# Building minimization optimization problem for 2nd-order elliptic PDE

I am solving elliptic PDE problem, for which, Euler scheme looks as following: $$\nabla [\gamma ( |\nabla u|^2) \nabla u] = 0,$$ where $$\gamma(|\nabla u|^2) = (1 + |\nabla u|^2)^{-1/2}.$$

I am aiming to define minimization problem for problem above. Does anybody have any suggestion how it should look like? Or some recommended literature?

## 2 Answers

For the particular equation you are solving (called the minimal surface equation), the functional you are trying to minimize is $$J(u) = \int_\Omega \sqrt{1 + |\nabla u|^2} \; dx.$$

You can find a derivation of the equations, as well as a discussion of solution approaches in lectures 31.5 and following here: http://www.math.tamu.edu/~bangerth/videos.html . Of course there are many other sources for the same material as well. For example, nearly every book about the calculus of variations will have your example.

• is there any example, how to embedded framework into trust region framework ? – bla_bla_bla Aug 29 '15 at 12:54
• No. But I also don't quite understand: you have an unconstrained optimization problem. Why not just use a line search? – Wolfgang Bangerth Aug 30 '15 at 3:08

Let's assume, that you have discretized problem and on every point you obtained evaluation $g_{ij}$. Therefore $g(u) = 0$ is your system of nonlinear equations.

Now let $\gamma$ to be norm function defined as $\gamma(x) = || g(x)||^2$.

Then your nonlinear eq. problem is equivalent to the problem $$\text{minimize} \ \ \gamma(x) .$$

• This is not a useful approach because it yields degenerate optimization problems (that do not satisfy the second order sufficient conditions for optima) at places where $g(x)=0$ and $g'(x)=0$. You need to use a better approach. – Wolfgang Bangerth Aug 24 '15 at 23:28