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 Fuˇc´ık spectrum and (b) possible variants and extensions of the monotonicity formula byAlt–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 Fuˇc´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

On a class of optimal partition problems related to the Fucik spectrum and to the monotonicity formulae

CONTI, MONICA;VERZINI, GIANMARIA
2005-01-01

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 Fuˇc´ık spectrum and (b) possible variants and extensions of the monotonicity formula byAlt–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 Fuˇc´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
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/555218
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