3,156 research outputs found
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application
We develop further the theory of symmetrization of fractional Laplacian
operators contained in recent works of two of the authors. The theory leads to
optimal estimates in the form of concentration comparison inequalities for both
elliptic and parabolic equations. In this paper we extend the theory for the
so-called \emph{restricted} fractional Laplacian defined on a bounded domain
of with zero Dirichlet conditions outside of .
As an application, we derive an original proof of the corresponding fractional
Faber-Krahn inequality. We also provide a more classical variational proof of
the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297
External optimal control of fractional parabolic PDEs
In this paper we introduce a new notion of optimal control, or source
identification in inverse, problems with fractional parabolic PDEs as
constraints. This new notion allows a source/control placement outside the
domain where the PDE is fulfilled. We tackle the Dirichlet, the Neumann and the
Robin cases. For the fractional elliptic PDEs this has been recently
investigated by the authors in \cite{HAntil_RKhatri_MWarma_2018a}. The need for
these novel optimal control concepts stems from the fact that the classical PDE
models only allow placing the source/control either on the boundary or in the
interior where the PDE is satisfied. However, the nonlocal behavior of the
fractional operator now allows placing the control in the exterior. We
introduce the notions of weak and very-weak solutions to the parabolic
Dirichlet problem. We present an approach on how to approximate the parabolic
Dirichlet solutions by the parabolic Robin solutions (with convergence rates).
A complete analysis for the Dirichlet and Robin optimal control problems has
been discussed. The numerical examples confirm our theoretical findings and
further illustrate the potential benefits of nonlocal models over the local
ones.Comment: arXiv admin note: text overlap with arXiv:1811.0451
Nonlocal elliptic equations in bounded domains: a survey
In this paper we survey some results on the Dirichlet problem for nonlocal operators of the form
We
start from the very basics, proving existence of solutions, maximum principles,
and constructing some useful barriers. Then, we focus on the regularity
properties of solutions, both in the interior and on the boundary of the
domain.
In order to include some natural operators in the regularity theory, we
do not assume any regularity on the kernels. This leads to some interesting
features that are purely nonlocal, in the sense that have no analogue for local
equations.
We hope that this survey will be useful for both novel and more experienced
researchers in the field
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