Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = e^h（x） dV（x） the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider...Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = e^h（x） dV（x） the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation △μ,pu=-λμ,p｜u｜^p-2ufor p ∈ （1, 2）. We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..展开更多
In this paper, using Perelman’s no local collapsing theorem and the geometric interpretation of Hamilton’s Harnack expressions along the Ricci flow introduced by R. Hamilton, we present a mathematical interpretation...In this paper, using Perelman’s no local collapsing theorem and the geometric interpretation of Hamilton’s Harnack expressions along the Ricci flow introduced by R. Hamilton, we present a mathematical interpretation of Hawking’s black hole theory in [1].展开更多
基金Supported by the National Natural Science Foundation of China (11171254, 11271209)
文摘Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = e^h（x） dV（x） the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation △μ,pu=-λμ,p｜u｜^p-2ufor p ∈ （1, 2）. We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..
文摘In this paper, using Perelman’s no local collapsing theorem and the geometric interpretation of Hamilton’s Harnack expressions along the Ricci flow introduced by R. Hamilton, we present a mathematical interpretation of Hawking’s black hole theory in [1].
