Equivalence of weak and viscosity solutions in fractional non-homogeneous problems
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Mathematische Annalen
Equivalence of weak and viscosity solutions in fractional non-homogeneous problems Begoña Barrios1 · Maria Medina2 Received: 29 June 2020 / Revised: 31 October 2020 / Accepted: 9 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We establish the equivalence between the notions of weak and viscosity solutions for non-homogeneous equations whose main operator is the fractional p-Laplacian and p the lower order term depends on x, u and Ds u, being the last one a type of fractional derivative.
1 Introduction In this work we study the equivalence between the notions of weak and viscosity solutions to the problem p (1.1) (−)sp u = f (x, u, Ds u), posed in a bounded domain ⊂ Rn , with s ∈ (0, 1) and 1 < p < +∞. By (−)sp we mean the fractional p-Laplacian, defined for a regular enough function u as
Communicated by Y. Giga. B. Barrios was partially supported by the MEC project PGC2018-096422-B-I00 (Spain) and the Ramón y Cajal fellowship RYC2018-026098-I (Spain). M. Medina was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement N 754446 and UGR Research and Knowledge Transfer Fund - Athenea3i. M. Medina wants to acknowledge Universidad de la Laguna’s hospitality, where this work was initiated during a visit in July 2019.
B
Maria Medina [email protected] Begoña Barrios [email protected]
1
Departamento de Análisis Matemático, Universidad de La Laguna, C/Astrofísico Francisco Sánchez s/n, 38271 La Laguna, Spain
2
Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain
123
B. Barrios, M. Medina
(−)sp u(x) := an,s, p p.v.
Rn
|u(x) − u(y)| p−2 (u(x) − u(y)) dy, |x − y|n+sp
(1.2)
where n > ps and an,s, p is a normalization constant that will be omitted from now on. The integral in (1.2) has to be understood in the principal value sense, that is, as the limit when ε → 0 of the same integral computed in Rn \Bε (x). This operator can be seen as a nonlocal counterpart of the p-Laplacian operator, − p u := −div(|∇u| p−2 ∇u), p
and Ds u stands for a sort of fractional gradient. Consider indeed the case p = 2, for which the nonlocal, and linear, operator (−)sp corresponds to the standard fractional Laplacian given by (−)s u(x) := p.v.
Rn
u(x) − u(y) dy. |x − y|n+2s
As stated in [4,10], the corresponding bilinear form Bs (u, v)(x) :=
Rn
(u(x) − u(y))(v(x) − v(y)) dy, |x − y|n+2s
can be seen as a fractional derivative of order s playing the role of ∇u · ∇v in the local case. Furthermore, Bs (u, u) can be thought as the analogue of |∇u|2 , the natural term appearing in the energy of problems driven by the classical Laplacian. Extending this idea to the whole range 1 < p < +∞ we define the form, linear only in the second variable, associated to (−)sp u as
p
Bs (u, v)(x) :=
Rn
|u(x) − u(y)| p−2 (u(x) − u(y))(v(x) − v(y)) dy, |x − y|n+sp
and the gradient p Ds u(x)
:=
p Bs (u, u)(x)
=
Rn
|u(x) − u(y
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