$\mathrm{L}^p$-extrapolation of non-local operators: Maximal regularity of elliptic integrodifferential operators with measurable coefficients

Regularity estimates for elliptic nonlocal operators

We study weak solutions to nonlocal equations governed by integrodifferential operators. Solutions are defined with the help of symmetric nonlocal bilinear forms. Throughout this work, our main emphasis is on operators with general, possibly singular, measurable kernels. We obtain regularity results which are robust with respect to the differentiability order of the equation. Furthermore, we provide a general tool for the derivation of Hölder a-priori estimates from the weak Harnack inequality. This tool is applicable for several local and nonlocal, linear and nonlinear problems on metric spac…

关于粗糙系数的广义 Stokes 算子的 $\mathrm{L}^p$ 理论

$\mathrm{L}^p$-extrapolation via non-local decay estimates: theory and applications to the generalized Stokes operator with rough coefficients

本文针对粗糙系数的广义 Stokes 算子，建立全范围尖锐的 $\mathrm{L}^p$ 映射性质，包括半群、梯度及 $\mathrm{H}^\infty$ 演算估计，并提出抽象框架统一超压缩性、非局部衰减与一致 $\mathrm{L}^p$ 界。

$\mathcal{R}$-sectoriality of higher-order elliptic systems on general bounded domains

On bounded domains $Ω\subset \mathbb{R}^d , d \geq 2$, reaching far beyond the scope of Lipschitz domains, we consider an elliptic system of order $2 m$ in divergence form with complex $\mathrm{L}^{\infty}$-coefficients complemented with homogeneous mixed Dirichlet/Neumann boundary conditions. We prove that the $\mathrm{L}^p$-realization of the corresponding operator $A$ is $\mathcal{R}$-sectorial of angle $ω\in [0 , \fracπ{2})$, where in the case $2m < d$, $p \in (\frac{2d}{d + 2 m} - \varepsilon , \frac{2d}{d - 2 m} + \varepsilon)$ for some $\varepsilon > 0$, and where $p \in (1 , \infty)$ i…

非光滑拟线性抛物系统的非自治极大抛物正则性

Nonautonomous maximal parabolic regularity for nonsmooth quasilinear parabolic systems

本文针对非光滑域上带混合边界条件的非自治拟线性抛物系统，建立了$L^p$-极大正则性，通过外推技术获得一致估计，仅需温和边界正则性及有界可测复系数，并允许时间非局部非线性，涵盖稳态不满足椭圆性的情形。

抛物型 $L^p$ 齐次边值问题的局部化与插值

Localization and interpolation of parabolic $L^p$ Neumann problems

研究抛物型方程局部解的局部估计，基于齐次边值问题和对偶算子的 $L^p$ 解的存在性，建立原子 Hardy 空间中 Neumann 问题的可解性，从而得到 $L^p$ Neumann 问题的可解性扩展。

Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted Sobolev classes. We establish mapping properties for the double and single layer potentials, as well as the Newton potential, on Besov spaces. We prove extrapolation-type solvability results: that is, we show that solvability of the Dirichlet or Neumann boundary value problem at any given L^p space automatically assures their solvability in an extended range of Besov spaces. We also esta…