793 research outputs found
Spectral properties of non-local uniformly-elliptic operators
In this paper we consider the spectral properties of a class of non-local uniformly elliptic operators, which arise from the study of non-local uniformly elliptic partial differential equations. Such equations arise naturally in the study of a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators, and the non-local perturbation is in the form of an integral term. We study the eigenvalues, the multiplicities of these eigenvalues, and the existence of corresponding positive eigenfunctions. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. However, we show that under suitable hypotheses, there still exists a principal eigenvalue of these operators
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity
This paper deals with the existence and the asymptotic behavior of
non-negative solutions for a class of stationary Kirchhoff problems driven by a
fractional integro-differential operator and involving a
critical nonlinearity. The main feature, as well as the main difficulty, of the
analysis is the fact that the Kirchhoff function can be zero at zero, that
is the problem is degenerate. The adopted techniques are variational and the
main theorems extend in several directions previous results recently appeared
in the literature
Spatial Hamiltonian identities for nonlocally coupled systems
We consider a broad class of systems of nonlinear integro-differential
equations posed on the real line that arise as Euler-Lagrange equations to
energies involving nonlinear nonlocal interactions. Although these equations
are not readily cast as dynamical systems, we develop a calculus that yields a
natural Hamiltonian formalism. In particular, we formulate Noether's theorem in
this context, identify a degenerate symplectic structure, and derive
Hamiltonian differential equations on finite-dimensional center manifolds when
those exist. Our formalism yields new natural conserved quantities. For
Euler-Lagrange equations arising as traveling-wave equations in gradient flows,
we identify Lyapunov functions. We provide several applications to
pattern-forming systems including neural field and phase separation problems.Comment: 39 pages, 1 figur
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