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We consider the problem concentrating along the whole of D , exponentially small in at any positive distance from it, provided that is small and away from certain critical numbers. In particular this establishes the validity of a conjecture raised in  in the two-dimensional case.
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 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)
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)
Light can change the orientation of a liquid crystal. This is the optical Freedericksz transition, discovered by Saupe. In the Janossy effect, the threshold intensity for the optical Freedericksz transition is dramatically reduced by the additon of a small amount of dye to the sample. This has been interpreted as an optically pumped orientational rachet… (More)