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Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius

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Abstract

Consider the class of optimal partition problems with long range interactions

$$\begin{aligned} \inf \left\{ \sum _{i=1}^k \lambda _1(\omega _i):\ (\omega _1,\ldots , \omega _k) \in \mathcal {P}_r(\Omega ) \right\} , \end{aligned}$$

where \(\lambda _1(\cdot )\) denotes the first Dirichlet eigenvalue, and \(\mathcal {P}_r(\Omega )\) is the set of open k-partitions of \(\Omega \) whose elements are at distance at least r: \({{\,\mathrm{dist}\,}}(\omega _i,\omega _j)\ge r\) for every \(i\ne j\). In this paper we prove optimal uniform bounds (as \(r\rightarrow 0^+\)) in \(\mathrm {Lip}\)–norm for the associated \(L^2\)-normalized eigenfunctions, connecting in particular the nonlocal case \(r>0\) with the local one \(r \rightarrow 0^+\). The proof uses new pointwise estimates for eigenfunctions, a one-phase Alt–Caffarelli–Friedman and the Caffarelli-Jerison-Kenig monotonicity formulas, combined with elliptic and energy estimates. Our result extends to other contexts, such as singularly perturbed harmonic maps with distance constraints.

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Via Edoardo Bonardi 9, 20133, Milan, Italy

    Nicola Soave

  2. CAMGSD and Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Hugo Tavares

  3. Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), 75006, Paris, France

    Alessandro Zilio

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  1. Nicola Soave
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  2. Hugo Tavares
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Correspondence to Alessandro Zilio.

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Communicated by Yoshikazu Giga.

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N. Soave is partially supported by the INdAM-GNAMPA group (Italy). H. Tavares is partially supported by the Portuguese government through FCT-Fundação para a Ciência e a Tecnologia, I.P., under the projects UID/MAT/04459/2020, PTDC/MAT-PUR/28686/2017 and PTDC/MAT-PUR/1788/2020. A. Zilio is partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR)

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Soave, N., Tavares, H. & Zilio, A. Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius. Math. Ann. 386, 551–585 (2023). https://doi.org/10.1007/s00208-022-02406-8

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  • DOI: https://doi.org/10.1007/s00208-022-02406-8

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