Michal Kowalczyk

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Intracellular transport in eukarya is attributed to motor proteins that transduce chemical energy into directed mechanical motion. Muscle myosin has been known since the mid nineteenth century and its role in muscle contraction demonstrated by A. F. Huxley and H. E. Huxley in the 1950's. Kinesins and their role in intracellular transport were discovered(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 homogenized constants and get optimal convergence rates towards equilibrium of the(More)
This paper is devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on(More)
A general class of nonlinear evolution equations is described, which support stable spatially oscillatory steady solutions. These equations are composed of an indefinite 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)
Abstract. We consider the problem ε2Δu+ (u− a(x))(1− u2) = 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where Ω is a smooth and bounded domain in R2, −1 < a(x) < 1. Assume that Γ = {x ∈ Ω, a(x) = 0} is a closed, smooth curve contained in Ω in such a way that Ω = Ω+ ∪ Γ ∪ Ω− and ∂a ∂n > 0 on Γ, where n is the outer normal to Ω+. Fife and Greenlee [Russian Math. Surveys, 29(More)
Let (M, g̃) be an N -dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation ε∆g̃u + (1− u)u = 0 in M, where ε is a small parameter. Let K ⊂M be an (N − 1)-dimensional smooth minimal submanifold that separatesM into two disjoint components. Assume that K is non-degenerate in the sense that it does not support(More)
We review some recent results on construction of entire solutions to the classical semilinear elliptic equation ∆u+u−u = 0 in R . 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 conjecture(More)
Tirapazamine is a hypoxia-activated prodrug which was shown to exhibit up to 300 times greater cytotoxicity under anoxic in comparison with aerobic conditions. Thus, the combined anticancer therapy of tirapazamine with a routinely used anticancer drug seems to be a promising solution. Because tirapazamine undergoes redox cycle transformation in this study,(More)