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