516 research outputs found
Inverse Problems of Determining Sources of the Fractional Partial Differential Equations
In this chapter, we mainly review theoretical results on inverse source
problems for diffusion equations with the Caputo time-fractional derivatives of
order . Our survey covers the following types of inverse
problems: 1. determination of time-dependent functions in interior source terms
2. determination of space-dependent functions in interior source terms 3.
determination of time-dependent functions appearing in boundary condition
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
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