Harnack inequalities on manifolds with boundary and applications

By Feng-Yu Wang. 2009

135 citations recorded by the source corpus.

Abstract On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup are proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for p t ( x , y ) the Neumann heat kernel w.r.t. a volume type measure μ and for K a constant, the curvature condition Ric − ∇ Z ⩾ K together with the convexity of the boundary is equivalent to the heat kernel entropy inequality: ∫ M p t ( x , z ) log p t ( x , z ) p t ( y , z ) μ ( d z ) ⩽ K ρ ( x , y ) 2 2 ( e 2 K t − 1 ) , t > 0 , x , y ∈ M , where ρ is the Riemannian distance. The main result is partly extended to manifolds with non-convex boundary and applied to derive the HWI inequality.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/a3626a3a2e05becc4cb007b3ba5f6e0af92dffc0

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In Applied Mathematics

Citations in the corpus

Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality (2011, background): “The proof of this result appears in Section 2 of [42] where a general study of these Harnack inequalities is done.”
Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift (2014, background): “7) is introduced in [19] as a limit version of the dimension-free Harnack inequality founded in [18].”
Harnack inequalities for stochastic equations driven by Levy noise (2012, background): “4] (see also [15, 20]), to save space we only mention here heat kernel estimates implied by Corollary 2.”
Equivalent semigroup properties for the curvature-dimension condition (2010, unknown): “[2, 10, 15, 19, 22] and references within for details.”