Coarse and Precise $L^p$-Green Potential Estimates on Noncompact Riemannian Manifolds

Details Coarse and Precise $L^p$-Green Potential Estimates on Noncompact Riemannian Manifolds Coarse and Precise $L^p$-Green Potential Estimates on Noncompact Riemannian Manifolds

2009-08-15

arxiv.org/abs/0908.2173v2

Journal of Mathematical Physics, 51, 1 (2010)

Mathematics

Analysis of PDEs

21 pages

Scientific paper

10.1063/1.3447730

We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and sharp Euclidean isoperimetric inequalities. Consequently, we are led to evaluate the critical limit of an induced monotone Green's functional using the asymptotic behavior of the Lorentz norm deficit of Green's function at the infinity, as well as the harmonic radius of a regular domain in the Riemannian manifold with nonnegative Ricci curvature.

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