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Journal Articles Archive for Rational Mechanics and Analysis Year : 2022

Shape optimization of a weighted two-phase Dirichlet eigenvalue

Abstract

Let $m$ be a bounded function and $\alpha$ a nonnegative parameter. This article is concerned with the first eigenvalue $\lambda_\alpha(m)$ of the drifted Laplacian type operator $\mathcal L_m$ given by $\mathcal L_m(u)= -\operatorname{div} \left((1+\alpha m)\nabla u\right)-mu$ on a smooth bounded domain, with Dirichlet boundary conditions. Assuming uniform pointwise and integral bounds on $m$, we investigate the issue of minimizing $\lambda_\alpha(m)$ with respect to $m$. Such a problem is related to the so-called ``two phase extremal eigenvalue problem'' and arises naturally, for instance in population dynamics where it is related to the survival ability of a species in a domain. We prove that unless the domain is a ball, this problem has no ``regular'' solution. We then provide a careful analysis in the case of a ball by: (1) characterizing the solution among all radially symmetric resources distributions, with the help of a new method involving a homogenized version of the problem; (2) proving in a more general setting, a stability result for the centered distribution of resources with the help of a monotonicity principle for second order shape derivatives which significantly simplifies the analysis.
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Dates and versions

hal-02432387 , version 1 (08-01-2020)
hal-02432387 , version 2 (18-10-2021)

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Cite

Idriss Mazari, Grégoire Nadin, Yannick Privat. Shape optimization of a weighted two-phase Dirichlet eigenvalue. Archive for Rational Mechanics and Analysis, 2022, 243, pp.95-137. ⟨10.1007/s00205-021-01726-4⟩. ⟨hal-02432387v2⟩
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