Approximation of Holder continuous homeomorphisms by piecewise affine homeomorphisms

Higher regularity of homeomorphisms in the Hartman-Grobman theorem for semilinear evolution equations

Hein and Prüss [J. Differential Equations, 261(2016)4709-4727] presented a version of Hartman-Grobman type $C^{0}$ linearization result for semilinear hyperbolic evolution equations. They showed that the linearising map (homomorphism) and its inverse are Hölder continuous. An important question: is it possible to improve the regularity of the homomorphisms? In the present paper, we prove that if the mild solutions of semilinear system are bounded, then the regularity of the homomorphisms is Lipchitzian, but the inverse is merely Hölder continuous. We also give a generalized local linearization…

Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$

We investigate a known problem whether a Sobolev homeomorphism between domains in $\mathbb{R}^n$ can change sign of the Jacobian. The only case that remains open is when $f\in W^{1,[n/2]}$, $n\geq 4$. We prove that if $n\geq 4$, and a sense-preserving homeomorphism $f$ satisfies $f\in W^{1,[n/2]}$, $f^{-1}\in W^{1,n-[n/2]-1}$ and either $f$ is Hölder continuous on almost all spheres of dimension $[n/2]$, or $f^{-1}$ is Hölder continuous on almost all spheres of dimensions $n-[n/2]-1$, then the Jacobian of $f$ is non-negative, $J_f\geq 0$, almost everywhere. This result is a consequence of a mo…

大集合上双 Lipschitz 映射的分解与分段仿射逼近

Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets

本文利用 corona 型分解定理，证明单位立方体的双 Lipschitz 嵌入在任意小测度例外集外可分解为有限多个小失真全局双 Lipschitz 映射，并得到球面分解及分段仿射逼近结果。

A Grobman-Hartman theorem for a differential equation with piecewise constant generalized argument

We obtain sufficient conditions ensuring the existence of a uniformly continuous and Hölder continuous homeomorphism between the solutions of a linear system of differential equations with piecewise constant argument of generalized type and the solutions of the quasilinear corresponding system. We use a definition (recently introduced by M. Akhmet) of exponential dichotomy for those systems combined with technical assumptions on the nonlinear part. Our result generalizes a previous work of G. Papaschinopoulos.