657 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
The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity
© 2018, The Author(s). In this paper, we focus on the convergence analysis and error estimation for the unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. By introducing a double iterative technique, in the case of the nonlinearity with singularity at time and space variables, the unique positive solution to the problem is established. Then, from the developed iterative technique, the sequences converging uniformly to the unique solution are formulated, and the estimates of the error and the convergence rate are derived
Variational Theory and Domain Decomposition for Nonlocal Problems
In this article we present the first results on domain decomposition methods
for nonlocal operators. We present a nonlocal variational formulation for these
operators and establish the well-posedness of associated boundary value
problems, proving a nonlocal Poincar\'{e} inequality. To determine the
conditioning of the discretized operator, we prove a spectral equivalence which
leads to a mesh size independent upper bound for the condition number of the
stiffness matrix. We then introduce a nonlocal two-domain variational
formulation utilizing nonlocal transmission conditions, and prove equivalence
with the single-domain formulation. A nonlocal Schur complement is introduced.
We establish condition number bounds for the nonlocal stiffness and Schur
complement matrices. Supporting numerical experiments demonstrating the
conditioning of the nonlocal one- and two-domain problems are presented.Comment: Updated the technical part. In press in Applied Mathematics and
Computatio
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