The Coercivity of uniformly positive definite Matrix of Sobolev function

Question

For $u=(u^1,\ldots, u^N)\in W^{1,2}(\Omega,R^N)$ where $\Omega$ is bounded. We define $$ E[u]=\int_\Omega g_{ij}(u)\nabla u^i\nabla u^jdx$$ where $G=(g_{ij})_{1\leq i,j\leq N}$ is an given uniformly positive definite matrix, i.e. $\xi^T\cdot G(u)\cdot \xi\geq \alpha\|\xi\|^2>0$ no matter the value of $u$. Now my text book conclude that $$ E[u]\geq \lambda\|\nabla u\|_{L^2}^2 $$ without prove. May it is a simple fact but I can not work it out. Could somebody help me to write done the details? Thx!

Answer 1

I think that you forgot $\|\xi\|^2$ in the definition the uniformly positive definite matrix. With this adjustment what you want to prove is straightforward.