A min-max theorem for multiple integrals of the Calculus of Variations and applications

Arcoya, David ; Boccardo, Lucio

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 6 (1995) no. 1, pp. 29-35.

Abstract (VO)

In this paper we deal with the existence of critical points for functionals defined on the Sobolev space \( W_{0}^{1,2} (\Omega) \) by \( J(v) = \int_{\Omega} \mathfrak{I} (x,v,Dv) \, dx \), \( v \in W_{0}^{1,2} (\Omega) \), where \( \Omega \) is a bounded, open subset of \( \mathbb{R}^{N} \). Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of Calculus of Variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions. Then we present some applications of this theorem to the study of the existence and multiplicity of nonnegative critical points for multiple integrals of the Calculus of Variations.

Riassunto

In questa Nota si studia l'esistenza di punti critici per funzionali definiti nello spazio di Sobolev \( W_{0}^{1,2} (\Omega) \) da \( J(v) = \int_{\Omega} \mathfrak{I} (x,v,Dv) \, dx \), \( v \in W_{0}^{1,2} (\Omega) \), dove \( \Omega \) è un aperto limitato di \( \mathbb{R}^{N} \). Esempi molto semplici (e ben noti) di funzionali mostrano che la differenziabilità può mancare anche se la funzione è molto regolare. Questo motiva l'interesse a un teorema astratto di punto critico del tipo di quello di Ambrosetti-Rabinowitz («Mountain Pass Theorem» o «Teorema del cratere») per funzionali che non sono differenziabili in tutte le direzioni. Presentiamo alcune applicazioni di tale teorema allo studio dell'esistenza e della molteplicità di punti critici positivi di integrali multipli del Calcolo delle Variazioni.

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