Geometric Analysis of Spectral Partition Problems with measure constraints

In this talk, we discuss a class of spectral partition problems with a measure constraint inside a given box \(\Omega \subset \mathbb{R}^N\). More concretely, we consider the problem

\begin{equation}
\inf \left\{\sum_{i=1}^k \lambda_1\left(\omega_i\right) \mid \omega_1, \ldots, \omega_k \subset \Omega \text { nonempty open sets, } \omega_i \cap \omega_j=\emptyset \text { for all } i \neq j, \text { and } \sum_{i=1}^k\left|\omega_i\right| \leq a\right\}
\end{equation}
which is a prototypical situation of a shape optimization problem in a “box" involving partitions with spectral costs .

We establish the existence of an optimal open partition \(\left(\omega_1^*, \ldots, \omega_k^*\right)\) that saturates the measure constraint, showing that the corresponding eigenfunctions are locally Lipschitz continuous, and obtain some qualitative properties for the partition. Then, we provide a full regularity result for the associate free boundary \(\cup_i \partial \omega_i^*\), for a particular solution.

The talk is based in two works, one joint with Ederson Moreira dos Santos (ICMC-USP), Pêdra Andrade and Makson Santos (IST-Lisboa), the other in collaboration with Dario Mazzoleni (Pavia) and Makson Santos. We will also show some numerical simulations performed by Pedro Antunes (IST-Lisboa), which lead to some open problems and support some conjectures.

Zoom: https://cuaieed-unam.zoom.us/j/89946525336

Meeting ID: 899 4652 5336