On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds

Mahmoudi, Fethi; Sánchez, Felipe Subiabre; Yao, Wei

doi:10.1016/j.jde.2014.09.010

Cited by 13 publications

(5 citation statements)

References 29 publications

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“…The condition n−k ≥ 7 appears also in many previous works like [16], it is a technical condition that seems essential for our method to work but we believe the phenomenon described should also be true for lower co-dimensions. We also point out that the resonance phenomenon has already been found in the analysis of higher dimensional concentration in other elliptic boundary value problems, in particular for Neumann singular perturbation problem in [22,25,26,27] and nonlinear Schrödinger equations on compact Riemannian manifolds without boundary or in R N , see [13], [24].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 70%

“…The first main ingredient in proving our main theorem is the construction of a very accurate approximate solution in powers of

ε

and

ρ = ε^{\frac{N - 1}{N - 2}}

, in a neighborhood of the scaled sub‐manifold

K_{ρ} = ρ^{- 1} K

. It is worth mentioning that concentration at higher dimensional sets for some related problem with Neumann boundary conditions or on manifolds has been extensively studied in the last decade, see [, , , , , ] and some references therein. In most of the above‐mentioned problems the profile has an exponential decay which is crucial in the construction of very accurate approximate solutions via an iterative scheme of Picard's type.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents

Deng

Mahmoudi

Musso

2018

Proc. London Math. Soc.

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Let normalΩ be an open bounded domain in Rn with smooth boundary ∂Ω. We consider the equation Δu+un−k+2n−k−2−ε=0inΩ, under zero Dirichlet boundary condition, where ε is a small positive parameter. We assume that there is a k‐dimensional closed, embedded minimal sub‐manifold K of ∂Ω, which is non‐degenerate, and along which a certain weighted average of sectional curvatures of ∂Ω is negative. Under these assumptions, we prove existence of a sequence ε=εj and a solution uε which concentrate along K, as ε→0+, in the sense that false|∇uε|2⇀Sn−kn−k2δKasε→0,where δK stands for the Dirac measure supported on K and Sn−k is an explicit positive constant.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 70%

“…The first main ingredient in proving our main theorem is the construction of a very accurate approximate solution in powers of

ε

and

ρ = ε^{\frac{N - 1}{N - 2}}

, in a neighborhood of the scaled sub‐manifold

K_{ρ} = ρ^{- 1} K

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents

Deng

Mahmoudi

Musso

2018

Proc. London Math. Soc.

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“…It is possible to find similar computations in [21], where Mahmoudi, Sànchez and Yao treat the more general case of a k dimensional submanifold in an N dimensional manifold. Lemma 6.…”

Section: Lemma 5 For a Functionmentioning

confidence: 88%

Clifford Tori and the singularly perturbed Cahn–Hilliard equation

Rizzi

2017

Journal of Differential Equations

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In this paper we construct entire solutions u ε to the Cahn-Hilliard equation −ε 2 ∆(−ε 2 ∆u + W ′ (u)) + W ′′ (u)(−ε 2 ∆u + W ′ (u)) = 0, under the volume constraint R 3 (1 − u ε )dx = 4 √ 2π 2 , whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/ √ 2 embedded in R 3 , as ε → 0. What is crucial is that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov-Schmidt reduction and on careful geometric expansions of the laplacian.Keywords: Lyapunov-Schmidt reduction; Cahn-Hilliard equation; Willmore surface; Clifford Torus. IntroductionThe Allen-Cahn equationarises in several physical contexts, such as the study of the stable configurations of two different fluids confined in a bounded container Ω. If u(x) is the density of one of the two fluids at a point x ∈ Ω and the energy per unit volume is given by a function W of u, it looks reasonable to obtain stable configurations by minimizing the energy functionalamong all distributions fulfilling the volume constraintIf, for instance, W (u) = (1 − u 2 ) 2 , and m ∈ (−1, 1), any piecewise constant function taking only the values ±1 and satisfying (2) is a minimizer, irrespectively of the shape of the interface. Therefore this model is unsatisfactory, since it is very far from the reasonable physical assumption that the interfaces are area minimizers, so one replaces the energy byWe can see that there is a competition between the potential energy, that forces u to be close to ±1, and the gradient term that penalizes the phase transition. By minimizing this functional, we are looking for the physical interfaces in which the phase transition can occur. The minimizers u ε of E ε are solutions to the Euler Lagrange equation, that is (1). In order to see if the interfaces are actually minimal surfaces, it is interesting to study the asymptotic behaviour of the level sets {u ε = c} as the parameter ε → 0. It is useful to exploit the variational structure of the problem. It was shown by Modica and 2 Mortola that the energy E ε , seen as a functional on L 1 (Ω) and extended to be +∞ when the integrand is not an L 1 function, Γ−converges to the functional E(u) = cP er Ω ({u = 1}) if u = ±1 a.e. in Ω +∞ otherwise in L 1 (Ω) in the strong topology of L 1 (Ω) (see [23]), where c > 0 is a suitable constant. Moreover, Modica showed that, if u ε are minimizers of F ε under the volume constraintfor some m ∈ (−1, 1), then there exists a sequence ε k → 0 such that u ε k converges to some function u in L 1 (Ω) (see proposition 3 of [22]). Furthermore, Theorem 1 of [22] asserts that u = ±1 a. e. in Ω, and the set E = {x ∈ Ω : u(x) = 1} is actually a perimeter minimizer between all the subsets F ⊂ Ω satisfying the volume constraintFurther results about the relation between the minimizers of E ε and the minimizers of the perimeter can be found in [22] and in [7], where Choksi and Sternberg also described the relation between phase transition theory and the ...

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“…We also mention [7], [10] for similar results obtained with different techniques and [8], [9] for problems with reduced symmetries. For general potentials (without symmetry restrictions), see [19], [38] and [58], especially for what concerns a conjecture in [3]. Concerning concentration at the boundary, it occurs for the Neumann problem provided M (1) > 0 or M (a) < 0: for the Dirichlet problem, opposite inequalities are needed.…”

Section: Theorem 33 ([4])mentioning

confidence: 99%

Perturbative techniques for the construction of spike-layers

Malchiodi¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper we survey some results concerning the construction of spike-layers, namely solutions to singularly perturbed equations that exhibit a concentration behaviour. Their study is motivated by the analysis of pattern formation in biological systems such as the Keller-Segel or the Gierer-Meinhardt's. We describe some general perturbative variational strategy useful to study concentration at points, and also at spheres in radially symmetric situations.

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On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds

Cited by 13 publications

References 29 publications

Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents

Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents

Clifford Tori and the singularly perturbed Cahn–Hilliard equation

Perturbative techniques for the construction of spike-layers

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