Variational analysis of nonlocal Dirichlet problems in periodically perforated domains

• Published in: Calculus of variations and partial differential equations Vol. 64; no. 7; p. 232
• Main Authors: Alicandro, Roberto, Gelli, Maria Stella, Leone, Chiara
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2025; Springer Nature B.V
• Subjects: Analysis; Asymptotic properties; Calculus of Variations and Optimal Control; Optimization; Control; Convergence; Dirichlet problem; Homogenization; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Order parameters; Sobolev space; Systems Theory; Theoretical; 49J45; 49J55; 35B27; 35B40; 74Q05; 45E10
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-025-03071-w

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter and -converging to local functionals defined on Sobolev spaces as . We study the asymptotic behaviour of the functionals when the order parameter is subject to Dirichlet conditions on a periodically perforated domains, given by a periodic array of small balls of radius centered on a –periodic lattice, being an additional small parameter and . We highlight differences and analogies with the local case, according to the interplay between the three scales , and . A fundamental tool in our analysis turns out to be a non local variant of the classical Gagliardo–Nirenberg–Sobolev inequality in Sobolev spaces which may be of independent interest and useful for other applications.