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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. 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)
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)
We consider the problem of existence of entire solutions to the Allen-Cahn equation Δu + u - u(3) = 0 in , usually regarded as a prototype for the modeling of phase transition phenomena. In particular, exploiting the link between the Allen-Cahn equation and minimal surface theory in dimensions N ≥ 9, we find a solution, u, with ∂(x(N))u > 0, such that its(More)
A general class of nonlinear evolution equations is described, which support stable spatially oscillatory steady solutions. These equations are composed of an indeenite self-adjoint linear operator acting on the solution plus a nonlinear function, a typical example of the latter being a double-well potential. Thus a Lyapunov functional exists. The linear(More)
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