5,378 research outputs found
Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
The fractional Laplacian is a non-local operator which
depends on the parameter and recovers the usual Laplacian as . A numerical method for the fractional Laplacian is proposed, based on
the singular integral representation for the operator. The method combines
finite difference with numerical quadrature, to obtain a discrete convolution
operator with positive weights. The accuracy of the method is shown to be
. Convergence of the method is proven. The treatment of far
field boundary conditions using an asymptotic approximation to the integral is
used to obtain an accurate method. Numerical experiments on known exact
solutions validate the predicted convergence rates. Computational examples
include exponentially and algebraically decaying solution with varying
regularity. The generalization to nonlinear equations involving the operator is
discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure
Lower bounds on nodal sets of eigenfunctions via the heat flow
We study the size of nodal sets of Laplacian eigenfunctions on compact
Riemannian manifolds without boundary and recover the currently optimal lower
bound by comparing the heat flow of the eigenfunction with that of an
artifically constructed diffusion process. The same method should apply to a
number of other questions; for example, we prove a sharp result saying that a
nodal domain cannot be entirely contained in a small neighbourhood of a
'reasonably flat' surface. We expect the arising concepts to have more
connections to classical theory and pose some conjectures in that direction
Hyperbolic Multi-Monopoles With Arbitrary Mass
On a complete manifold, such as Euclidean 3-space or hyperbolic 3-space, the
limit at infinity of the norm of the Higgs field is called the mass of the
monopole. We show the existence, on hypebolic 3-space, of monopoles with given
magnetic charge and arbitrary mass. Previously, aside from charge one
monopoles, existence was known only for monopoles with integral mass (since
these arise from U(1) invariant instantons on Euclidean 4-space). The method of
proof is based on Taubes' gluing procedure, using well-separated, explicit,
charge one monopoles. The analysis is carried out in a weighted Sobolev space
and necessitates eliminating the possibility of point spectra.Comment: 20 page
Charged-Surface Instability Development in Liquid Helium; Exact Solutions
The nonlinear dynamics of charged-surface instability development was
investigated for liquid helium far above the critical point. It is found that,
if the surface charge completely screens the field above the surface, the
equations of three-dimensional (3D) potential motion of a fluid are reduced to
the well-known equations describing the 3D Laplacian growth process. The
integrability of these equations in 2D geometry allows the analytic description
of the free-surface evolution up to the formation of cuspidal singularities at
the surface.Comment: latex, 5 pages, no figure
On the Cauchy problem for a general fractional porous medium equation with variable density
We study the well-posedness of the Cauchy problem for a fractional porous
medium equation with a varying density. We establish existence of weak energy
solutions; uniqueness and nonuniqueness is studied as well, according with the
behavior of the density at infinity
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