What corresponds to this Lagrangian density? Is there a physical example of a field that would have the following Lagrangian density
$$
L= \sqrt{1+\phi_x^2 +\phi_y^2+\phi_z^2}
$$ 
where the subscripts denote partial derivatives and $\phi$ is a scalar field?
 A: This looks a lot like soap film statics, but with an extra dimension. Consider a soap film glued to a ring. The film is described by a function $z = \phi(x,y)$, with $z$ the height of the film above the xy-plane. We want to minimise the potential energy of the film, which means to a good approximation minimising the surface area. The total area of the film is given by
$ A = \int \sqrt{1 + \phi_x^2 + \phi_y^2}$
where the integration is over the entire xy-plane. This functional looks very similar to your Lagrangian density. The solution can be derived by applying the Euler-Lagrange equations and the solution is confusingly also called Lagrange's equation
$ (1 + \phi_y)^2 \phi_{xx} + 2 \phi_x \phi_y \phi_{xy} + (1 + \phi_x)^2 \phi_{yy} = 0 $
To give you an idea of what the solutions look like, if the field doesn't change too rapidly we can make some approximations ($\phi_x \approx \phi_y \approx \phi_{xy} \approx 0$) and get Laplace's equation
$ \phi_{xx} + \phi_{yy} = \Delta \phi = 0$
This also provides an interesting interpretation for solutions to the Laplace equation, as they are approximate minimal-area surfaces.
Most of this carries over three dimensions but the minimal-surface interpretation isn't so clear anymore.
