Project SpectralOPs
Spectral optimal partitions: geometric and numerical analysis
Supported by Fundação para a Ciência e Tecnologia, project with reference 2023.13921.PEX, with DOI identifier https://doi.org/10.54499/2023.13921.PEX
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Brief Description
This project, with starts February 1, 2025 and will last 1 year and a half, addresses contemporary questions in spectral partition problems (SPP), a subject within the area of shape optimization and spectral theory, where one generally seeks to minimize a certain cost functional depending on eigenvalues of a Schrödinger operator among a class of partitions (collections of mutually disjoint subsets of Euclidean spaces, manifolds or graphs).
Such problems find mathematical application in several areas of mathematical physics and in the analysis of partial differential equations (PDE), such as the study of the nodal sets of eigenfunctions of Schrödinger operators, as well as in the proof of monotonicity formulas, crucial for the regularity theory of free boundary problems, and are also important for characterizing the limiting behavior of solutions to semilinear elliptic systems of PDE with competition terms. They also appear quite naturally in physics (e.g. in liquid crystals or Cahn-Hilliard fluids), image processing, and (in the discrete case) machine learning, via their geometric interpretation, in particular in searching for clusters within data sets.
We propose to investigate three related, cutting-edge topics:
(1) Spectral partitions with volume and inclusion constraints.
(2) Spectral partitions for Schrödinger operators on unbounded domains.
(3) Spectral partitions for the Robin Laplacian.
These and related questions in SPP naturally bring challenges, not only for analytical studies but also for the development of fast and highly effective numerical methods. On the one hand, the development of numerical analysis of SPP has its own interest. On the other hand - and this is one of the pillars of this proposal - we intend to develop both theoretical and numerical aspects side by side, in such a way that both benefit from the advances of the other.
Extensive numerical simulations will allow us to explore, and support or refute, possible conjectures about the asymptotic behavior of optimizers with respect to measure constraints or the Robin boundary parameter, or the interaction between the fixed and free boundaries of optimizers of SPP. At the same time, the analytical developments will provide new mathematical tools for rigorous proofs in the numerical analysis of SPP, such as convergence and stability results for the numerical methods that are proposed.
The Research Team
Faculdade de Ciências da Universidade de Lisboa
FernUniversität in Hagen
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