Why do we use the word "energy" for this integral? I came across the following and I would like to know the motivation or idea behind using the word "energy".
Suppose $u(x)$ is a $C^1$ function on some domain $\Omega \subset R^3$.
Then, we can think of $\int_\Omega |\nabla u|^2$dx as the energy associated to the function u.
Why can we think of this integral as energy? With little background in physics, I am puzzled that the norm of the gradient has anything to do with energy of something.
My best (and naive) guess would be that if $u(x)$ is a temperature in a domain, then the integral above is the total heat energy or something like that.
Would you able to explain or possibly give me some examples on this idea of energy??
 A: For example: take $u$ to be the scalar field of electric-potential (i.e. voltage).  Then the electric field is given by $-\nabla u$.  The energy stored in a (steady) electric field within a domain $\Omega$ is given by
$$
U = \frac 12 \varepsilon \int_\Omega |E|^2\,dx = C \int_\Omega |\nabla u|^2\,dx
$$
you might be able to come up with a similar expression for temperature.  However, I would bet that electromagnetism is what motivated the terminology.
A: I tend to think of a function's "energy" as its deviation from equilibrium. For example, if $u$ measures the height of a stretched membrane, $|\nabla u|$ measures the amount of stretching at a point. Integrate $|\nabla u|^2$ over your domain and you measure the total amount of stretching.
The local measure of deviation from equilibrium is the Laplace operator, and you will find that if you integrate by parts, this is exactly the quadratic form that corresponds to the Laplacian on your domain (once you have chosen boundary conditions). Harmonic functions are those whose stretching is dictated entirely by their boundary conditions, i.e., have no "energy" to push themselves away from equilibrium.
