p-泊松方程解的贝塞空间正则性

Regularity in Sobolev and Besov spaces for parabolic problems on domains of polyhedral type

This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale $B^r_{τ,τ}$, $\frac{1}τ=\frac rd+\frac 1p$ of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.

Anisotropic Besov regularity of parabolic PDEs

This paper is concerned with the regularity of solutions to parabolic evolution equations. Special attention is paid to the smoothness in the specific anisotropic scale $\ B^{r\mathbf{a}}_{τ,τ}, \ \frac{1}τ=\frac{r}{d}+\frac{1}{p}\ $ of Besov spaces where $\mathbf{a}$ measures the anisotropy. The regularity in these spaces determines the approximation order that can be achieved by fully space-time adaptive approximation schemes. In particular, we show that for the heat equation our results significantly improve previous results by Aimar and Gomez [3].

非凸多面体域上非线性抛物型偏微分方程的Besov正则性

Besov regularity of non-linear parabolic PDEs on non-convex polyhedral domains

该文研究非凸域上半线性抛物方程的Besov空间正则性，证明其正则性足够支撑自适应算法应用，证明基于Schauder不动点定理。

Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains

We prove Besov regularity estimates for the solution of the Dirichlet problem involving the integral fractional Laplacian of order $s$ in bounded Lipschitz domains $Ω$: \[ \begin{aligned} \|u\|_{\dot{B}^{s+r}_{2,\infty}(Ω)} \le C \|f\|_{L^2(Ω)}, & \quad r = \min\{s,1/2\}, & \quad \mbox{if } s \neq 1/2, \\ \|u\|_{\dot{B}^{1-ε}_{2,\infty}(Ω)} \le C \|f\|_{L^2(Ω)}, & \quad ε\in (0,1), & \quad \mbox{if } s = 1/2, \end{aligned} \] with explicit dependence of $C$ on $s$ and $ε$. These estimates are consistent with the regularity on smooth domains and show that there is no loss of regularity due to L…

On the lack of interior regularity of the $p$-Poisson problem with $p>2$

In this note we are concerned with interior regularity properties of the $p$-Poisson problem $Δ_p(u)=f$ with $p>2$. For all $0<λ\leq 1$ we constuct right-hand sides $f$ of differentiability $-1+λ$ such that the (Besov-) smoothness of corresponding solutions $u$ is essentially limited to $1+λ/ (p-1)$. The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savaré [J. Funct. Anal. 152:176-201, 1998], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130:298-329, 2016]. Keywords:…

有界Lipschitz域上随机抛物方程的Lq(Lp)-正则性与Besov光滑性

On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

研究有界Lipschitz域上带零Dirichlet边界的线性随机抛物方程在Krylov加权Sobolev空间中的存在唯一性、时间Hölder正则性，并通过嵌入定理得到Besov尺度下的正则性结果。