2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69544In this paper we give an unified approach to some questions arising in different fields of nonlinear analysis, namely: (a) the study of the structure of the Fuč\'ık spectrum and (b) possible variants and extensions of the monotonicity formula by Alt--Caffarelli--Friedman \cite{acf}. In the first part of the paper we present a class of optimal partition problems involving the first eigenvalue of the Laplace operator. Beside establishing the existence of the optimal partition, we develop a theory for the extremality conditions and the regularity of minimizers. As a first application of this approach, we give a new variational characterization of the first curve of the Fuč\'ık spectrum for the Laplacian, promptly adapted to more general operators. In the second part we prove a monotonicity formula in the case of many subharmonic components and we give an extension to solutions of a class of reaction--diffusion equation, providing some Liouville--type theorems.21 pagesAnalysis of PDEsFunctional Analysis35J65; 58E05text