Let D be a domain in R n , n U 2. Let H 0 be a divergence form uniformly elliptic operator with Dirichlet boundary conditions on S and Neumann boundary conditions on @D n S, where S is a closed subset of @D. We prove intrinsic ultracontractivity for the semigroup associated to the Schrödinger operator H ¼ H 0 þ V, where V is a potential in the Kato class, provided that @D n S is locally Lipschitz and S is given by the boundary of either a Hölder domain of order 0 or a uniformly Hölder domain of order , 0 < < 2. Our results extend to the mixed boundary case the results of Bañuelos, Bass and Burdzy, Bass and Hsu, and Davies and Simon. (2000): 60J45.
Mathematics Subject Classification
≥ 2 be the unbounded domain above the graph of a bounded Lipschitz function. We study the asymptotic behavior of the transition density p D (t, x, y) of killed Brownian motions in D and, where u is a minimal harmonic function corresponding to the Martin point at infinity and C 1 is a positive constant.
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