logo AFST
On divergent fractional Laplace equations
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 2, pp. 255-265.

Nous considérons le Laplacien fractionnaire divergent introduit dans [5] et démontrons trois types de résultats.

Premièrement, nous montrons que toute fonction donnée peut être approchée localement par une solution d’une équation de Laplace fractionnaire divergente, dont les valeurs sont de plus prescrites au voisinage de l’infini.

Deuxièmement, nous démontrons l’existence de solutions au problème de Dirichlet pour le Laplacien fractionnaire divergent, et caractérisons leur multiplicité.

Enfin, nous obtenons des résultats d’approximation dans le cadre d’équations non linéaires, résultats qui sont nouveaux même lorsque le Laplacien fractionnaire peut être défini au sens usuel.

We consider the divergent fractional Laplace operator presented in [5] and we prove three types of results.

Firstly, we show that any given function can be locally shadowed by a solution of a divergent fractional Laplace equation which is also prescribed in a neighborhood of infinity.

Secondly, we take into account the Dirichlet problem for the divergent fractional Laplace equation, proving the existence of a solution and characterizing its multiplicity.

Finally, we take into account the case of nonlinear equations, obtaining a new approximation results.

These results maintain their interest also in the case of functions for which the fractional Laplacian can be defined in the usual sense.

Publié le :
DOI : 10.5802/afst.1673
Serena Dipierro 1 ; Ovidiu Savin 2 ; Enrico Valdinoci 1

1 Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA6009 Crawley, Australia
2 Department of Mathematics, Columbia University, 2990 Broadway, New York NY 10027, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AFST_2021_6_30_2_255_0,
     author = {Serena Dipierro and Ovidiu Savin and Enrico Valdinoci},
     title = {On divergent fractional {Laplace} equations},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {255--265},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 30},
     number = {2},
     year = {2021},
     doi = {10.5802/afst.1673},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1673/}
}
TY  - JOUR
AU  - Serena Dipierro
AU  - Ovidiu Savin
AU  - Enrico Valdinoci
TI  - On divergent fractional Laplace equations
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2021
SP  - 255
EP  - 265
VL  - 30
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1673/
DO  - 10.5802/afst.1673
LA  - en
ID  - AFST_2021_6_30_2_255_0
ER  - 
%0 Journal Article
%A Serena Dipierro
%A Ovidiu Savin
%A Enrico Valdinoci
%T On divergent fractional Laplace equations
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2021
%P 255-265
%V 30
%N 2
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1673/
%R 10.5802/afst.1673
%G en
%F AFST_2021_6_30_2_255_0
Serena Dipierro; Ovidiu Savin; Enrico Valdinoci. On divergent fractional Laplace equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 2, pp. 255-265. doi : 10.5802/afst.1673. https://afst.centre-mersenne.org/articles/10.5802/afst.1673/

[1] Claudia Bucur Local density of Caputo-stationary functions in the space of smooth functions, ESAIM, Control Optim. Calc. Var., Volume 23 (2017) no. 4, pp. 1361-1380 | DOI | MR | Zbl

[2] Alessandro Carbotti; Serena Dipierro; Enrico Valdinoci Local density of solutions to fractional equations, De Gruyter Studies in Mathematics, 74, Walter de Gruyter, 2019 | Zbl

[3] Alessandro Carbotti; Serena Dipierro; Enrico Valdinoci Local density of Caputo-stationary functions of any order, Complex Var. Elliptic Equ., Volume 65 (2020), pp. 1115-1138 | DOI | MR | Zbl

[4] Serena Dipierro; Ovidiu Savin; Enrico Valdinoci All functions are locally s-harmonic up to a small error, J. Eur. Math. Soc., Volume 19 (2017) no. 4, pp. 957-966 | DOI | MR | Zbl

[5] Serena Dipierro; Ovidiu Savin; Enrico Valdinoci Definition of fractional Laplacian for functions with polynomial growth, Rev. Mat. Iberoam., Volume 35 (2019) no. 4, pp. 1079-1122 | DOI | MR | Zbl

[6] Serena Dipierro; Ovidiu Savin; Enrico Valdinoci Local approximation of arbitrary functions by solutions of nonlocal equations, J. Geom. Anal., Volume 29 (2019) no. 2, pp. 1428-1455 | DOI | MR | Zbl

[7] Serena Dipierro; Enrico Valdinoci Elliptic partial differential equations from an elementary viewpoint (2021) (https://arxiv.org/abs/2101.07941)

[8] N. V. Krylov On the paper “All functions are locally s-harmonic up to a small error” by Dipierro, Savin, and Valdinoci (2018) (https://arxiv.org/abs/1810.07648)

[9] Xavier Ros-Oton; Joaquim Serra The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., Volume 101 (2014) no. 3, pp. 275-302 | DOI | MR | Zbl

[10] Angkana Rüland; Mikko Salo Exponential instability in the fractional Calderón problem, Inverse Probl., Volume 34 (2018) no. 4, 045003, 21 pages | DOI | Zbl

[11] Enrico Valdinoci All functions are (locally) s-harmonic (up to a small error) – and applications, Partial differential equations and geometric measure theory (Lecture Notes in Mathematics), Volume 2211, Springer, 2018, pp. 197-214 | DOI | MR | Zbl

Cité par Sources :