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.