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:
|
E(u)= |
∫ |
|
f(x,u(x),∇ u(x)) dx
(1) |
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.
Click on a picture to see the evolution
Three examples of minimization under two volume constraints (the two
first different initial states on the left converge to the same local
minima)
One example of minimization under three volume constraints (on the
left) and an example of nontrivial local minima (the two pictures on
the right)