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

By Monica Conti, S. Terracini, Gianmaria Verzini. 2003

114 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/279408e771d03db3cbe4d367381e986fa58703e5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Bernard Helffer
2. T. Hoffmann-Ostenhof
3. S. Terracini
4. Gianmaria Verzini
5. Nils Berglund

In Computational Theory and Mathematics

1. Bernard Helffer
2. T. Hoffmann-Ostenhof
3. S. Terracini
4. Nils Berglund
5. Fabrice Baudoin
6. Cédric Villani
7. Frédéric Hérau

In Mathematical Physics

1. Bernard Helffer
2. T. Hoffmann-Ostenhof
3. S. Terracini
4. Nils Berglund
5. Cédric Villani
6. Frédéric Hérau
7. Fabrice Baudoin

Citations in the corpus

• Nodal Domains and Spectral Minimal Partitions (2006, methodology): “It has been proved1 by Conti-Terracini-Verzini [7, 8, 9] that Theorem 1.11 For any k, there exists a k– minimal regular strong partition.”
• On Spectral Minimal Partitions: A Survey (2010, methodology): “It has been proved by Conti-Terracini-Verzini [CTV1, CTV2, CTV3 ] (existence) and Helffer–Hoffmann-Ostenhof–Terracini [HHOT1] (regularity) the following theorem:”