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no. 2
On divergent fractional Laplace equations
Serena Dipierro1; Ovidiu Savin2; Enrico Valdinoci1
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
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 2, pp. 255-265.

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.

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.

Published online:
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Serena Dipierro; Ovidiu Savin; Enrico Valdinoci. On divergent fractional Laplace equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 2, pp. 255-265. doi : 10.5802/afst.1673. https://afst.centre-mersenne.org/articles/10.5802/afst.1673/

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