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

Abstract

In this work, given p ∈ (1, ∞), we prove the existence and simplicity of the first eigenvalue λp and its corresponding eigenvector (up, vp), for the following local/nonlocal PDE system (0.1)where Ω ⊂ IR N is a bounded open domain, 0 < r, s < 1 and α(p) + β(p) = p. Moreover, we address the asymptotic limit as p → ∞, proving the explicit geometric characterization of the corresponding first ∞−eigenvalue, namely λ∞, and the uniformly convergence of the pair (up, vp) to the ∞−eigenvector (u∞, v∞). Finally, the triple (u∞, v∞, λ∞) verifies, in the viscosity sense, a limiting PDE system.

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

“…The following result for the fractional Sobolev spaces with zero boundary value follows from [12,Lemma 2.1].…”

“…Much less is known in the nonlinear case p = 2 of (1.16). For this, we refer to Buccheri-da Silva-Miranda [12], da Silva-Salort [17], Biagi-Mugnai-Vecchi [8], Garain-Kinnunen [28] and Garain-Ukhlov [30]. Concerning mixed parabolic equation, Barlow-Bass-Chen-Kassmann [3] obtained Harnack inequality for the linear equation (1.17) ∂ t u + (−∆) s u − ∆u = 0.…”

A Brezis-Oswald approach for mixed local and nonlocal operators

“…in Ω, see Theorem 3.1 for the precise statement. We believe that this preliminary result is of independent interest, and we stress that Theorem 3.1 cannot be deduced as a corollary of the maximum principles proved in [5] nor in [11].…”

“…Here, ∆ p u = div(|∇u| p−2 ∇u) is the classical p−Laplacian operator and, for fixed s ∈ (0, 1) and up to a multiplicative positive constant, the fractional p−Laplacian is defined as (−∆) s p u(x) := 2 P.V. Problems driven by operators like L p,s have raised a certain interest in the last few years, both for the mathematical complications that the combination of two so different operators imply and for the wide range of applications, see for instance [5,4,6,11,12,13,14] and the references therein. A common feature of the aforementioned papers is to deal with weak solutions, in contrast with other results existing in the literature where viscosity solutions have been considered, see e.g.…”

A Faber-Krahn inequality for mixed local and nonlocal operators

“…We want to stress that one can get almost everywhere positivity of u 0 by means of the strong maximum principle for weak solutions proved in [14,9].…”