Let $\Omega \subset \mathbb R^d$ be a $C^1$ domain or, more generally, Lipschitz with small constant and $A(x)$ $d \times d$ uniformly elliptic, symmetric matrix coefficients. Assume $u$ is harmonic in $\Omega$, or greater generality solves $\operatorname{div}(A(x)\nabla u)=0$ vanishes on $\Sigma = \partial\Omega \cap B$ for some ball $B$. We study the dimension of singular set $\Sigma$, partic...

We consider an autonomous conservative second order system defined by a potential which admits a connected set Γ of critical points at level zero. We prove the existence of multiple periodic solutions of large period which are located near Γ.

We shall consider slightly more general problems. Namely, we shall investigate the existence of continuous: piecewise monotone, piecewise strictly monotone, and piecewise linear n-th roots of interval maps which have a continuous n-th root. Here by an n-th root of f we mean a map g such that f = g (g is the n-th iterate of g). A continuous map f : I → J , where I, J are closed intervals, is pie...

We consider the behavior of piecewise isometries in Euclidean spaces. We show that if n is odd and the system contains no orientation reversing isometries then recurrent orbits with rational coding are not expected. More precisely, a prevalent set of piecewise isometries do not have recurrent points having rational coding. This implies that when all atoms are convex no periodic points exist for...

This note fixes a small gap in the proof in [4] that the limit set of the handlebody set has measure zero.