I came across this paper Variational Surface Cutting and a key concept introduced is the one of shape derivative. Is there a definition or reference I can use to get a better understanding? Is this related to Riemannian geometry, I own a couple of books on the subject there's no such definition.
Is there a reference you can suggest to look this up, together with main theorems involved in the theory maybe.
Thank you.
The global ideas are instead of defining the derivative of your functional ( which is defined for example on a domain $\Omega_0 \subset \mathbb{R}^n$ ) in a neighborhood of $\Omega_0$ "which have no sense because the subset of $\mathbb{R}^n$ is not a normed vector space "
you define the neighborhood of $\Omega_0$ as $\Omega= (Id+V)(\Omega_0)$ where $V$ is a vector field and $||V||\le\epsilon$ and by changes of variables you can define the derivative of your functional in the neighborhood of the identity, this idea belongs to Hadamard. I think you can find this notion in any book of shape optimization please look at these references:
https://www.amazon.fr/Shape-Variation-Optimization-Geometrical-Analysis/dp/3037191783 https://books.google.co.ma/books?id=XXvNAQAACAAJ&dq=antoine+henrot&hl=fr&sa=X&ved=0ahUKEwjcmu6i_vfpAhVKRBUIHfMJBeIQ6AEIRjAD https://www.amazon.fr/Shapes-Geometries-Analysis-Differential-Optimization/dp/0898719364