Ahlfors-regularity for minimizers of a multiphase optimal design problem

2025/06/18 by Esposito, Luca, Lamberti, Lorenzo, Pisante, Giovanni
#49N60 #49Q10 #49Q20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2506.15861

Abstract

We establish an Alhfors-regularity result for minimizers of a multiphase optimal design problem. It is a variant of the classical variational problem which involves a finite number of chambers E(i) of prescribed volume that partition a given domain Ω⊂ℝⁿ. The cost functional associated with a configuration (\E(i)\_i,u) is made up of the perimeter of the partition interfaces and a Dirichlet energy term, which is discontinuous across the interfaces. We prove that the union of the optimal interfaces is (n-1)-Alhfors-regular via a penalization method and decay estimates of the energy.

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