Nonlocal elliptic equations in bounded domains: a survey

By Xavier Ros-Oton. 2015

243 citations recorded by the source corpus.

In this paper we survey some results on the Dirichlet problem ( Lu = f in u = g in R n n for nonlocal operators of the form Lu(x) = PV Z Rn u(x) u(x + y) K(y)dy: 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 L 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 they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the eld. 2010 Mathematics Subject Classication: 47G20, 60G52, 35B65.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/5217adc74c2466c9c07609bb825bbd1ce76f69b7

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Citations in the corpus

Fractional Laplace Operator and Meijer G-function (2015, background): “This operator arises as a Markov generator of an isotropic stable process in R, it is of increasing interest for the partial differential equations community and it is used in various areas of applied mathematics, see [6, 22].”
Fractional order Orlicz-Sobolev spaces (2017, background): “Up to date is almost impossible to give a comprehensive list of references and we refer the interested reader, for instance, to the surveys [8, 23, 27].”
The Calderón problem for variable coefficients nonlocal elliptic operators (2017, unknown): “See the references [4, 37] for subsequent discussion.”
The fractional Calderón problem: Low regularity and stability (2017, background): “ kqwk H s( ) k( ) swk H s(Rn) + kqwk H s( ) kwk H s + kqwk ( ); where we have used the compact support assumption of win the last line. Duality combined with a fractional Poincare inequality (c.f. [RO16] and the references therein) yields kqwk H s( ) = sup k’k Hs =1 (qw;’) L2(R n) sup k’k Hs =1 kqk”
Fractional thoughts (2017, methodology): “For these important aspects of the theory we refer the reader to the paper [Ka11’] and the nice introductory presentations in [BuV16], [RO15], but also to the original paper by Riesz [R38] and the book [La72], as well as the works [Bo97], [Bo99], [BB99], [BB99’], [Si07] and [FKV15].”
Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview (2018, background): “We refer interested readers to the excellent surveys for other nonlocal problems and applications, namely, on problem involving fractional (spectral and integral) Laplacian [21, 22], on inverse problems with anomalous diffusion [18, 23], on regularity theory of nonlocal elliptic equations in bounded”
Ten equivalent definitions of the fractional laplace operator (2015, result): “We also refer to a survey article [40] and the references”
Nonlocal Diffusion and Applications (2015, background): “Also, a complete discussion of the nonlocal equations in bounded domains is not available here (for this, we refer to the recent survey [78]).”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, background): “and nonlocal diusion, and also in the formulation of fractional quantum mechanics where the fractional Schrodinger equation arises naturally as a generalization of the ordinary Schrodinger equation [3, 6, 17, 27, 48, 49, 53, 66]. Since the fractional Laplacian is a non-local operator, it is more nat”
Global uniqueness for the fractional semilinear Schrödinger equation (2017, background): “The result of this section is motivated by [11] in which the author considers elliptic integro-differential operators.”
The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, background): “Fractional Laplacians appear in the study of anomalous and nonlocal diffusion, and these diffusion phenomena can be used in many areas such as continuum mechanics, graph theory and ecology just to mention a few [2, 6, 15, 34, 41].”
Quantitative approximation properties for the fractional heat equation (2017, background): “The maximum principle for example in general only holds on global scales, which does not rule out local oscillations [20, 34].”
A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian (2016, background): “alized Hopf lemma for fractional harmonic functions. Whereas, in [30], under the assumption Ω is a smooth bounded domain, it is proven a Hopf lemma for weak solutions of a Dirichlet problem. See also [15, 29]. For a Hopf lemma with mixed boundary condition, see [2]. Finally, Theorems 1.4 and 1.5 are”