Existence, Lipschitz regularity, unique continuation, nodal set estimates, and C^{1,α} free boundary regularity are proved for optimal torsional partitions and segregated configurations.
arXiv preprint arXiv:2412.00781 , year=
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abstract
We consider energy-minimizing harmonic maps into trees and we prove the regularity of the singular part of the free interface near triple junction points. Precisely, by proving a new epiperimetric inequality, we show that around any point of frequency $3/2$, the free interface is composed of three $C^{1,\alpha}$-smooth $(d-1)$-dimensional manifolds (composed of points of frequency $1$) with common $C^{1,\alpha}$-regular boundary (made of points of frequency $3/2$) that meet along this boundary at 120 degree angles. Our results also apply to spectral optimal partition problems for the Dirichlet eigenvalues.
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math.AP 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Minimizers of the multiphase vectorial Bernoulli functional exist, are locally Lipschitz, avoid triple points on the free boundary, and have C^{1,η} regularity near two-phase and branching points.
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Optimal partition and segregation problems driven by torsional rigidity
Existence, Lipschitz regularity, unique continuation, nodal set estimates, and C^{1,α} free boundary regularity are proved for optimal torsional partitions and segregated configurations.
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