Convexity of a functional using the Hessian

1. Feb 19, 2013

lmedin02

1. The problem statement, all variables and given/known data
Consider the functional $I:W^{1,2}(\Omega)\times W^{1,2}(\Omega)\rightarrow \mathbb{R}$ such that $I(f_1,f_2)=\int_{\Omega}{\dfrac{1}{2}|\nabla f_1|^2+\dfrac{1}{2}|\nabla f_1|^2+e^{f_1+f_2}-f_1-f_2}dx$. I would like to show that the functional is strictly convex by using the Hessian matrix.

2. Relevant equations

3. The attempt at a solution
Well, I think that clearly the functional is convex since each function inside the integrand is convex. However, I need to show strict convexity using the Hessian. But I am totally not sure of how to approach this derivative. Am I going to take a directional derivative in which I vary the first component only. Also, it seems like it would be rather challenging to show that the Hessian matrix is positive definite when its entries all are integrals. Any help or references towards getting started would be greatly appreciated. I definitely would like to understand how to attack such a question using the Hessian.