12 research outputs found
Dirichlet conditions in Poincar\'e-Sobolev inequalities: the sub-homogeneous case
We investigate the dependence of optimal constants in Poincar\'e- Sobolev
inequalities of planar domains on the region where the Dirichlet condition is
imposed. More precisely, we look for the best Dirichlet regions, among closed
and connected sets with prescribed total length (one-dimensional Hausdorff
measure), that make these constants as small as possible. We study their
limiting behaviour, showing, in particular, that Dirichler regions homogenize
inside the domain with comb-shaped structures, periodically distribuited at
different scales and with different orientations. To keep track of these
information we rely on a -convergence result in the class of varifolds.
This also permits applications to reinforcements of anisotropic elastic
membranes. At last, we provide some evidences for a conjecture.Comment: arXiv admin note: text overlap with arXiv:1412.294
Approximation of length minimization problems among compact connected sets
In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some
classical minimization problems involving the length of an unknown
one-dimensional set, with an additional connectedness constraint, in dimension
two. We introduce a term of new type relying on a weighted geodesic distance
that forces the minimizers to be connected at the limit. We apply this approach
to approximate the so-called Steiner Problem, but also the average distance
problem, and finally a problem relying on the p-compliance energy. The proof of
convergence of the approximating functional, which is stated in terms of
Gamma-convergence relies on technical tools from geometric measure theory, as
for instance a uniform lower bound for a sort of average directional Minkowski
content of a family of compact connected sets
Where best to place a Dirichlet condition in an anisotropic membrane?
We study a shape optimization problem for the first eigenvalue of an elliptic
operator in divergence form, with non constant coefficients, over a fixed
domain . Dirichlet conditions are imposed along and,
in addition, along a set of prescribed length (-dimensional
Hausdorff measure). We look for the best shape and position for the
supplementary Dirichlet region in order to maximize the first
eigenvalue. The limit distribution of the optimal sets, as their prescribed
length tends to infinity, is characterized via -convergence of suitable
functionals defined over varifolds: the use of varifolds, as opposed to
probability measures, allows one to keep track of the local orientation of the
optimal sets (which comply with the anisotropy of the problem), and not just of
their limit distribution.Comment: 23 pages, 2 figure
Partial regularity for the optimal -compliance problem with length penalization
We establish a partial regularity result for minimizers of the
optimal -compliance problem with length penalization in any spatial
dimension , extending some of the results obtained in
[Chambolle-Lamboley-Lemenant-Stepanov 17], [Bulanyi-Lemenant 20]. The key
feature is that the regularity of minimizers for some free
boundary type problem is investigated with a free boundary set of codimension
. We prove that every optimal set cannot contain closed loops, and it is
regular at -a.e. point for every .Comment: 42 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1911.0924