A property of Sobolev spaces on complete Riemannian manifolds
Let $(M,g)$ be a complete Riemannian manifold with metric $g$ and the Riemannian volume form $d u$. We consider the $mathbb{R}^{k}$-valued functions $Tin [W^{-1,2}(M)cap L_{loc}^{1}(M)]^{k}$ and $uin [W^{1,2}(M)]^{k}$ on $M$, where $[W^{1,2}(M)]^{k}$ is a Sobolev space on $M$ and $[W^{-1,2}(M)]^{k}...
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| Format: | Article |
| Language: | English |
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Texas State University
2005-07-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2005/77/abstr.html |
| Summary: | Let $(M,g)$ be a complete Riemannian manifold with metric $g$ and the Riemannian volume form $d u$. We consider the $mathbb{R}^{k}$-valued functions $Tin [W^{-1,2}(M)cap L_{loc}^{1}(M)]^{k}$ and $uin [W^{1,2}(M)]^{k}$ 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 $langle T, u angle$ and the integral of $(Tcdot u)$ over $M$, where $langlecdot,cdot angle$ 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. |
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| ISSN: | 1072-6691 |