# Physical interpretation of Dirichlet energy for a membrane.

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 $$1$$).

• It is probably a small $u$ approximation to the area functional. $$\verb/Area/(u) - \verb/Area/(u \equiv 0) = \int_\Omega \left( \sqrt{1 + |\nabla u|^2} - 1\right) dx \approx \frac12 \int_\Omega |\nabla u|^2 dx$$ – achille hui Apr 27 at 4:38
• Perhaps you can try the physics site. For what it's worth, this post there is probably an overkill for your inquiry. Might or might not be helpful to you. – Lee David Chung Lin Apr 27 at 6:33
• Crossposted to physics.stackexchange.com/q/476334/2451 – Qmechanic Apr 27 at 21:19