Solutions of the Allen-Cahn equation which are invariant under screw-motion

Pino, Manuel del; Musso, Monica; Pacard, Frank

doi:10.1007/s00229-011-0492-3

Cited by 14 publications

(7 citation statements)

References 10 publications

(12 reference statements)

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“…, where R 2 + ≡ {y = (y 1 , y 2 ) : y 1 > 0, y 2 > 0}, for small ε > 0, we obtain a positive solution in R 2 + of the equation in (22) − which is zero on the coordinate axis and approaches zero as |y| → ∞ (we can see that the minimizer is nontrivial, if ε is small, by adapting Example 5.11 in [21] or Lemma 2.1 in [82]). A solution u 2 defined in the entire space is then obtained using odd reflections through the lines y 1 = 0 and y 2 = 0.…”

Section: Open Problems and Future Directionsmentioning

confidence: 76%

“…As before, minimizing the functional Very recently, del Pino, Musso, and Pacard [82] studied entire solutions of the Allen-Cahn equation (49) which are defined in 3-dimensional Euclidean space and which are invariant under screw-motion. In particular, their nodal set is a helicoïd of R 3 .…”

Section: Open Problems and Future Directionsmentioning

confidence: 99%

“…What is the asymptotic behavior of these solutions as ε → 0? Can some results of [82] be generalized in our context?…”

Section: Open Problems and Future Directionsmentioning

confidence: 99%

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The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

Karali

Sourdis

2015

Arch Rational Mech Anal

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We study the ground state which minimizes a Gross-Pitaevskii energy with general non-radial trapping potential, under the unit mass constraint, in the ThomasFermi limit where a small parameter ε tends to 0. This ground state plays an important role in the mathematical treatment of recent experiments on the phenomenon of Bose-Einstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrödinger equations. Many of these applications require delicate estimates for the behavior of the ground state near the boundary of the condensate, as ε → 0, in the vicinity of which the ground state has irregular behavior in the form of a steep corner layer. In particular, the role of this layer is important in order to detect the presence of vortices in the small density region of the condensate, to understand the superfluid flow around an obstacle, and it also has a leading order contribution in the energy. In contrast to previous approaches, we utilize a perturbation argument to go beyond the clas-

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Section: Open Problems and Future Directionsmentioning

confidence: 76%

Section: Open Problems and Future Directionsmentioning

confidence: 99%

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The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

Karali

Sourdis

2015

Arch Rational Mech Anal

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“…, X n−1 of the basis B are all horizontal, whereas X n = T = ∇ξ/ ∇ξ . As above, we have (7) ℓ ij = µa ij ∀i, j = 1, . .…”

Section: Basic Lemmasmentioning

confidence: 98%

“…Their methods, however, are different from ours. A distinguished property of these a-twisted cones is that, for sufficiently large a > 0, they constitute nodal sets of the solutions of the Allen-Cahn differential equation (see [7]).…”

Section: Construction and Local Characterization Of Vertical Helicoidsmentioning

confidence: 99%

Helicoids and Catenoids in $M\times\mathbb{R}$

Lima¹,

Roitman²

2019

Preprint

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Given an arbitrary C ∞ Riemannian manifold M n , we consider the problem of introducing and constructing minimal hypersurfaces in M ×R which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space R 3 = R 2 × R. Such hypersurfaces are defined by imposing conditions on their height functions and horizontal sections, and then called vertical helicoids and vertical catenoids. We establish that vertical helicoids in M ×R have the same fundamental uniqueness properties of the helicoids in R 3 . We provide several examples of vertical helicoids in the case where M is one of the simply connected space forms. Vertical helicoids which are entire graphs of functions on Nil 3 and Sol 3 are also presented. We give a local characterization of hypersurfaces of M × R which have the gradient of their height functions as a principal direction. As a consequence, we prove that vertical catenoids exist in M × R if and only if M admits families of isoparametric hypersurfaces. If so, they can be constructed through the solutions of a certain first order linear differential equation. Finally, we give a complete classification of the hypersurfaces of M × R whose angle function is constant.

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Periodic Solutions to a Cahn–Hilliard–Willmore Equation in the Plane

Malchiodi

Mandel²,

Rizzi

2017

Arch Rational Mech Anal

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In this paper we construct entire solutions to the Cahn-Hilliard equation −∆(−∆u + W (u)) + W (u)(−∆u + W (u)) = 0 in the Euclidean plane, where W (u) is the standard double-well potential 1 4(1 − u 2 ) 2 . Such solutions have a non-trivial profile that shadows a Willmore planar curve, and converge uniformly to ±1 as x2 → ±∞. These solutions give a counterexample to the counterpart of Gibbons' conjecture for the fourth-order counterpart of the Allen-Cahn equation. We also study the x2-derivative of these solutions using the special structure of Willmore's equation.

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Solutions of the Allen-Cahn equation which are invariant under screw-motion

Abstract: We study entire solutions of the Allen-Cahn equation which are defined in the 3-dimensional Euclidean space and which are invariant under screw-motion. In particular, we discuss the existence and non existence of nontrivial solutions whose nodal set is a helicoid of R 3 .

Cited by 14 publications

References 10 publications

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

Helicoids and Catenoids in $M\times\mathbb{R}$

Periodic Solutions to a Cahn–Hilliard–Willmore Equation in the Plane

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