Let $\Omega \subset \mathbb{R}^3$ be a sheared waveguide, i.e., $\Omega$ is built by translating a cross-section (an arbitrary bounded connected open set of $\mathbb{R}^2$ ) in a constant direction along an unbounded spatial curve. Consider $-\Delta_{\Omega}^D$ the Dirichlet Laplacian operator in $\Omega$. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-\Delta_{\Omega}^D$. After that, we state sufficient conditions that give rise to a non-empty discrete spectrum for $-\Delta_{\Omega}^D$. Finally, in case the cross section translates along a broken line in $\mathbb{R}^3$, we prove that the discrete spectrum of $-\Delta_{\Omega}^D$ is finite, furthermore, we show a particular geometry for $\Omega$ which implies that the total multiplicity of the discrete spectrum is equals 1.[-]

In this talk we deal with the regularity of optimal sets for a shape optimization problem involving a combination
of eigenvalues, under a fixed volume constraints. As a model problem, consider
\[
\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\},
\]
where $\langle_i(\cdot)$ denotes the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue measure.
We prove that any minimizer $_{opt}$ has a regular part of the topological boundary which is relatively open and
$C^{\infty}$ and that the singular part has Hausdorff dimension smaller than $d-d^*$, where $d^*\geq 5$ is the minimal
dimension allowing the existence of minimal conic solutions to the blow-up problem.

We mainly use techniques from the theory of free boundary problems, which have to be properly extended to the case of
vector-valued functions: nondegeneracy property, Weiss-like monotonicity formulas with area term; finally through the
properties of non tangentially accessible domains we shall be in a position to exploit the ''viscosity'' approach recently proposed by De Silva.

This is a joint work with Dario Mazzoleni and Bozhidar Velichkov.[-]