We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1

We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L 2 Poincaré inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass, see [2]. It satisfies the volume regularity and not the Poincaré inequality. We construct another example that does not satisfy the volume regularity.

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We study the spectral asymptotics of the Dirichlet-to-Neumann operator on a multiply-connected, bounded, domain in IR , d 3, associated with the uniformly elliptic operator L = P d i;j=1 @ i ij @ j , where is a smooth, positive-de nite, symmetric matrix-valued function on We prove that the operator is approximately diagonal in the sense that = D +R , where D is a direct sum of operators, each of which acts on one boundary component only, and R is a smoothing operator. This representation follows from the fact that the -harmonic extensions of eigenfunctions of vanish rapidly away from the boundary. Using this representation, we study the inverse problem of determining the number of holes in the body, that is, the number of the connected components of the boundary, by using the high-energy spectral asymptotics of . September 12, 2001 Supported in part by NSF grant DMS-9707049. 1

We show that the expected lifetime of a conditioned elliptic diffusion in a bounded domain in R^n is bounded above by an absolute constant times the sum of the volume of the domain and the p-th power of the L^p-norm of the quasihyperbolic metric all raised to the power 2/n, with p ≥ 2/n. It follows in an L^p-averaging domain that this lifetime is bounded above by a constant times the volume to the power 2/n.