One-dimensional symmetry of bounded entire solutions of some elliptic equations

By Henri Berestycki, François Hamel, Régis Monneau. 2000

164 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/8e26506c5e993f7c5c22d7021e982311373c4532

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones (2011, unknown): “A P] 1 6 Fe b 20”
MULTIPLE-END SOLUTIONS TO THE ALLEN-CAHN EQUATION IN R2 (2010, background): “Gibbons’ conjecture for all dimensions is proved independently in [3], [4] and [10].”
Nonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions (2011, methodology): “Indeed, we will see that the proof follows that of [5] and thus only requires f to be Lipschitz in [−1, 1] and nonincreasing in a neighborhood of −1 and of 1.”
Asymptotic properties and classification of bistable fronts with Lipschitz level sets (2005, unknown): “Then from Theorem 2 in [3], we get that v(X̃, Ỹ ) only depends on Ỹ , and then one concludes that ∀(x, y) ∈ R, v(x, y) = U(−x cos α + y sin α + t−).”
On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property (2001, background): “With the additional assumption that the2 limits in (1.3) are uniform in x0 2 Rn 1, the question of De Giorgi is known as \Gibbonsconjecture", and it is by now completely settled.”
Hamiltonian identities for elliptic partial differential equations (2008, unknown): “Related results can also be found in [2,5,7,19,22], etc.”
On De Giorgi's conjecture in dimensions 4 and 5 (2003, methodology): “Recent developments on the conjecture can be found in [15], [4], [7], [14], [2], [1].”
1D symmetry for solutions of semilinear and quasilinear elliptic equations (2011, methodology): “... and (1.12), also satises (1.2) (see, for instance, [GG98, Far99, Far01]) and so (1.6) (see, e.g., [AC00, AAC01] and Lemma 9.1 in [VSS06]): therefore, Theorem 1.8 contains, as a particular case, the fact that, for p = 2, solutions of (1.1) with uniform limits 1 are 1D in any dimension N. This sta”
Existence and qualitative properties of multidimensional conical bistable fronts (2005, background): “Let us now notice that, if α = π/2, then u only depends on y (this result follows for instance from Theorem 2 in [4]).”
On Phase-Separation Models: Asymptotics and Qualitative Properties (2013, methodology): “Gibbons’ Conjecture has been proved in all dimensions with different methods by Caffarelli and Córdoba [12], Farina [30], Barlow, Bass and Gui [6], and Berestycki, Hamel, and Monneau [7].”
Layer solutions in a half‐space for boundary reactions (2005, background): “This is the exact analogue of a result for interior reactions due to Berestycki, Hamel, and Monneau [11].”