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Minimization under volume constraints

in collaboration with O. Rieger

Here are some results obtained in collaboration with Marc Oliver Rieger. All details can be found in our paper "Local minimizers of functionals with multiple volume constraints " (see also the publication section).

We study variational problems with volume constraints, i.e., with level sets of prescribed measure. The general form of a variational problem with two level set constraints is given by the minimization of: under the volume constraints |{x∈Ω, u(x)=a}|=α, |{x∈Ω, u(x)=b}|=β, where u ∈ H¹(Ω) and α + β <|Ω| . Problems of this class have been encountered in the context of immissible fluids and mixtures of micromagnetic materials. The difficulty of such problems is the special structure of their constraints: A sequence of functions satisfying these constraints can have a limit which fails to satisfy the constraints.

In this work we are introducing a numerical method for the approximation of local minimizers of (1). We apply this method to various examples and obtain a first picture of the shape of local and global minimizers for some simple domains in R². Guided by the numerical results, we prove rigorously that even on the unit square solutions are not depending continuously on the parameter α and β and illustrate this with numerical results. Moreover, we show that even on convex domains in R² nontrivial local minimizers can exist.