606 research outputs found
New Liouville theorems for linear second order degenerate elliptic equations in divergence form
No full textIn this paper we give conditions on the positive function phi(2) under which every bounded solution sigma of the elliptic equation V.(phi(2)delsigma) = 0 in R-n must be constant. The case when phi(2) only depends on one or two variables is discussed at length. Moreover the asymptotic behavior of possibly unbounded solutions is characterized improving in such a way a Liouville theorem due to Berestycki, Caffarelli and Nirenberg. (C) 2004 Elsevier SAS. All rights reserved
Harnack inequality and heat kernel estimates for the Schroedinger operator with Hardy potential
No full textIn this preliminary note we outline some results of the forthcoming paper [11], concerning positive solutions of the equation \partial_t u=\Delta u+ c/|x|^2 u (0<c<(n-2)^2/4; n\ge 3). A parabolic Harnack inequality is proved, which in particular implies a sharp two-sided estimate for the associated heat-kernel. Our approach relies on the unitary equivalence of the Schroedinger operator Hu=-\Delta u -c/|x|^2 u with the opposite of the weighted Laplacian \Delta_\lambda v=1/|x|^2 div(|x|^\lambda \nabla v) when \lambda=2-n+2\sqrt[c_0-c}
SHARP TWO-SIDED HEAT KERNEL ESTIMATES FOR CRITICAL SCHRODINGER OPERATORS ON BOUNDED DOMAINS
No full textOn a smooth bounded domain \Omega \subset R^N we consider the Schrödinger operators ?\Delta? V, with V being either the critical borderline potential V(x) = (N -2)^2/4 |x|^(?2) or V(x) = (1/4) dist(x, ?\Omega)^(?2), under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev,Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies
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