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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.

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    $\begingroup$ 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 $L^2$? $\endgroup$
    – Igor Khavkine
    Commented Nov 19, 2016 at 8:34
  • $\begingroup$ With respect to $m \otimes P$ where $m$ is a lebesgue measure and $P$ is a probability measure. $\endgroup$
    – Rombaldo Meniscus
    Commented Nov 20, 2016 at 21:13
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    $\begingroup$ 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). $\endgroup$
    – Igor Khavkine
    Commented Nov 20, 2016 at 22:54

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The book of Struwe "Variational Methods", in the first chapter, has an abstract configuration in Banach Spaces. The so called variational lemmas. Then he gives some examples applied to partial differential equations. This is the book: http://www.springer.com/gp/book/9783662026243

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