On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property

By Giovanni S. Alberti, Luigi Ambrosio, X. Cabré. 2001

198 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/5163a8717e06e95c7db1027fc33abaa8c65f6183

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones (2011, background): “We refer to [1] where this condition is discussed at length.”
Local and global minimizers for a variational energy involving a fractional norm (2011, unknown): “e literature, (9) is generally referred by saying that uis a local minimizer for F in the domain Br. The proof of Theorem 1 follows a classical sliding argument (see, e.g., Lemma 9.1 in [27] and also [2] and [5] for a diﬀerent variational approach for the classical local functional), but we need her”
Nonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions (2011, background): “This statement on the 1D symmetry of u(·, 0) is closely related to a conjecture of De Giorgi on 1D symmetry for interior reactions, proved in [2, 3, 15] in low dimensions and partially settled by Savin [20] up to dimension 8.”
ENERGY ESTIMATES AND 1-D SYMMETRY FOR NONLINEAR EQUATIONS INVOLVING THE HALF-LAPLACIAN (2010, methodology): “R2 logR, and this is enough to carry out the proof of Theorem 2.1 in the present setting. Before giving the proof of Theorem 1.3, we need the following proposition. It is the analog of Theorem 4.4 of [1] and asserts that the monotonicity of a solution implies its minimality among a suitable family o”
Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result (2008, methodology): “, [1,20,31] and Section 7 of [23] for further details).”
Layer solutions in a half‐space for boundary reactions (2005, methodology): “The upper bound has been proven by two different methods: a quite elementary one in [4] and a variational one in [1] that uses the local minimality property of layer solutions.”