Variational Problems with Volume Constraints

We study variational problems with level set constraints of the form

where and .

In the one-dimensional case sharp conditions for the existence of global

and local minimizers are derived. Moreover some existence results are

provided when the energy E(u) is not of integral form, but instead

satisifes some abstract conditions like additivity, translation invariance

and solvability of a transition problem.

In the general n-dimensional case we consider the -limit when

. The result turns out to be a nonlocal functional

with minimizers satisfying (in the isotropic case) the minimal interface

criterion.