I am looking for a book/paper which has the proof of the Rellich-Nicas identity.


**[EDIT by Yemon Choi]** It seems that what was meant is "the Rellich-Necas identity", although the original poster hasn't really clarified or expanded on the request.



Rellich-Necas identity
(Boo! I tried to post this in a comment to Ady, but the HTML Math won't parse right. So here goes. Sorry about the really long equation being broken up not very neatly.)


Googling Rellich-Necas turns up a bunch of recent papers by LUIS ESCAURIAZA in which the identities are used. But as far as I can tell the identity is just a simple differential equality obtained from symbolic manipulation of terms. The following seems to be a straight-forward version of the identity: let $A = (A\_{ij})$ be a symmetric bilinear form (with variable coefficients) on RN, $v$ a vector field, $u$ a function, and $\delta$ denoting the Euclidean divergence, we have


$ \delta( A(\nabla u,\nabla u) v) = 2 \delta( v(u) A(\nabla u)) + \delta(v) A(\nabla u,\nabla u)$
$- 2A(\nabla u) \cdot \nabla v \cdot \nabla u - 2 v(u) \delta(A(\nabla u)) + v(A)(\nabla u,\nabla u)$


Where $v(u)$ is the partial derivative of $u$ in the direction of $v$, and $A(\nabla u)\cdot\nabla v \cdot \nabla u$ is, in coordinates, $\partial\_i u A\_{ij} \partial\_j v\_k \partial\_k u$ with implied summation, and $v(A)$ is the symmetric bilinear form obtained by taking the $v$ partial derivative of the coefficients of $A$. 


Verifying that the identity is true should just be a basic application of multivariable calculus. 


