Hölder Continuity of the Data to Solution Map for HR in the Weak Topology

H$\mathbf{ö}$lder continuity of QCH mappings from the unit ball to a domain with $C^1$ boundary

We prove that every quasiconformal mapping from the harmonic $β$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $α$-Hölder continuous for $α<1-β$, with the Hölder coefficient that does not depend neither on the mapping nor on $β$. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for $α<1$. This extends some results from the complex plane obtained by Warschawski in \cite{Warschawski} for conformal mappings and Kalaj in \cite{Kalaj6} for quasiconformal harmonic mappings.

Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift

We consider non degenerate Brownian SDEs with H{ö}lder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial H{ö}lder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.

Leray-Hopf and Continuity Properties for All Weak Solutions for the 3D~Navier-Stokes Equations

In this note we prove that each weak solution for the 3D Navier-Stokes system satisfies Leray-Hopf property. Moreover, each weak solution is rightly continuous in the standard phase space $H$ endowed with the strong convergence topology.

演化方程的弱算子连续性

Weak Operator Continuity for Evolutionary Equations

本文研究了 Picard 意义下演化方程的解算子连续性，提出了一种针对材料律的向量值全纯函数拓扑，使得解算子在弱算子拓扑下连续，并应用于时间依赖偏微分方程的非局部均匀化及细胞迁移模型。

Interior Hölder continuity for singular-degenerate porous medium type equations with an application to a biofilm model

We show interior Hölder continuity for a class of quasi-linear degenerate reaction-diffusion equations. The diffusion coefficient in the equation has a porous medium type degeneracy and its primitive has a singularity. The reaction term is locally bounded except in zero. The class of equations we analyse is motivated by a model that describes the growth of biofilms. Our method is based on the original proof of interior Hölder continuity for the porous medium equation. We do not restrict ourselves to solutions that are limits in the weak topology of a sequence of approximate continuous solution…

The weak Cartan property for the p-fine topology on metric spaces

We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality, 1 < p< oo. We establish a weak Cartan property, which yields characterizations of the p-thinness and the p-fine continuity, and allows us to show that the p-fine topology is the coarsest topology making all p-superharmonic functions continuous. Our p-harmonic and superharmonic functions are defined by means of scalar-valued upper gradients and do not rely on a vector-valued differentiable structure.