Electron. J. Diff. Eqns., Vol. 2005(2005), No. 77, pp. 1-10.

A property of Sobolev spaces on complete Riemannian manifolds

Ognjen Milatovic

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.

Submitted June 25, 2005. Published July 8, 2005.
Math Subject Classifications: 58J05.
Key Words: Complete Riemannian manifold; Sobolev space

Show me the PDF file (234K), TEX file, and other files for this article.

Ognjen Milatovic
Department of Mathematics and Statistics
University of North Florida
Jacksonville, FL 32224, USA
email: omilatov@unf.edu

Return to the EJDE web page