Eigenvalues for A Combination Between Local and Nonlocal p-Laplacians

Abstract

In this paper we study the Dirichlet eigenvalue problemHere ∆pu is the standard local p−Laplacian, ∆ J,p u is a nonlocal, p−homogeneous operator of order zero and Ω is a bounded domain in R N . We show that the first eigenvalue (that is isolated and simple) satisfies (λ 1 ) 1/p → Λ as p → ∞ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = limp→∞ up, and find the limit problem that is satisfied in the limit.

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

“…where g is nonsingular have been studied recently. In this regard, Del Pezzo-Ferreira-Rossi [33] studied the following eigenvalue problem…”

A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

“…The case of mixed operators. The study of mixed local/nonlocal operators has been recently received an increasing level of attention, both in view of their intriguing mathematical structure, which combines the classical setting and the features typical of nonlocal operators in a framework that is not scale-invariant [40,45,46,5,32,10,21,4,20,24,23,22,39,7,1,18,30,27,28,35,36,37,38,19,9,6,54], and of their importance in practical applications such as the animal foraging hypothesis [29,51].…”

A System of Local/Nonlocal $p$-Laplacians: The Eigenvalue Problem and Its Asymptotic Limit as $p\to\infty$

“…We remark that the study of the aforementioned issues may give some insight on the connection between problems that admit distributional formulation with their limiting counterpart without this kind of structure. For a better comprehension on this subject we refer the reader to [5], [10], [11], [12], [13], [15], [16], [17], [18], [20], [22], [29] or [37].…”

“…It is interesting to notice that the geometric characterization of λ ∞ strongly depends on a simple geometric property of the domain: the radius of the largest ball contained in Ω (cf. [17,Theorem 1.2]). Indeed, if R < 1, the limiting eigenvalue is neither affected by the presence of the nonlocal diffusion nor by the behaviour of the sequences α n , β n (and one recover the same result of [5,Theorem 1.1]).…”