4,887 research outputs found
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations
We deal with a class of equations driven by nonlocal, possibly degenerate,
integro-differential operators of differentiability order and
summability growth , whose model is the fractional -Laplacian with
measurable coefficients. We state and prove several results for the
corresponding weak supersolutions, as comparison principles, a priori bounds,
lower semicontinuity, and many others. We then discuss the good definition of
-superharmonic functions, by also proving some related properties. We
finally introduce the nonlocal counterpart of the celebrated Perron method in
nonlinear Potential Theory.Comment: To appear in Math. An
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
Parallel algorithm with spectral convergence for nonlinear integro-differential equations
We discuss a numerical algorithm for solving nonlinear integro-differential
equations, and illustrate our findings for the particular case of Volterra type
equations. The algorithm combines a perturbation approach meant to render a
linearized version of the problem and a spectral method where unknown functions
are expanded in terms of Chebyshev polynomials (El-gendi's method). This
approach is shown to be suitable for the calculation of two-point Green
functions required in next to leading order studies of time-dependent quantum
field theory.Comment: 15 pages, 9 figure
Charge and current oscillations in Fractional quantum Hall systems with edges
Stationary solutions of the Chern-Simons effective field theory for the
fractional quantum Hall systems with edges are presented for Hall bar, disk and
annulus. In the infinitely long Hall bar geometry (non compact case), the
charge density is shown to be monotonic inside the sample. In sharp contrast,
spatial oscillatory modes of charge density are found for the two circular
geometries, which indicate that in systems with compact geometry, charge and
current exist also far from the edges.Comment: 16 pages, 6 figures Revte
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