Regularity of flat level sets in phase transitions

By Ovidiu Savin. 2009

308 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/cdddff30f35c9776689ec4312f2cb75a56c06a01

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Hamiltonian identities for elliptic partial differential equations (2008, unknown): “For dimensions up to 8, the conjecture is essentially proved in [ 20 ], provided that u satisfies the limiting condition”
Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones (2011, background): “The extra condition (1.6), used by Savin, is related to the so-called : Gibbons’ conjecture : Assume that u is a bounded solution of (1.1) satisfying (1.7) lim xn+1→±∞ u(x′, xn+1) = ±1, uniformly in x′ ∈ Rn.”
MULTIPLE-END SOLUTIONS TO THE ALLEN-CAHN EQUATION IN R2 (2010, background): “We recall here that the heteroclinic is stable, and from [5] we know that in dimension N = 2 stability of a solution implies that it is necessarily a one dimensional solution (for the related minimality conjecture, see for example [28] and [26] and the reference therein).”
Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces. (2009, background): “Savin [37] proved its validity for 4 ≤ N ≤ 8 under a mild additional assumption.”
On Phase-Separation Models: Asymptotics and Qualitative Properties (2013, unknown): “Savin [42] established its validity for 4 ≤ N ≤ 8 under the following mild additional assumption lim xN→±∞ u(x ′ , xN) = ±1 (7.”
De Giorgi type results for elliptic systems (2012, background): “However, Ghoussoub and Gui also showed in [12] that it is true for N = 4 or N = 5 for solutions that satisfy certain antisymmetry conditions, and Savin [16] established its validity for 4 ≤ N ≤ 8 under the following additional natural hypothesis on the solution, lim xN→±∞ u(x′, xN ) → ±1.”
Serrin’s overdetermined problem and constant mean curvature surfaces (2013, background): “hat entire solutions, monotone in one direction must have level sets which are parallel hyperplanes. This is true in dimensions N = 2,3 [13, 2], and under a certain additional condition for 4 ≤ N ≤ 8 [20]. This statement is indeed false for N ≥ 9 as proven in [8] by the construction of an example of”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “See [13, 60, 10, 11, 12] for progresses in this direction and [32, 3, 56, 25, 30, 31] for the state-of-the-art results on the conjecture in the classical case, formally obtained by taking s = 1.”
Saddle-shaped solutions of bistable diffusion equations in all of ℝ2m (2009, methodology): “For n ≤ 8, a weaker version of the conjecture was proven recently by Savin [22].”
Solutions to overdetermined elliptic problems in nontrivial exterior domains (2016, result): “ontrivial epigraphs if N≥ 9, see [8]. In [33] Wang and Wei prove that this type of solutions do not exist if N≤ 8, a result that can be put in analogy with that of Savin for the De Giorgi conjecture ([22]). Finally, the notion of stability plays an important role in the De Giorgi conjecture, and als”
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.”