In 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 Fučík spectrum and (b) possible variants and extensions of the monotonicity formula by Alt-Caffarelli-Friedman [1]. 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 Fučík 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. © Springer-Verlag 2004.
Conti, M., Terracini, S., Verzini, G. (2005). On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 22(1), 45-72 [10.1007/s00526-004-0266-9].
On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae
TERRACINI, SUSANNA;
2005
Abstract
In 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 Fučík spectrum and (b) possible variants and extensions of the monotonicity formula by Alt-Caffarelli-Friedman [1]. 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 Fučík 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. © Springer-Verlag 2004.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.