Results 1  10
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102
Sobolev inequalities in disguise
 Indiana Univ. Math. J
, 1995
"... We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff argu ..."
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Cited by 39 (4 self)
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We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff arguments. This method has interesting consequences concerning Trudinger type inequalities. 1. Introduction. On R n, the classical Sobolev inequality [27] indicates that, for every smooth enough function f with compact support,
Ricci Flow And The Uniformization On Complete Noncompact Kähler Manifolds
 J. Differential Geom
, 1997
"... this paper proved the following result in his Ph.D. thesis [43] in 1990: ..."
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Cited by 36 (0 self)
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this paper proved the following result in his Ph.D. thesis [43] in 1990:
Uniformly elliptic operators on Riemannian manifolds
 J. Diff. Geom
, 1992
"... Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiiso ..."
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Cited by 32 (2 self)
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Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiisometric to g. We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we use Moser's iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of g. Some of them are new even when applied to the Laplace operator of (M, g). 1.
Nonlocal Dirichlet forms and symmetric jump processes
 Transactions of the American Mathematical Society
, 1999
"... We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x ..."
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Cited by 30 (16 self)
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We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x  −d−α ≤ J(x, y) ≤ κ2y − x  −d−β for 0 < α < β < 2, x − y  < 1. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E, F). We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.
Harnack inequalities and subGaussian estimates for random walks
 Math. Annalen
, 2002
"... We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the m ..."
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Cited by 29 (6 self)
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We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.
Bernstein Type Theorems for Higher Codimension
, 1998
"... We show a Bernstein theorem for minimal graphs of arbitrary dimension and codimension under a bound on the slope that improve previous results and is independent of the dimension and codimension. The proof depends on the regularity theory for the harmonic Gauss map and the geometry of Grassmann mani ..."
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Cited by 21 (6 self)
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We show a Bernstein theorem for minimal graphs of arbitrary dimension and codimension under a bound on the slope that improve previous results and is independent of the dimension and codimension. The proof depends on the regularity theory for the harmonic Gauss map and the geometry of Grassmann manifolds.
A new approach to absolute continuity of elliptic measure, with applications to nonsymmetric equations
 Adv. in Math
"... In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc ..."
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Cited by 14 (3 self)
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In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc
Metric based upscaling
 Communications on Pure and Applied Mathematics
, 2007
"... We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the med ..."
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Cited by 11 (2 self)
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We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators. 1 Introduction and main results Let Ω be a bounded and convex domain of class C2. We consider the following benchmark PDE
Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations
 Publ. Mat
, 1996
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