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On the criticality and the principal eigenvalue of almost periodic elliptic operators
[Sur la criticité et la valeur propre principale des opérateurs elliptiques presque périodiques]
Luca Rossi1
1Istituto G. Castelnuovo, Sapienza Università di Roma, Rome, Italy
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 321-331

Cet article fait partie du numéro thématique From Generation to Generation: The Mathematical Legacy of Haïm Brezis coordonné par : Henri Berestycki et al..  

We review the notion and the properties of the generalized principal eigenvalue for elliptic operators in unbounded domains, and we relate it to the criticality theory. We focus on operators with almost periodic coefficients. We present a Liouville-type result in dimension $N\le 2$. Next, we show with a counterexample that criticality is not equivalent to the existence of an almost periodic principal eigenfunction, even for self-adjoint operators. Finally, we exhibit an almost periodic operator which is subcritical but which admits a critical limit operator. This is a manifestation of the instability character of the criticality property in the almost periodic setting.

Nous passons en revue la notion et les propriétés de la valeur propre principale généralisée pour des opérateurs elliptiques dans des domaines non bornés, et nous la mettons en relation avec la théorie de la criticité. Nous nous concentrons sur des opérateurs à coefficients presque périodiques. Nous présentons un résultat de type Liouville en dimension $N\le 2$. Ensuite, nous montrons, au moyen d’un contre-exemple, que la criticité n’est pas équivalente à l’existence d’une fonction propre principale presque périodique, même pour des opérateurs auto-adjoints. Enfin, nous exhibons un opérateur presque périodique qui est sous-critique mais qui admet un opérateur limite critique. Cela illustre le caractère instable de la propriété de criticité dans le cadre presque périodique.

Reçu le :
Révisé le :
Accepté le :
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DOI : 10.5802/crmath.829
Classification : 35B15, 35B09, 35J10, 35B53, 35P05
Keywords: Generalized principal eigenvalue, elliptic operators, almost periodic coefficients, criticality theory, Liouville-type theorem
Mots-clés : Valeur propre principale généralisée, opérateurs elliptiques, coefficients presque périodiques, théorie de la criticité, théorème de Liouville
Note : Article soumis sur invitation

Luca Rossi  1

1 Istituto G. Castelnuovo, Sapienza Università di Roma, Rome, Italy
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Luca Rossi. On the criticality and the principal eigenvalue of almost periodic elliptic operators. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 321-331. doi: 10.5802/crmath.829
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     language = {en},
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