A property of Sobolev spaces on complete Riemannian manifolds

Milatovic, Ognjen

Electronic Journal of Differential Equations, Tome 2005 (2005).

Résumé

Summary: Let $(M,g)$ be a complete Riemannian manifold with metric $g$ and the Riemannian volume form $d\nu$. We consider the $\mathbb{R}^{k}$-valued functions $T\in [W^{-1,2}(M)\cap L_{loc}^{1}(M)]^{k}$ and $u\in [W^{1,2}(M)]^{k}$ on $M$, where $[W^{1,2}(M)]^{k}$ is a Sobolev space on $M$ and $[W^{-1,2}(M)]^{k}$ is its dual. We give a sufficient condition for the equality of $\langle T, u\rangle$ and the integral of $(T\cdot u)$ over $M$, where $\langle\cdot,\cdot\rangle$ is the duality between $[W^{-1,2}(M)]^{k}$ and $[W^{1,2}(M)]^{k}$. This is an extension to complete Riemannian manifolds of a result of H. Brezis and F. E. Browder.

Détail

Classification : 58J05
Keywords: complete Riemannian manifold, Sobolev space