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12
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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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
On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains. Algebra i Analiz 24:53– 94 (English summary); translation in
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"... II. Theory 30 A. Anisotropic molecular reorientation in an ordering potential 32 B. Restricted rotation in a cone model.. 33 ..."
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II. Theory 30 A. Anisotropic molecular reorientation in an ordering potential 32 B. Restricted rotation in a cone model.. 33
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
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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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
Harnack Inequalities: an introduction
, 2007
"... The aim of this article is to give an introduction to certain inequalities named after Carl ..."
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The aim of this article is to give an introduction to certain inequalities named after Carl
On the principal eigenvalue of elliptic operators in RN and applications
"... Two generalizations of the notion of principal eigenvalue for elliptic operators in RN are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; selfadjoint operators; and “limit periodic ” operators. These results ..."
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Two generalizations of the notion of principal eigenvalue for elliptic operators in RN are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; selfadjoint operators; and “limit periodic ” operators. These results apply to questions of existence and uniqueness for some semilinear problems in all of space. We also indicate several outstanding open problems and formulate some conjectures. 1
The Lp Dirichlet Problem and . . .
, 2002
"... 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 ..."
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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 < ∞.
Research Article Hölder Regularity of Solutions to SecondOrder Elliptic Equations in Nonsmooth Domains
"... We establish the global Hölder estimates for solutions to secondorder elliptic equations, which vanish on the boundary, while the righthand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying an exterior cone condition, similar results were obtained by J. H ..."
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We establish the global Hölder estimates for solutions to secondorder elliptic equations, which vanish on the boundary, while the righthand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying an exterior cone condition, similar results were obtained by J. H. Michael, who in turn relied on the barrier techniques due to K. Miller. Our approach is based on special growth lemmas, and it works for both divergence and nondivergence, elliptic and parabolic equations, in domains satisfying a general “exterior measure ” condition. Copyright © 2007 S. Cho and M. Safonov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.