establish that a minimization problem is indeed a convex minimization problem and then solve it I have a minimization problem of the following form :
$$\underset{\lambda (x) \in [0,1]}{min} \int_\Omega (1- \lambda(x))C_s(x)dx +  \int_\Omega \lambda(x)C_t(x)dx + \alpha\int_\Omega |\nabla\lambda(x)|dx$$
$ \Omega$ is a region in $\mathbb{R}^2$ ( for example a rectangular region )
$ C_s, C_t$ are some kind of cost functions.
First I want to be able to establish that this problem is indeed a convex minimization problem. Having done that I want to find out various methods that can be used to minimize. I have gone through the artice on wikipedia about convex optimization and I don't find it much helpful to my cause. I would like to know what are the various methods I should read that could help me solve this kind of a minimization problem. I have with me a book on convex optimization by Stephen Boyd. But the book seems large and I don't know where to start.
 A: Your problem looks convex to me. But if you are talking about Convex Optimization by Boyd and Vandenberghe, it only seems useful if you want
to consider approximations of $\lambda$ with a finite number of variables.
Otherwise you may want to look into calculus of variations. However, as the
problem stands, it may not have an optimum in the space of continuously
differentiable functions, or even in the continuous, almost everywhere
differentiable functions. In fact, for many choices of $\Omega, C_s, C_t$
I think the solutions will tend to an infeasible functions of the form
$\lambda(x) = 1$ if $C_s > C_t$, $\lambda(x) = 0$ otherwise.
If you change $|\nabla \lambda(x)|$ to something differentiable and strictly convex like $|\nabla \lambda(x)|^2$ it might be a different story.

From a quick look at the article you mention, I gather that this variational
problem is just an intermediate step in turning a non-convex discrete problem into a convex numerical problem. The "degenerate" solutions I warned about are
in fact precisely what is ultimately wanted.
So, to apply the techniques developed there you would definitely need to
understand convex (numerical) optimization problems, but I doubt that that
will be enough. A degree in physics would probably be useful.
Maybe someone else can recommend literature about variational calculus. For a quick introduction you might take a look at this PDF.
