Optimization of the principal eigenvalue under mixed boundary conditions (Q2877641)

Let \(\Omega\subset \mathbb{R}^2\) be a smooth and bounded domain and let \(\Gamma\) be a portion of \(\partial\Omega\) with positive \(1\)-Lebesgue measure. The paper under review deals with optimization of the principal eigenvalue \(\lambda\) of the Laplacian operator under mixed boundary conditions NEWLINE\[NEWLINE \begin{cases} \Delta u+\lambda g(x)u=0 & \text{in}\;\Omega,\\ u=0 & \text{on}\;\Gamma,\\ \dfrac{\partial u}{\partial\nu}=0 & \text{on}\;\partial\Omega\setminus \Gamma, \end{cases} NEWLINE\]NEWLINE in the case when the weight \(g(x)\) has indefinite sign and varies in a class of rearrangements. From a biological point of view, these optimization problems are motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive or to decline. The authors prove existence and uniqueness results, and present some features of the optimizers. In some special cases, results on symmetry and symmetry breaking for the minimizer are proved as well.