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11
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant 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 que ..."
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Cited by 27 (3 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant 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
Graphs between the elliptic and parabolic Harnack inequalities.
"... 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 ..."
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Cited by 7 (0 self)
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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.
Spectral Asymptotics Of The DirichletToNeumann Map On Multiply Connected Domains In R^d
 in R d , Inverse Probl
, 2001
"... We study the spectral asymptotics of the DirichlettoNeumann operator on a multiplyconnected, 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, positivede nite, symmetric matrixvalued function on We prove that t ..."
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Cited by 2 (0 self)
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We study the spectral asymptotics of the DirichlettoNeumann operator on a multiplyconnected, 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, positivede nite, symmetric matrixvalued 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 highenergy spectral asymptotics of . September 12, 2001 Supported in part by NSF grant DMS9707049. 1
Elliptic Diffusions, Exit Times and the Quasihyperbolic Metric
, 2002
"... 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 pth power of the L^pnorm of the quasihyperbolic metric all raised to the power 2/n, with p ≥ 2/n. It f ..."
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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 pth power of the L^pnorm of the quasihyperbolic metric all raised to the power 2/n, with p &ge; 2/n. It follows in an L^paveraging domain that this lifetime is bounded above by a constant times the volume to the power 2/n.
HARNACK INEQUALITY FOR A CLASS OF DEGENERATE ELLIPTIC OPERATORS
, 2003
"... Abstract. We prove a Harnack inequality for a class of twoweight degenerate elliptic operators. The metric distance is induced by continuous Grushintype vector fields. It is not know whether there exist cutoffs fitting the metric balls. This obstacle is bypassed by means of a covering argument tha ..."
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Abstract. We prove a Harnack inequality for a class of twoweight degenerate elliptic operators. The metric distance is induced by continuous Grushintype vector fields. It is not know whether there exist cutoffs fitting the metric balls. This obstacle is bypassed by means of a covering argument that allows the use of rectangles in the Moser iteration. 1.
HARNACK'S INEQUALITY FOR A NONLINEAR EIGENVALUE PROBLEM ON METRIC SPACES
"... Abstract: We prove Harnack's inequality for ¯rst eigenfunctions of the pLaplacian in metric measure spaces. The proof is based on the famous Moser iteration method, which has the advantage that it only requires the (1; p)Poincar¶e inequality. As a byproduct we obtain the continuity and the f ..."
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Abstract: We prove Harnack's inequality for ¯rst eigenfunctions of the pLaplacian in metric measure spaces. The proof is based on the famous Moser iteration method, which has the advantage that it only requires the (1; p)Poincar¶e inequality. As a byproduct we obtain the continuity and the fact that ¯rst eigenfunctions do not change sign in bounded domains. AMS subject classi¯cations: 35P30, 35J20
MOSER'S METHOD FOR MINIMIZERS ON METRIC MEASURE SPACES
"... Niko Marola: Moser's method for minimizers on metric measure spaces; Helsinki ..."
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Niko Marola: Moser's method for minimizers on metric measure spaces; Helsinki