The Liouville Property and a Conjecture of De Giorgi

By Martin T. Barlow, Richard F. Bass, Changfeng Gui. 2000

159 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3d960b4158dfec497a329feccf33178ae1f3a44e

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

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].”
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].”
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].”
De Giorgi type results for elliptic systems (2012, methodology): “Their proof used the following linear Liouville type theorem for elliptic equations in divergence form, which (only) holds in dimensions 1 and 2 ([5, 11]).”
Regularity of flat level sets in phase transitions (2009, background): “Barlow, Bass and Gui proved in [3] that monotonic solutions in Rn with Lipschitz level sets are planar in all dimensions.”
Asymptotic properties and classification of bistable fronts with Lipschitz level sets (2005, methodology): “Also note that this condition already appears in Barlow, Bass, Gui [2]: in this paper, in addition to a proof of the de Giorgi Conjecture in spatial dimension 2, there is a Liouville theorem for the solutions of the Allen-Cahn equation in all space dimensions; it is commonly conjectured - although n”
Hamiltonian identities for elliptic partial differential equations (2008, unknown): “Related results can also be found in [2,5,7,19,22], etc.”
Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones (2011, methodology): “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].”
Harnack Inequalities for Jump Processes (2002, background): “In [ BBG ] a Liouville property, which is closely related to a Harnack inequality, was proved.”