Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm asking if that condition is also true $W^{1,2}([0,\pi])$, because i don't find find an example for wich this property is not verified.
If i consider a othonormal complete system $\{\phi_{i}\}_{i\in N}$ for our Sobolev space, and also fixed the parameter $i \in N$, for each sequence $u_n \in span\{\phi_i\}$ the condition of coercivity it satisfied, can i conclude that the condition it's in general true?
Similar question (coercivity) for the following $$ I(\rho)=\int_{0}^{\pi}{\sqrt{\dot\rho^2+\rho^2}\,dx} $$ with $\rho \in W^{1,2}([0,\pi])$.
To be coercive you need to have an estimate of the form $J(\rho) > C\|\rho\|_{W^{1,2}}^2$. If you consider (say) $\rho_k(x) = \sin kx$ you have $J(\rho_k) = \dfrac{\pi}{4}$ and $\|\rho_k\|_{W^{1,2}}^2 = \dfrac{\pi}{4} ( 1 + k^2)$. The coercivity condition would force $C$ to satsify $ 0 < C < \dfrac{1}{1 + k^2}$ for all natural numbers $k$.