Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimizing the first Dirichlet eigenvalue of the Laplacian with an
  obstacle   Inside a fixed bounded domain $\Omega$ of the plane, we look for the best
compact connected set $K$, of given perimeter, in order to maximize the first
Dirichlet eigenvalue $\lambda_1(\Omega\setminus K)$. We discuss some of the
qualitative properties of the maximizers, moving toward existence, regularity
and geometry. Then we study the problem in specific domains: disks, rings, and,
more generally, disks with convex holes. In these situations, we prove symmetry
and, in some cases non symmetry results, identifying the solution.
  We choose to work with the outer Minkowski content as the "good" notion of
perimeter. Therefore, we are led to prove some new properties for it as its
lower semicontinuity with respect to the Hausdorff convergence and the fact
that the outer Minkowski content is equal to the Hausdorff lower semicontinuous
envelope of the classical perimeter.

12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph