Geodesic
A counterexample to the Liouville property of some nonlocal problems
Brasseur, Julien1 ; Coville, Jérôme2 
1 EHESS, CAMS, 54 Boulevard Raspail, F-75006 Paris, France
2 BioSP, INRAE, 84914, Avignon, France
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 549-579

Voir la notice de l'article provenant de la source Numdam

In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form

∫RN∖KJ(x−y)(u(y)−u(x))dy+f(u(x))=0,x∈RN∖K,
where K⊂RN is a bounded compact set, called an “obstacle”, and f is a bistable nonlinearity. When K is convex, it is known that solutions ranging in [0,1] and satisfying u(x)→1 as |x|→∞ must be identically 1 in the whole space. We construct a nontrivial family of simply connected (non-starshaped) obstacles as well as data f and J for which this property fails.

DOI : 10.1016/j.anihpc.2019.12.003
Keywords: Nonlocal diffusion, Liouville property, Bistable nonlinearity, Counterexample, Positive stationary solution, Calculus of variations

Brasseur, Julien 1 ; Coville, Jérôme 2

1 EHESS, CAMS, 54 Boulevard Raspail, F-75006 Paris, France
2 BioSP, INRAE, 84914, Avignon, France 
@article{AIHPC_2020__37_3_549_0,
     author = {Brasseur, Julien and Coville, J\'er\^ome},
     title = {A counterexample to the {Liouville} property of some nonlocal problems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {549--579},
     publisher = {Elsevier},
     volume = {37},
     number = {3},
     year = {2020},
     doi = {10.1016/j.anihpc.2019.12.003},
     mrnumber = {4093620},
     zbl = {1439.35107},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.12.003/}
}
TY  - JOUR
AU  - Brasseur, Julien
AU  - Coville, Jérôme
TI  - A counterexample to the Liouville property of some nonlocal problems
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2020
SP  - 549
EP  - 579
VL  - 37
IS  - 3
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.12.003/
DO  - 10.1016/j.anihpc.2019.12.003
LA  - en
ID  - AIHPC_2020__37_3_549_0
ER  - 
%0 Journal Article
%A Brasseur, Julien
%A Coville, Jérôme
%T A counterexample to the Liouville property of some nonlocal problems
%J Annales de l'I.H.P. Analyse non linéaire
%D 2020
%P 549-579
%V 37
%N 3
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.12.003/
%R 10.1016/j.anihpc.2019.12.003
%G en
%F AIHPC_2020__37_3_549_0
Brasseur, Julien; Coville, Jérôme. A counterexample to the Liouville property of some nonlocal problems. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 549-579. doi: 10.1016/j.anihpc.2019.12.003

Cité par Sources :