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A celebrated conjecture due to De Giorgi states that any bounded solution of the equation ∆u+(1−u 2)u = 0 in R N with ∂ y N u > 0 must be such that its level sets {u = λ} are all hyperplanes, at least for dimension N ≤ 8. A counterexample for N ≥ 9 has long been believed to exist. Based on a minimal graph Γ which is not a hyperplane, found by Bombieri, De(More)
A celebrated conjecture due to De Giorgi states that any bounded solution of the equation ∆u+(1−u 2)u = 0 in R N with ∂y N u > 0 must be such that its level sets {u = λ} are all hyperplanes, at least for dimension N ≤ 8. A counterexample for N ≥ 9 has long been believed to exist. Starting from a minimal graph Γ which is not a hyperplane, found by Bombieri,(More)
— We consider minimal surfaces M which are complete, embedded and have finite total curvature in R 3 , and bounded, entire solutions with finite Morse index of the Allen-Cahn equation ∆u+f (u) = 0 in R 3. Here f = −W with W bistable and balanced, for instance W (u) = 1 4 (1 − u 2) 2. We assume that M has m ≥ 2 ends, and additionally that M is(More)
We review some recent results on construction of entire solutions to the classical semilinear elliptic equation ∆u + u − u 3 = 0 in R N. In various cases, large dilations of an embedded, complete minimal surface approximate the transition set of a solution that connects the equilibria ±1. In particular, our construction answers negatively a celebrated(More)
Let (M, ˜ g) be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation ε 2 ∆ ˜ g u + (1 − u 2)u = 0 in M, where ε is a small parameter. Let K ⊂ M be an (N − 1)-dimensional smooth minimal submanifold that separates M into two disjoint components. Assume that K is non-degenerate in the sense that it does(More)
These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spectral properties of the Laplace-Beltrami(More)