# when does the “Laplacian” of a $W^{1,2}$ function exist?

$\Omega$ is an open domain in $R^n$. Given a function $u \in W_{loc}^{1,2}\left( \Omega \right)$, we define a functional ${L_u}$ on $Li{p_0}\left( \Omega \right)$(Lipschitz functions with compact support) by $${L_u}\left( \phi \right) = - \int_\Omega {\left\langle {\nabla u,\nabla v} \right\rangle } dvol,\forall \phi \in Li{p_0}\left( \Omega \right)$$ If u is convex(concave), then $L_u$ is a signed radon measure? With non-negative(non-positive) singular part respectively?

Let $f \in {L^2}\left( \Omega \right)$ and $u \in W_{loc}^{1,2}\left( \Omega \right)$. If $${L_u}\left( \phi \right) \ge \int_\Omega {f\phi dvol} \left( {or{L_u}\left( \phi \right) \le \int_\Omega {f\phi dvol} } \right)$$ for all nonnegative $\phi \in Li{p_0}\left( \Omega \right)$，then $L_u$ is a signed radon measure?