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We study a weighted eigenvalue problem with anisotropic diffusion in boundedLipschitz domains $\Omega\subset \mathbb{R}^{N} $, $N\ge1$, under Robinboundary conditions, proving the existence of two positive eigenvalues$\lambda^{\pm}$ respectively associated with a positive and a negativeeigenfunction. Next, we analyze the minimization of $\lambda^{\pm}$ withrespect to the sign-changing weight, showing that the optimal eigenvalues$\Lambda^{\pm}$ are equal and the optimal weights are of bang-bang type, namelypiece-wise constant functions, each one taking only two values. As aconsequence, the problem is equivalent to the minimization with respect to thesubsets of $\Omega$ satisfying a volume constraint. Then, we completely solvethe optimization problem in one dimension, in the case of homogeneous Dirichletor Neumann conditions, showing new phenomena induced by the presence of theanisotropic diffusion. The optmization problem for $\lambda^{+}$ naturallyarises in the study of the optimal spatial arrangement of resources for aspecies to survive in a heterogeneous habitat.

Spectral optimization for weighted anisotropic problems with Robin conditions