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| An Integral Equality on the Laplace-Beltrami Operator |
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Citation： |
Shen Yibing,Shui Naixiang.An Integral Equality on the Laplace-Beltrami Operator[J].Chinese Annals of Mathematics B,1982,3(3):375~380 |
| Page view： 801
Net amount： 1059 |
Authors： |
Shen Yibing; Shui Naixiang |
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| Abstract： |
Let M be an n-dimensional Riemanniaii manifold of class C^r(r> 2)> and D \subset M
be a regular oriented domain on M with boundary \partial D, and W_2,0^2(D) denote the
Sobolev space defined on D with the property that every u \in W_2,0^2(D) vanishes at \partial D,i. e., $u|_\partial D=0$. Let (\alpha_=\alpha \beta) be the symmetric matrix of the positive definite metric of M,and $\nabla _\alpha$ denote the operator of the covariant derivative with respect to $\alpha _\alpha \beta$. For any
$u \in C^\infty(M)$, it is convenient to define
$u_\alpha=\nabla_\alpha u,u_\alpha \beta=\nabla_\alpha u_\beta$
$u^\alpha=a^\alpha\beta=a^\alpha\gamma \nabla_\gamma u^\beta.(1 \leq \alpha,\beta,\gamma \leq n)$
In this paper we establish the following
Theorem. Let $u \in W_2,0^2(D)$ . Then
$\int_D{u^\alpha \beta u_\alpha \beta dV}=\int_D{(\Delta u)^2dV}+\int_D{Ric(du,du)dV}-\int_\partial D{(\nabla_Nu)^2\Omega ds}$
where $\Delta$ is the Laplace-Beltrami operator, Ric (du, du) is the Ricci curvature of M with respect to the vector field du, \nabla_N is the directional derivative in the direction of the exterior normal vector N at $\partial D$ is the mean curvature of \partial D in M. |
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