Michal Kowalczyk

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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 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)
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)
The purpose of this study was to examine physiological and physical determinants of ice-hockey performance in order to assess their impact on the result during a selection for ice hockey. A total of 42 ice hockey players took part in the selection camp. At the end of the camp 20 best players were selected by team of expert coaches to the ice hockey team and(More)