10 de Mayo de 2025
10:00 hrs

Geometric Analysis of Spectral Partition Problems with measure constraints

Hugo Tavares
Universidade de Lisboa

In this talk, we discuss a class of spectral partition problems with a measure constraint inside a given box Ω⊂ℝN. More concretely, we consider the problem: inf{∑i=1kλ1(ωi)∣ω1,…,ωk⊂Ω nonempty open sets, ωi∩ωj=∅ for all i≠j, and ∑i=1k∣∣ωi∣∣≤a} 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 (ω∗1,…,ω∗k) 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 ∪i∂ω∗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