Convexity of a functional using the Hessian

[lmedin02]

Homework Statement

Consider the functional [itex]I:W^{1,2}(\Omega)\times W^{1,2}(\Omega)\rightarrow \mathbb{R}[/itex] such that [itex]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[/itex]. I would like to show that the functional is strictly convex by using the Hessian matrix.

Homework Equations

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.

 

[mighty2000]

Thank you for your post. It is great that you are considering the use of the Hessian matrix to show strict convexity of the given functional. As you have correctly pointed out, the individual components of the integrand are convex, but we need to show that the functional as a whole is strictly convex.

To do this, we need to show that the Hessian matrix is positive definite. As you have mentioned, this can be challenging since the entries of the Hessian are integrals. One approach could be to use the Cauchy-Schwarz inequality to bound the integrals and show that the resulting Hessian matrix is positive definite.

Another approach could be to use the definition of positive definiteness, which states that a matrix A is positive definite if and only if for all nonzero vectors x, x^T A x > 0. In this case, we can consider the directional derivatives in the direction of a nonzero vector and show that they are all strictly positive.

I hope this helps and guides you in the right direction. Good luck with your work! If you need further assistance, please do not hesitate to ask.