Nonlocal eigenvalue problems and superposition operators
Serena Dipierro, Edoardo Proietti Lippi, Caterina Sportelli, Enrico Valdinoci
TL;DR
The paper develops a spectral theory for mixed local and nonlocal operators formed by a signed superposition $L_\mu u = \int_{[0,1]} (-\Delta)^s u\, d\mu(s)$, with lower-order terms allowed to carry the opposite sign. It establishes a perturbation framework showing convergence to classical local eigenvalue problems as the measure concentrates and characterizes sign-definiteness and simplicity of eigenfunctions on connected domains. A detailed analysis on disconnected domains reveals that first eigenfunctions must change sign and that unions of components can have strictly smaller first eigenvalues, contrasting with classical Laplacian theory. Additionally, a regularity theory is developed to support these spectral results, including $W^{2,p}$ and $C^{1,\alpha}$ estimates for solutions of the perturbed problems, with implications for nonlocal operator perturbations and population dynamics models.
Abstract
We study the spectral theory of mixed local and nonlocal operators with lower-order terms in the right-hand side of the equation. This kind of problems is motivated by the analysis of superposition operators of mixed order and with the "wrong sign" of the lower-order terms with respect to the classical elliptic theory. Our results include: -convergence to classical cases when the right-hand side of the eigenvalye equations "localizes", recovering the simplicity and sign-definiteness of eigenfunctions in the limit; -a detailed analysis of disconnected domains, showing that, unlike the classical case, any eigenfunction associated with the first eigenvalue must change sign, and that the first eigenvalue of a union of disconnected domains is strictly smaller than that of its individual components; -examples in which the first eigenvalue is either simple or non-simple in disconnected domains; -a regularity theory that underpins these results.