Michal Kowalczyk

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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)
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)
A periodic perturbation of a Gaussian measure modifies the sharp constants in Poincaré and logarithmic Sobolev inequalities in the homogenization limit, that is, when the period of a periodic perturbation converges to zero. We use variational techniques to determine the ho-mogenized constants and get optimal convergence rates towards equilibrium of the(More)