semiconvex domains, J. Funct. Anal. 258 (2010), no. 8, 2507–2585. MR 2593333, DOI 10.1016/j.jfa.2010.01.012
El Maati Ouhabaz, Analysis of heat equations on domains, London Mathematical Society Monographs Series, vol. 31, Princeton University Press, Princeton, NJ, 2005. MR 2124040
Michael Röckner and Feng-Yu Wang, Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), no. 1, 27–37. MR 2646789, DOI 10.1142/S0219025710003936
N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884
Feng-Yu Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997), no. 3, 417–424. MR 1481127, DOI 10.1007/s004400050137
Feng-Yu Wang, Gradient estimates and the first Neumann eigenvalue on manifolds with boundary, Stochastic Process. Appl. 115 (2005), no. 9, 1475–1486. MR 2158015, DOI 10.1016/j.spa.2005.04.004
Feng-Yu Wang, Estimates of the first Neumann eigenvalue and the log-Sobolev constant on non-convex manifolds, Math. Nachr. 280 (2007), no. 12, 1431–1439. MR 2344874, DOI 10.1002/mana.200410555
Feng-Yu Wang, Semigroup properties for the second fundamental form, Doc. Math. 15 (2010), 527–543. MR 2679065
Feng-Yu Wang, Gradient and Harnack inequalities on noncompact manifolds with boundary, Pacific J. Math. 245 (2010), no. 1, 185–200. MR 2602689, DOI 10.2140/pjm.2010.245.185
Jiaping Wang, Global heat kernel estimates, Pacific J. Math. 178 (1997), no. 2, 377–398. MR 1447421, DOI 10.2140/pjm.1997.178.377
Toshio Yamada and Shinzo Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167. MR 278420, DOI 10.1215/kjm/1250523691