# Reference Request: Calculus of Variations in Hilbert Space

I'm looking for a good reference to a book on calculus of variations in the setting of Banach Spaces.

If it helps, I'm working with a particular functional acting on Fr\'{e}chet-differentiable operators between a particular $L^2$-space and itself and am trying to minimize it.

Thank you very much.

• Just to be clear, you want to minimize a functional $S \colon C^k(L^2,L^2) -> \mathbb{R}$, for some $k$, where $C^k(L^2,L^2)$ is the space of $k$-times Fréchet differentiable functions $L^2 \to L^2$? Usually, in the calculus of variations, the functional $S$ is defined by integrating over some domain, which for you would be a domain in $L^2$. How are you integrating over $L^2$? Nov 19 '16 at 8:34
• With respect to $m \otimes P$ where $m$ is a lebesgue measure and $P$ is a probability measure. Nov 20 '16 at 21:13
• I'm not quite sure how to read your notation (you are taking a tensor product of measures?). Moreover, as far as I know, the Lebesgue measure (in the usual straight forward sense) does not exist on any infinite dimensional Hilbert space (or any infinite dimensional normed space). Nov 20 '16 at 22:54