Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 77, pp. 1-10.

Title: A property of Sobolev spaces on complete Riemannian manifolds

Author: Ognjen Milatovic (Univ. of North Florida, Jacksonville, FL, USA)

Abstract: 
 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
 $T\in [W^{-1,2}(M)\cap L_{\rm loc}^{1}(M)]^{k}$ 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. Br\'ezis and
 F. E. Browder.
  
Submitted June 25, 2005. Published July 8, 2005.
Math Subject Classifications: 58J05.
Key Words: Complete Riemannian manifold; Sobolev space