WEAK AND VISCOSITY SOLUTIONS OF THE FRACTIONAL LAPLACE EQUATION

By Raffaella Servadei, Enrico Valdinoci. 2014

218 citations recorded by the source corpus.

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation ( ( ) s u = f in u = g in R n n are also continuous solutions (up to the boundary) of this problem in the viscosity sense. Here s2 (0; 1) is a xed parameter, is a bounded, open subset of R n (n > 1) with C 2 -boundary, and ( ) s is the fractional Laplacian operator, that may be dened as

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6933cb913e734f06b1cbdda94a2cec595b0971be

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Local behavior of fractional $p$-minimizers (2015, unknown): “, the local behavior of the fractional minimizers, it is worth mentioning the very relevant contributions for the case p = 2 by Kassmann ([12, 13]); see also [31, 30].”
Fractional elliptic problems with critical growth in the whole of $\R^n$ (2015, background): “2 and Proposition 5 in [46], being u a solution to (2.”
Mixed local and nonlocal elliptic operators: regularity and maximum principles (2020, background): “the proof of Proposition 9 in [39] and of Theorem 2.”
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum (2014, background): “This means that ψ̄ is a weak solution to (7-47), and so a strong solution, thanks to [Servadei and Valdinoci 2014].”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “In the quadratic case p = 2, spaces of this kind have been for instance considered in [57, 58].”
The effect of the Hardy potential in some Calder\'on-Zygmund properties for the fractional Laplacian (2015, unknown): “−∆)sφ= ϕ,ϕ∈ L∞(Ω) ∩ Cα(Ω),0 <α<1,φ= 0 in RN\Ω}. Notice that every φ∈ T belongs in particular to L∞(Ω) (see [28]) and moreover it is a strong solution to the equation (−∆)sφ= ϕ. See for instance [30] and [31]. Deﬁnition 2.7. Assume f∈ L1(Ω). We say that u∈ L1(Ω) is a weak supersolution (subsolu”
A logistic equation with nonlocal interactions (2016, background): “s dy+ ν1 s1 (1−s1) 2 Ω 1 u(x)−u(y) |x− y|n+2s1 dy + ν2 s1 (1−s2) 2 Z Rn\Ω2 u(x)− u(y) |x− y|n+2s2 dy= (σ− µu)u in Ω2, (16) inthe weak sense (andalsopointwise, by Theorem 5.5(3)in[Kri15]and Theorem 1in[SV14]). The biological interpretation of equation (16) is that the population has local behavior in”
On the spectrum of two different fractional operators (2014, unknown): “…delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy (enrico@mat.uniroma3.it) (MS received 20 October 2012; accepted 22 May 2013) In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type.”