u/Wudulala

Hi guys, I am studying an integral functional where the integration domain is an upper contour set whose boundary is defined by a level set of a smooth function. I am interested in taking a functional derivative—something like a Gateaux or Hadamard derivative—with respect to the underlying smooth function.

That is, I am interested in an object like: \int_{h_0(x)>=0} w(x)h_0(x)dx,
or something more complicated as
\int_{h_0(x)=0} w(x)h_0(x)dH(x)^{d-1}.

And I’m taking a derivative with respect to h_0 in a certain direction.

Do you know of any good books or references on this topic? Thank you very much.

Or at least not emphasize on it enough. Feel like making this distinction explicit early on would prevent a lot of back-and-forth later.