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

A property of Sobolev spaces on complete Riemannian manifolds

Ognjen Milatovic

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 $T\in [W^{-1,2}(M)\cap L_{loc}^{1}(M)]^{k}$ 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

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

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