Linear theory for a mixed operator with Neumann conditions

Article type: Research Article

Authors: Dipierro, Serena^{a; *} | Proietti Lippi, Edoardo^b | Valdinoci, Enrico^a

Affiliations: [a] Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009, Australia | [b] Department of Mathematics and Computer Science, University of Florence, Viale Morgagni 67/A, 50134 Firenze, Italy

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We consider here a new type of mixed local and nonlocal equation under suitable Neumann conditions. We discuss the spectral properties associated to a weighted eigenvalue problem and present a global bound for subsolutions. The Neumann condition that we take into account comprises, as a particular case, the one that has been recently introduced in (Rev. Mat. Iberoam. 33(2) (2017), 377–416). Also, the results that we present here find a natural application to a logistic equation motivated by biological problems that has been recently considered in (Dipierro, Proietti Lippi and Valdinoci (2021)).

Keywords: Long-range interactions, zero-flux condition, spectral theory, boundedness of subsolutions

DOI: 10.3233/ASY-211718

Journal: Asymptotic Analysis, vol. 128, no. 4, pp. 571-594, 2022