In the following model of a membrane with a mass particle in it, why does the integral represents the elastic energy of the system?
Let $\Omega$ be an open connected region (the membrane) in $\Re^2$, $x\in \Omega$ and $u(x)$ be the profile of the membrane with $u=0$ at $\partial\Omega$. If $P$ is a unit mass particle we put in the membrane in position $q$, then $$\Delta u=\delta_{q} $$ is satisfied, where $\delta_{q}$ is the Dirac measure. The problem of finding the equilibrium position of the particle $P$ can be reduced to find the function u that minimizes the energy functional: $$ E(u,q)=\frac{1}{2}\int_{\Omega}|\nabla u(x)|^2dx +u(q)$$ which is the sum of the elastic energy and the gravitational energy (considering all physical constants equal to $1$).