Variational Problems with Volume Constraints

We study variational problems with level set constraints of the form where u ∈ H¹() and α + β < ||.

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 Γ-limit when α + β ⭢ ||. The result turns out to be a nonlocal functional with minimizers satisfying (in the isotropic case) the minimal interface criterion.