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arXiv · 2506.15861

Ahlfors-regularity for minimizers of a multiphase optimal design problem

Luca Esposito·Lorenzo Lamberti·Giovanni Pisante
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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 $\mathcal{E}(i)$ of prescribed volume that partition a given domain $\Omega\subset\mathbb{R}^n$. The cost functional associated with a configuration $\left(\{\mathcal{E}(i)\}_i,u\right)$ 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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Luca Esposito, Lorenzo Lamberti, Giovanni Pisante. 2025-06-18. Ahlfors-regularity for minimizers of a multiphase optimal design problem. https://arxiv.org/abs/2506.15861

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