# Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

## Abstract

In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalue problem in a fixed box $$\Omega$$ and we consider the general case of Robin boundary conditions on $$\partial \Omega$$. It is well known that it suffices to consider bang-bang weights taking two values of different signs, that can be parametrized by the characteristic function of the subset E of $$\Omega$$ on which resources are located. Therefore, the optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to E, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. Namely, we completely solve the problem in dimension 1, we prove the counter-intuitive result that the ball is almost never a solution in dimension 2 or higher, despite what suggest the numerical simulations. We also introduce a new rearrangement in the ball allowing to get a better candidate than the ball for optimality when Neumann boundary conditions are imposed. We also provide numerical illustrations of our results and of the optimal configurations.

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1. 1.

Indeed, by comparing the Rayleigh quotients for $$m_1$$ and $$m_2$$, one gets

\begin{aligned} m_1>m_2 \ \Longrightarrow \ \lambda (m_1)<\lambda (m_2). \end{aligned}

Footnote 1 continued

One can refer for instance to [37, Lemma 2.3].

2. 2.

Here the uniqueness must be understood up to some subset of zero Lebesgue measure. In other words if $$E^*$$ is optimal then the union of $$E^*$$ with any subset of zero measure is also a solution.

3. 3.

This means that E satisfies the necessary first order optimality conditions of Problem (7), in other words that E is an upper level set of the eigenfunction $$\varphi$$ associated with the principal eigenvalue $$\lambda (E)$$ , more precisely that there exists $$\alpha$$ such that $$E=\{\varphi >\alpha \}$$, see also Sect. 2.4.

4. 4.

The Wallis integrals are the terms of the sequence $$(W_n)_{n \in \mathbb {N}}$$ defined by

\begin{aligned} W_n = \int \limits _0^{\frac{\pi }{2}} \sin ^nx\,\mathrm{d}x. \end{aligned}

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Correspondence to Yannick Privat.