On De Giorgi's conjecture in dimensions 4 and 5

By Nassif Ghoussoub, Changfeng Gui. 2003

128 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/14873b616e0c77ebdd3d5e5a831128c3f406af28

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

On Phase-Separation Models: Asymptotics and Qualitative Properties (2013, background): “One may refer to Chang-Lin-Lin-Lin [17] for the pointwise convergence of (u Λ , v Λ ) away from the interface γ ≡ {x ∈ Ω : u ∞ (x) = v ∞ (x) = 0}; Wei-Weth [46] for the uniform equicontinuity of (u Λ , v Λ ); and Noris-Tavares-Terracini-Verzini [39] for the uniform Hölder continuity of (u Λ , v Λ ).”
Layer solutions in a half‐space for boundary reactions (2005, unknown): “This statement on two-dimensional symmetry is closely related to a conjecture of De Giorgi on one-dimensional symmetry for interior reactions, partially proven in [1, 4, 9, 22, 23] and completely settled by Savin [34].”
Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian (2015, unknown): “These results are analogues of a well studied phenomenon for the classical Allen–Cahn equation and usually referred to as the De Giorgi conjecture (see [32,45,3, 46,79,36]).”
De Giorgi type results for elliptic systems (2012, methodology): “Also, Ghoussoub and Gui in [12] used a slightly stronger version to show that the De Giorgi’s conjecture is true in dimensions four and five for a special class of solutions that satisfy an antisymmetry condition.”
Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones (2011, result): “Gibbons’ Conjecture has been proved in all dimensions with different methods by Caffarelli and Córdoba [12], Farina [22], Barlow, Bass and Gui [6], and Berestycki, Hamel, and Monneau [7].”
Existence and qualitative properties of multidimensional conical bistable fronts (2005, background): “Also, our problem has close connections to some questions around a conjecture of De Giorgi [1], [3], [14], [24].”