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.
@article{AFST_2021_6_30_2_255_0,
author = {Dipierro, Serena and Savin, Ovidiu and Valdinoci, Enrico},
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 = {http://archive.numdam.org/articles/10.5802/afst.1673/}
}
TY - JOUR AU - Dipierro, Serena AU - Savin, Ovidiu AU - Valdinoci, Enrico 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 - http://archive.numdam.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 Dipierro, Serena %A Savin, Ovidiu %A Valdinoci, Enrico %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 http://archive.numdam.org/articles/10.5802/afst.1673/ %R 10.5802/afst.1673 %G en %F AFST_2021_6_30_2_255_0
Dipierro, Serena; Savin, Ovidiu; Valdinoci, Enrico. 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. http://archive.numdam.org/articles/10.5802/afst.1673/
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