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Gaussian Estimates in Lipschitz Domains

Published online by Cambridge University Press:  20 November 2018

N. Th. Varopoulos
N. Th. Varopoulos*
Affiliation:
Institut Universitaire de France, Université Paris VI, Département de mathématiques, 4, Place Jussieu, 75005 Paris, France
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Abstract

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We give upper and lower Gaussian estimates for the diffusion kernel of a divergence and nondivergence form elliptic operator in a Lipschitz domain.

Résumé

Résumé

On donne des estimations Gaussiennes pour le noyau d'une diffusion, réversible ou pas, dans un domaine Lipschitzien.

Keywords

39A7035-0265M06
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 2 , 01 April 2003 , pp. 401 - 431
Copyright
Copyright © Canadian Mathematical Society 2003

References

[1] Aronson, D. G., Non negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa 22 (1968), 607–694.Google Scholar
[2] Bass, R. F. and Burdzy, K., A boundary Harnack principle in twisted Hçlder domains. Ann. of Math. (134) 134 (1991), 253–276.Google Scholar
[3] Bass, R. F. and Burdzy, K., The boundary Harnack principle for nondivergence form elliptic operators. J. LondonMath. Soc. (2) 50 (1994), 157–169.Google Scholar
[4] Bauman, P., Positive solutions of elliptic equations in nondivergence form and their adjoints. Ark. Mat. 22 (1984), 153–173.Google Scholar
[5] Bauman, P., A Wiener test for nondivergence structure, second-order elliptic equations. Indiana Univ. Math. J. 34 (1985), 825–844.Google Scholar
[6] Bensoussan, A., Lions, J.-L. and Papanicolaou, G. C., Asymptotic Analysis of Periodic Structures. North-Holland Publ., 1978.Google Scholar
[7] Calderon, A. P., Lebesgue spaces of differentiable functions and distributions. Proc. Sympos. Pure Math. 5 (1961), 33–49.Google Scholar
[8] Carleson, L., On the existence of boundary values for harmonic functions in several variables. Ark. Mat. 4 (1962), 339–393.Google Scholar
[9] Carne, T. K., A transmutation formula for Markov chains. Bull. Sci. Math. (2) 109 (1985), 399–405.Google Scholar
[10] Chung, S. Y. A., Wilson, J. M. and Wolf, T. H., Some weighted norm inequalities concerning the Schrçdinger Operator. Comm. Math. Helv. 60 (1985), 217–246.Google Scholar
[11] Saloff-Coste, L., A note on Poincaré Sobolev and Harnack inequalities. Duke Math. J. IMRN 2 (1992), 27–28.Google Scholar
[12] Delmotte, T., Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana (1) 15 (1999), 181–232.Google Scholar
[13] Doob, J. L., Classical Potential Theory and its Probabilistic Counterpart. Springer-Verlag.Google Scholar
[14] Doob, J. L., Stochastic Processes. J. Wiley.Google Scholar
[15] Fabes, E. B., Garofalo, N. and Salsa, , A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations. Illinois J. Math. 30 (1986), 536–565.Google Scholar
[16] Fabes, E. and Safonov, M. V., Behavior near the boundary of positive solutions of second order parabolic equations. J. Fourier Anal. Appl. 3 (1997), 871–882.Google Scholar
[17] Fabes, E. B., Safonov, M. V. and Yuan, Y., Behavior near the boundary of positive solutions of second order parabolic equations. II. Trans. Amer.Math. Soc. (12) 351 (1999), 4947–4961.Google Scholar
[18] Fabes, E., Garofalo, N., Marin-Malava, S. and Salsa, S., Fatou theorems for some nonlinear elliptic equations. Rev. Mat. Iberoamericana (2) 4 (1988), 227–251.Google Scholar
[19] Fabes, E. B. and Stroock, D., The Lp-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations. Duke Math. J. 51 (1984), 977–1016.Google Scholar
[20] Fabes, E. B. and Stroock, D., A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. (4) 96 (1986), 327–338.Google Scholar
[21] Fukushima, M., Dirichlet forms and Marlov Processes. North-Holland, 1980.Google Scholar
[22] Feller, W., An Introduction to Probability Theory. Volumes I and II, Wiley.Google Scholar
[23] Garofalo, N., Second order parabolic equations in nonvariational forms: boundary Harnack principle and comparison theorems for nonnegative solutions. Ann.Mat. Pura Appl. 138 (1984), 367–296.Google Scholar
[24] Grigoryan, A., Gaussian upper bounds for the heat kernel on arbitrary manifolds. J. Differential Geom. 45 (1997), 33–52.Google Scholar
[25] Hebisch, W., Saloff-Coste, L.. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. (2) 21 (1993), 673–709.Google Scholar
[26] Jerison, D. and Kenig, C., Boundary behaviour of harmonic functions in nontangentially accessible domains. Adv. in Math. 146 (1982), 80–147.Google Scholar
[27] Kenig, C. E., Potential Theory of Non-Divergence Form Elliptic Equations. Proceedings of C.I.M.E. Course in Dirichlet forms, 1992.Google Scholar
[28] Kenig, C. E., Harmonic analysis techniques for second order elliptic boundary value problems. C.B.M.S. 83, Amer.Math. Soc., 1994.Google Scholar
[29] Krylov, N. V. and Safonov, M. V., A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR 44, English Translation Math. USSR-Izv. 16 (1981), 151–164.Google Scholar
[30] Kuo, H.-J. and Trudinger, N. S., Evolving monotone difference operators on general space-time meshes. Duke Math. J. (3) 91 (1998), 587–607.Google Scholar
[31] Lawler, G. F., Estimates for differences and Harnack inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous, increments. Proc. LondonMath. Soc. (3) 63 (1991), 552–568.Google Scholar
[32] Mackenhoupt, B., The equivalence of two conditions for weight functions. Studia Math. 49 (1974), 101–106.Google Scholar
[33] McKean, H. P. Jr., Stochastic Integrals. Academic Press, 1969.Google Scholar
[34] Moser, J., On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 (1961), 557–591.Google Scholar
[35] Moser, J., A Harnack inequality for parabolic differential equations. Comm. Pure and Appl. Math. 17 (1964), 101–134.Google Scholar
[36] Moser, J., A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 20 (1967), 232–236.Google Scholar
[37] Safonov, M. V. and Yuan, Y., Doubling properties for second order parabolic equations. Ann. of Math. 150 (1999), 313–327.Google Scholar
[38] Stein, E. M., Singular Integrals and Differentiation Properties of Functions. Princeton Univ. Press.Google Scholar
[39] Ušakov, V. I., Stabilization of solutions of the third mixed problem for a second-order parabolic equation in a noncylindrical domain. Mat. Sb. 153(1980), Translation: Math. USSR-Sb. (1) 39 (1981), 87–105.Google Scholar
[40] Varopoulos, N. Th., Information theory and harmonic functions. Bull. Sci. Math. (2) 109 (1985), 225–252.Google Scholar
[41] Varopoulos, N. Th., Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63 (1985), 240–260.Google Scholar
[42] Varopoulos, N. Th., Geometric and potential theoretic results on Lie groups. Canad. J. Math. (2) 52 (2000), 412–437.Google Scholar
[43] Varopoulos, N. Th., Potential theory in Lipchitz domains. Canad. J. Math. (5) 53 (2001), 1057–1120.Google Scholar
[44] Varopoulos, N. Th., Saloff-Coste, L. and T. Coulhon, Analysis and Geometry on Groups. Cambridge Tracts in Math. 100(1992).Google Scholar