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Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions. (English) Zbl 1366.49004
The authors analyze a free boundary/shape optimization governed by the linear eigenvalue with indefinite weight $\Delta \varphi + \lambda m\varphi = 0\;\text{in}\,\Omega,\;\;\partial_n\varphi+\beta \varphi=0\;\text{on}\,\partial\Omega,$ where $$\Omega$$ is a bounded open and connected set in $$\mathbb{R}^N$$ with Lipschitz boundary $$\partial\Omega$$ and the weight $$m$$ is a bounded measurable function changing sign in $$\Omega$$ i.e. $$\Omega_m^+= \{x\in \Omega:\,m(x)>0\}$$ has a measure strictly between $$0$$ and $$|\Omega|$$ and fulfils $$-1\leq m(x)\leq \kappa\;\text{a.e.}\,x\in \Omega,\;\kappa>0.$$ The optimization problem has the form $\inf_{m\in \mathcal{M}}\lambda(m),\;\mathcal{M}=\left\{m\in L^{\infty}:-1\leq m\leq \kappa,\;|\Omega_m^+|>0,\;\int_\Omega m\geq -m_0|\Omega|\right\}.$ The biological motivation for considering such a problem is formulated.

##### MSC:
 49J15 Existence theories for optimal control problems involving ordinary differential equations 49K20 Optimality conditions for problems involving partial differential equations 49Q10 Optimization of shapes other than minimal surfaces 49R05 Variational methods for eigenvalues of operators 49M05 Numerical methods based on necessary conditions
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