Entire solutions of semilinear elliptic equations in R^3 and a conjecture of De Giorgi

By Luigi Ambrosio. 2000

352 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e87bf78a230e02e9f5967dfd598a28057109b4d5

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Saddle-shaped solutions of bistable diffusion equations in all of ℝ2m (2009, methodology): “The proof of the following lemma, which follows from integrating the ODE ¨ u −G′(u) = 0, can be found in [ 5 ] —see also a sketch of the proof below, after”
Towards a counter-example to a conjecture of De Giorgi in high dimensions (2004, background): “The conjecture of De Giorgi C5 was proved by Ghoussoub and Gui [37] in dimension n = 2 and by Ambrosio and Cabr´ e [ 4 ] in dimension n = 3. This conjecture is still open in dimensions n ≥ 4.”
De Giorgi type results for elliptic systems (2012, methodology): “We introduce and exploit the concept of an orientable system, which seems to be key for dealing with systems of three or more equations.”
Layer solutions in a half‐space for boundary reactions (2005, methodology): “It is an analogue of a Liouville theorem from [4], used in that paper to prove a conjecture of De Giorgi for reactions in the interior when n = 3.”
Local and global minimizers for a variational energy involving a fractional norm (2011, neutral): “1 in [27] and also [2] and [5] for a different variational approach for the classical local functional), but we need here to operate some modifications due to the non-locality of the fractional operators (−∆)s.”
Existence and qualitative properties of multidimensional conical bistable fronts (2005, background): “…1), this condition (1.5) was used in [17], where the existence of solutions to (1.1), (1.5) is proved in 2 space dimensions; furthermore, because of curvature effects, this condition (1.5) – which in particular implies that the lines with slopes ± cotα are asymptotic to the level lines of the…”
Hamiltonian identities for elliptic partial differential equations (2008, neutral): “The conjecture has been proved for n = 2 in [14] and n = 3 in [4].”
Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces. (2009, neutral): “1”
Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian (2015, neutral): “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]).”
On the stability of radial solutions of semilinear elliptic equations in all of R n (2003, background): “Dans un premier résultat de cette Note, on utilise des méthodes récemment développées dans [2] [1] pour établir que toute solution de (1) non constante, bornée et dans H 1 (R”
Nonlocal Diffusion and Applications (2015, methodology): “We recall that this conjecture by De Giorgi was proved for n ď 3, see [58, 8, 4].”