Fractional diffusion with Neumann boundary conditions: The logistic equation

Montefusco, Eugenio; Pellacci, Benedetta; Verzini, Gianmaria

doi:10.3934/dcdsb.2013.18.2175

Cited by 50 publications

(55 citation statements)

References 30 publications

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“…We point out that for the equation (−∆ N ) 1/2 u = f parallel results can be found in [22], while for the case of the fractional Dirichlet Laplacian (−∆ D ) 1/2 u = f one can see [5]. Section 4 introduces the proper concept of weak solution for the semilinear problem (1.2) and shows the existence of a positive nonconstant smooth solution u ε for small ε.…”

Section: /2mentioning

confidence: 99%

“…Then the system (1.9) is equivalent to find a solution to (1.4), where ε = D 2 /a, g(t) = t p for t ≥ 0 and p = χ/D 1 and v Ω = v. Since the appearance of the papers by L. Caffarelli, L. Silvestre and collaborators [6,7,8,9,27,28,29] nonlocal PDEs with fractional Laplacians became a topic which is nowadays deserving a lot of attention, also because of the various applications in several fields. As far as the Neumann Laplacian is concerned, some problems were studied in [2,19,22]. A fractional Keller-Segel model was considered in [11], though the author there proposes to model the concentration of the chemical with the usual local diffusion while the bacteria satisfy a nonlocal diffusion in dimension one.…”

Section: /2mentioning

confidence: 99%

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Fractional semilinear Neumann problems arising from a fractional Keller–Segel model

Stinga

Volzone

2014

Calc. Var.

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Abstract. We consider the following fractional semilinear Neumann problem on a smooth boundedwhere ε > 0 and 1 < p < (n + 1)/(n − 1). This is the fractional version of the semilinear Neumann problem studied by Lin-Ni-Takagi in the late 80's. The problem arises by considering steady states of the Keller-Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variational techniques, we prove existence of nonconstant smooth solutions for small ε, which are obtained by minimizing a suitable energy functional. In the case of large ε we obtain nonexistence of nonconstant solutions. It is also shown that as ε → 0 the solutions uε tend to zero in measure on Ω, while they form spikes in Ω. The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest.

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Section: /2mentioning

confidence: 99%

Section: /2mentioning

confidence: 99%

Fractional semilinear Neumann problems arising from a fractional Keller–Segel model

Stinga

Volzone

2014

Calc. Var.

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“…Finally, let us mention that the optimization of λ(m) has been investigated also in different, although related, settings: with pointwise constraints for positive weights with Dirichlet boundary conditions, see [24,Chapter 9] and references therein; in the framework of composite membranes [2,14,13]; in the case of the p-Laplace operator in [19]; when analyzing best dispersal strategies in spatially heterogeneous environments, where also non-local diffusion is allowed [38,40].…”

Section: The Optimal Survival Threshold Problemmentioning

confidence: 99%

Asymptotic spherical shapes in some spectral optimization problems

Mazzoleni

Pellacci

Verzini

2020

Journal de Mathématiques Pures et Appliquées

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We study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box Ω ⊂ R N , which arises in the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the weight, one is led to consider a shape optimization problem, which is known to admit no spherical optimal shapes (despite some previously stated conjectures). We investigate whether spherical shapes can be recovered in some singular perturbation limit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limit the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. Using α-symmetrization techniques on cones, we prove that, for suitable choices of the box Ω, the optimal shapes for this second problem are indeed spherical. Moreover, for general Ω, we show that small volume spectral drops are asymptotically spherical, centered near points of ∂Ω having largest mean curvature. RésuméOn etudie l'optimisation de la valeur propre principale positive d'un problème avec un poids indéfini, associé au Laplacien avec conditions au bord de Neumann dans une boîte Ω ⊂ R N , qui apparaît dans l'étude naturellement comme valeur limite de survie en dynamique des populations. En essayant de minimiser une telle valeur propre par rapport au poids, on est amenéà considérer un problème d'optimisation de forme, qui n'admet pas de formes sphériques comme solutions (chose qui contredit certaines conjecturesénoncées précédemment). Nousétudions si des formes sphériques peuventêtre recouvrées comme solutions de certaines perturbations singulières du problème. Notamment, on démontre que, si la partie négative du poids diverge, le problème d'optimisation de forme se rapprocheà la limiteà un problème de "goutte spectrale", cetteà dire un problème de minimisation de la première valeur propre du Laplacien avec conditions mixtes Dirichletet-Neumann au bord. En utilisant techniques de α-symétrisation sur les cônes, on démontre que, pour des choix opportunes de la boîte Ω, les formes optimales pour ce second problème sont bien sphériques. De plus, pour Ω générique, on démontre que les gouttes spectrales s'approchent asymptotiquement -lorsque leur volume devient infinitésimal-à des gouttes sphériques, centrées près des points de ∂Ω ayant courbure moyenne maximale.

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“…In recent years, there has been great interest in using nonlocal integro-differential equations (IDEs) as a means to describe physical systems, due to their natural ability to describe physical phenomena at small scales and their reduced regularity requirements which lead to greater flexibility [54,8,64,29,30,63,27,25,49,42,41,13,22,18,37,24,3,17,52,14,31]. In particular, nonlocal problems with Neumann-type boundary constraints have received particular attention [15,16,20,34,47,7,21,23,26,55,51,1,46,63] due to their prevalence in describing problems related to: interfaces [2], free boundaries, and multiscale/multiphysics coupling problems [38,53,62,5,6]. Unlike classical PDE models, in the nonlocal IDEs the boundary conditions must be defined on a region with non-zero volume outside the surface [16,21,55], in contrast to more traditional engineering scenarios where boundary conditio...…”

Section: Introductionmentioning

confidence: 99%

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

You

Task

et al. 2020

ESAIM: M2AN

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In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, we present a new generalization of classical local Neumann conditions that recovers the local case as O(δ 2 ) in the L ∞ (Ω) norm. This convergence rate is optimal considering the O(δ 2 ) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ 2 ) convergence.

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Fractional diffusion with Neumann boundary conditions: The logistic equation

Cited by 50 publications

References 30 publications

Fractional semilinear Neumann problems arising from a fractional Keller–Segel model

Fractional semilinear Neumann problems arising from a fractional Keller–Segel model

Asymptotic spherical shapes in some spectral optimization problems

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

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