A property of Sobolev spaces on complete Riemannian manifolds
Electronic journal of differential equations, Tome 2005 (2005)
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.
@article{EJDE_2005__2005__a251,
author = {Milatovic, Ognjen},
title = {A property of {Sobolev} spaces on complete {Riemannian} manifolds},
journal = {Electronic journal of differential equations},
year = {2005},
volume = {2005},
zbl = {1072.58011},
language = {en},
url = {http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a251/}
}
Milatovic, Ognjen. A property of Sobolev spaces on complete Riemannian manifolds. Electronic journal of differential equations, Tome 2005 (2005). http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a251/