A counterexample to the Liouville property of some nonlocal problems

Brasseur, Julien; Coville, Jérôme

doi:10.1016/j.anihpc.2019.12.003

Cited by 5 publications

(8 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Yet,

u

is not identically 1 in the whole set

{double-struckR}^{N} ∖ K

. The question of whether it is possible to construct nontrivial counterexamples will be addressed in a forthcoming paper , where it will be shown that smooth simply connected obstacles can be constructed so that the Liouville type property fails.…”

Section: Resultsmentioning

confidence: 99%

Liouville type results for a nonlocal obstacle problem

Brasseur

Coville

Hamel

et al. 2019

Proc. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

This paper is concerned with qualitative properties of solutions to nonlocal reaction–diffusion equations of the form 0true∫RN∖KJ(x−y)0.16em(ufalse(yfalse)−ufalse(xfalse))0.16emnormaldy+f(ufalse(xfalse))=0,1emx∈RN∖K, set in a perforated open set double-struckRN∖K, where K⊂double-struckRN is a bounded compact ‘obstacle’ and f is a bistable nonlinearity. When K is convex, we prove some Liouville‐type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on K.

show abstract

“…Yet,

u

is not identically 1 in the whole set

{double-struckR}^{N} ∖ K

Section: Resultsmentioning

confidence: 99%

Liouville type results for a nonlocal obstacle problem

Brasseur

Coville

Hamel

et al. 2019

Proc. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We point out, however, that in such domains (see Figure 3) there might exist stable patterns which do not satisfy (22) but satisfy lim |x|→+∞ u(x) = 0. See also [10] for similar conclusions if the domain is a cylinder with varying cross section, and [14,15] for the case of non-local diffusion.…”

Section: Link With Minimal Surfaces Consider Equation (1) With the Rescaled Allenmentioning

confidence: 79%

Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions

Nordmann

2021

Ann. Inst. H. Poincaré Anal. Non Linéaire

View full text Add to dashboard Cite

We call pattern any non-constant solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [20] and Matano [50] states that stable patterns do not exist in convex domains. In this article, we show that the assumptions of convexity of the domain and stability of the pattern in this theorem can be relaxed in several directions. In particular, we propose a general criterion for the non-existence of patterns, dealing with possibly non-convex domains and unstable patterns. Our results unfold the interplay between the geometry of the domain, the stability of patterns, and the C 1 norm of the nonlinearity.In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincaré inequality), stating that the L 2 -norm of the gradient of a solution is controlled by the L 2 -norm of f (u), with a constant that only depends on the domain. This inequality holds for non-homogeneous equations. We also give several flatness estimates.Our approach relies on the introduction of what we call the Robincurvature Laplacian. This operator is intrinsic to the domain and contains much information on how the geometry of the domain affects the shape of the solutions.Finally, we extend our results to unbounded domains. It allows us to improve the results of our previous paper [54] and to extend some results on De Giorgi's conjecture to a larger class of domains.

show abstract

“…This is a consequence of the fact that (P ∞ ) satisfies a Liouville type property, see Figure 3. (e) t = 600 (f) t = 750 (g) t = 900 (h) t = 1000 However, the authors have shown in [18] that there exist obstacles K as well as a datum (J, f ) for which this property is violated, i.e. such that (P ∞ ) admits a non-trivial solution u ∞ ∈ C(Ω) with 0 < u ∞ < 1 in Ω.…”

Section: Large Time Behaviourmentioning

confidence: 99%

“…Remark 2.8. The distance δ ∈ Q(Ω) in Corollary 2.7 may be chosen to be either the Euclidean or the geodesic distance, see [18]. See Figure 4 for an example illustrating the conclusion of Corollary 2.7.…”

Section: Large Time Behaviourmentioning

confidence: 99%

See 1 more Smart Citation

Propagation Phenomena with Nonlocal Diffusion in Presence of an Obstacle

Brasseur

Coville

2021

J Dyn Diff Equat

Self Cite

View full text Add to dashboard Cite

We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form R N \ K, where K ⊂ R N is a compact "obstacle". The model we study is motivated by applications in biology and takes into account long range dispersal events that may be anisotropic depending on how a given population perceives the environment. To formulate this in a meaningful manner, we introduce a new theoretical framework which is of both mathematical and biological interest. The main goal of this paper is to construct an entire solution that behaves like a planar travelling wave as t → −∞ and to study how this solution propagates depending on the shape of the obstacle. We show that whether the solution recovers the shape of a planar front in the large time limit is equivalent to whether a certain Liouville type property is satisfied. We study the validity of this Liouville type property and we extend some previous results of Hamel, Valdinoci and the authors. Lastly, we show that the entire solution is a generalised transition front. JULIEN BRASSEUR AND JÉRÔME COVILLE 7. On the impact of the geometry 40 7.1. A Liouville type result 40 7.2. The stationary solution encodes the geometry 41 8. Large time behaviour of super-level sets 44 8.1. An upper bound: the super-level sets move at speed at most c 44 8.2. A lower bound: the super-level sets move asymptotically at speed c 46 8.3. Conclusion: the super-level sets move exactly at speed c 51 9. The entire solution u(t, x) is a generalised transition wave 55 Acknowledgments 57 Appendix A. The J-covering property 58 References 61

show abstract

A counterexample to the Liouville property of some nonlocal problems

Cited by 5 publications

References 19 publications

Liouville type results for a nonlocal obstacle problem

Liouville type results for a nonlocal obstacle problem

Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions

Propagation Phenomena with Nonlocal Diffusion in Presence of an Obstacle

Contact Info

Product

Resources

About