Abstract
In this paper, we prove two-sided pointwise estimates for the Green function of a parabolic operator with singular first order term on a C1,1-cylindrical domain Ω. Basing on these estimates, we establish the equivalence of the parabolic measure, the adjoint parabolic measure and the surface measure on the lateral boundary of Ω. These results are first studied by some authors for certain elliptic and less general parabolic operators.
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Ancona, A.: ‘Comparaison des mesures harmoniques et des fonctions de Green pour des opérateurs elliptiques sur un domaine lipschitzien’, C. R. Acad. Sci. Paris 294(1) (1982), 505–508.
Aronson, D.G.: ‘Bounds for the fundamental solution of a parabolic equation’, Bull. Amer. Math. Soc. 73 (1967), 890–896.
Aronson, D.G.: ‘Nonnegative solution of linear parabolic equations’, Ann. Sci. Norm. Sup. Pisa 22 (1968), 607–694.
Chung, K.L. and Zhao, Z.: From Brownien Motion to Schrödinger’s Equation, Springer, Berlin, 1995.
Cranston, M. and Zhao, Z.: ‘Conditional transformation of drift formula and potential theory for \({1\over 2}\Delta+b(\cdot)\cdot\nabla\) ’, Comm. Math. Phys. 112 (1987), 613–625.
Dahlberg, B.: ‘Estimates of harmonic measure’, Arch. Rational Mech. Anal. 65(3) (1978), 275–288.
Davies, E.: ‘The equivalence of certain heat kernel and Green function bounds’, J. Funct. Anal. 71 (1987), 88–103.
Davies, E.: Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.
Eklund, N.A.: ‘Existence and representation of solutions of parabolic equations’, Proc. Amer. Math. Soc. 47(1) (1975), 137–142.
Fabes, E.B. and Salsa, S.: ‘Estimates of caloric measure and the initial Dirichlet problem for the heat equation in cylinders’, Trans. Amer. Math. Soc. 279(2) (1983), 635–650.
Fabes, E.B. and Stroock, D.W.: ‘A new proof of Moser’s parabolic Harnack inequality using the old idea of Nash’, Arch. Rational Mech. Anal. 96 (1986), 326–338.
Fabes, E.B., Garofalo, N. and Salsa, S.: ‘A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations’, Illinois J. Math. 30(4) (1986), 536–565.
Garofalo, N.: ‘Second order parabolic equations in nonvariational form: Boundary Harnack principle and comparison theorems for nonnegative solutions’, Ann. Mat. Pura Appl. 138 (1984), 267–296.
Grüter, M. and Widman, K.O.: ‘The Green function for uniformly elliptic equations’, Manuscripta Math. 37 (1982), 303–342.
Heurteaux, Y.: ‘Solutions positives et mesures harmoniques pour des opérateurs paraboliques dans des ouverts lipschitziens’, Ann. Inst. Fourier (Grenoble) 41(3) (1991), 601–649.
Hofmann, S. and Lewis, J.: ‘The Dirichlet problem for parabolic operators with singular drift terms’, Mem. Amer. Math. Soc. 151(719) (2001).
Hueber, H.: ‘A uniform estimate for Green functions on C1,1-domains’, Bibos. Publication, Universität Bielefeld, 1986.
Hueber, H. and Sieveking, M.: ‘Uniform bounds for quotients of Green functions on C1,1-domains’, Ann. Inst. Fourier (Grenoble) 32(1) (1982), 105–117.
Hui, K.M.: ‘A Fatou theorem for the solution of the heat equation at the corner points of a cylinder’, Trans. Amer. Math. Soc. 333 (1992), 607–642.
Kaufman, R. and Wu, J.M.: ‘Singularity of parabolic measures’, Compositio Math. 40(2) (1980), 243–250.
Lieberman, G.M.: Second Order Parabolic Differential Equations, World Scientific, 1996.
Riahi, L.: ‘Green function bounds and parabolic potentials on a half-space’, Potential Anal. 15 (2001), 133–150.
Riahi, L.: ‘Boundary behaviour of positive solutions of second order parabolic operators with lower order terms in a half-space’, Potential Anal. 15 (2001), 409–424.
Widman, K.O.: ‘Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations’, Math. Scand. 21 (1967), 17–37.
Wu, J.M.: ‘On parabolic measures and subparabolic functions’, Trans. Amer. Math. Soc. 251 (1979), 171–185.
Zhang, Q.S.: ‘A Harnack inequality for the equation ∇(a∇u)+b∇u=0 when |b|∈Kn+1’, Manuscripta Math. 89 (1995), 61–77.
Zhang, Q.S.: ‘Gaussian bounds for the fundamental solutions of ∇(A∇u)+B∇u−ut=0’, Manuscripta Math. 93 (1997), 381–390.
Zhao, Z.: ‘Green function for Schrödinger operator and conditioned Feynman–Kac-gauge’, J. Math. Anal. Appl. 116 (1986), 309–334.
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Mathematics Subject Classifications (2000)
31B25, 35B05, 35K10, 58J35.
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Riahi, L. Comparison of Green Functions and Harmonic Measures for Parabolic Operators. Potential Anal 23, 381–402 (2005). https://doi.org/10.1007/s11118-005-2606-6
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DOI: https://doi.org/10.1007/s11118-005-2606-6