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SUMMARY:Free boundary regularity for a spectral optimal  partition problem
  with volume and inclusion constraints - Dario Mazzoleni (Università degl
 i Studi di Pavia)
DTSTART:20260203T155000Z
DTEND:20260203T163000Z
UID:TALK239617@talks.cam.ac.uk
DESCRIPTION:In this talk we will deal with the characterization of the fre
 e boundary of the solutions to the following spectral $k$-partition proble
 m with measure and inclusion constraints:&nbsp\; &nbsp\; \\begin{equation*
 }\\inf \\left\\{\\sum_{i=1}^k \\lambda_1(\\omega_i)\\\; : \\\;\\begin{arra
 y}{c}\\omega_i \\subset \\Omega \\mbox{ are nonempty open sets for all } i
 =1\,\\ldots\, k\, \\\\&nbsp\;\\omega_i \\cap \\omega_j = \\emptyset \\\, \
 \: \\text{for all}\\: i \\not=j \\mbox{ and } \\sum_{i=1}^{k}|\\omega_i| =
  &nbsp\;a\\\\\\end{array}\\right\\}\,\\end{equation*}where $\\Omega$ is a 
 bounded domain of $\\mathbb{R}^N$\, $a\\in (0\,|\\Omega|)$. In particular\
 , we will show free boundary optimality conditions\, classify contact poin
 ts\, characterize the regular and singular part of the free boundary (incl
 uding branching points)\, and describe the interaction of the partition wi
 th the fixed boundary $\\partial \\Omega$.\nThe proof is based on a pertur
 bed version of the problem\, combined with monotonicity formulas\, blowup 
 analysis and classification of blowups\, suitable deformations of optimal 
 sets and eigenfunctions\, as well as the improvement of flatness of [Russ-
 Trey-Velichkov\, CVPDE 58\, 2019] for the one-phase points\, and of [De Ph
 ilippis-Spolaor-Velichkov\, Invent. Math. 225\, 2021] at two-phase points.
 \nFinally\, if time permits\, we will discuss some related problems that w
 e are currently investigating\, in particular the case when each component
  of the partition has a different measure constraint.\nThis is a joint pro
 ject with Makson S. Santos (Univ. Lisbon) and Hugo Tavares (IST Lisbon).
LOCATION:Seminar Room 1\, Newton Institute
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