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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
Harnack inequality for nondivergent elliptic operators on Riemannian manifolds
 Pacific J. Math
, 2004
"... We consider secondorder linear elliptic operators of nondivergence type which are intrinsically defined on Riemannian manifolds. Cabré proved a global KrylovSafonov Harnack inequality under the assumption that the sectional curvature is nonnegative. We improve Cabré’s result and, as a consequence, ..."
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Cited by 3 (0 self)
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We consider secondorder linear elliptic operators of nondivergence type which are intrinsically defined on Riemannian manifolds. Cabré proved a global KrylovSafonov Harnack inequality under the assumption that the sectional curvature is nonnegative. We improve Cabré’s result and, as a consequence, we give another proof to the Harnack inequality of Yau for positive harmonic functions on Riemannian manifolds with nonnegative Ricci curvature using the nondivergence structure of the Laplace operator. 1. Introduction and
In memory of E.Fabes
, 2002
"... Abstract. We consider the Dirichlet problem Lu = 0 in D u = g on ∂D for two second order elliptic operators Lku = ∑n i,j=1 ai,j k (x) ∂iju(x), k = 0, 1, in a bounded Lipschitz domain D ⊂ IR n. The coefficients a i,j k belong to the space of bounded mean oscillation BMO with a suitable small BMO modu ..."
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Abstract. We consider the Dirichlet problem Lu = 0 in D u = g on ∂D for two second order elliptic operators Lku = ∑n i,j=1 ai,j k (x) ∂iju(x), k = 0, 1, in a bounded Lipschitz domain D ⊂ IR n. The coefficients a i,j k belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L0 is regular in Lp (∂D, dσ) for some p, 1 < p < ∞, that is, ‖Nu‖Lp ≤ C ‖g‖Lp for all continuous boundary data g. Here σ is the surface measure on ∂D and Nu is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients εi,j (x) = a i,j 1 (x) − ai,j 0 (x) that will assure the perturbed operator L1 to be regular in Lq (∂D, dσ) for some q, 1 < q < ∞. 1.
BOUNDARY ESTIMATES FOR POSITIVE SOLUTIONS TO SECOND ORDER ELLIPTIC EQUATIONS
, 810
"... Abstract. Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which guarantee the HopfOleinik type estimates and the bo ..."
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Abstract. Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which guarantee the HopfOleinik type estimates and the boundary Lipschitz estimates for solutions. These conditions are sharp even for harmonic functions. 1. Introduction. Formulation of