Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur [1] showed that a certain class of weak solutions to the drift diffusion equation with initial data in L gain Hölder continuity provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on BMO norm of a smooth velocity implies uniform bound on the C norm of the solutio...

We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step small enough for the method to be stable. We find sharp uniform bounds for the converge...

A formal definition of ǫ-complexity of an individual continuous function defined on a unit cube is proposed. This definition is consistent with the Kolmogorov’s idea of the complexity of an object. A definition of ǫ-complexity for a class of continuous functions with a given modulus of continuity is also proposed. Additionally, an explicit formula for the ǫ-complexity of a functional class is o...

We study mappings satisfying the so-called inverse Poletsky inequality. Under integrability of corresponding majorant, it is proved that these are logarithmic Hölder continuous in neighborhood boundary points. In particular, indicated properties hold for homeomorphisms whose satisfy weighted