Distributional framework for solving fractional differential equations

关于具有有限分布分数周长的集合

On sets with finite distributional fractional perimeter

本文研究局部有限分布分数周长集合的精细性质，改进其爆破刻画，证明其与有限分数周长集合相交的Leibniz法则，并描述相关非局部边界。

分布分数阶Stockwell变换与分数阶小波变换的渐近结果

Asymptotic results for the distributional fractional Stockwell and fractional wavelet transforms

本文研究Lizorkin分布的拟渐近行为，通过Abelian和Tauberian型结果用Stockwell变换刻画，证明分数阶小波变换及其合成算子在Schwartz空间上的连续性，并建立分数阶Stockwell与小波变换的联系及渐近结果。

A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up

We introduce the new space $BV^α(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $α\in (0, 1)$ via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical $BV$ theory, we give a new notion of set $E$ of (locally) finite fractional Caccioppoli $α$-perimeter and we define its fractional reduced boundary $\mathscr{F}^α E$. We are able to show that $W^{α,1}(\mathbb{R}^n)\subset BV^α(\mathbb{R}^n)$ continuously and, similarly, that sets with (…

分布分数空间中的莱布尼茨法则与高斯-格林公式

Leibniz rules and Gauss-Green formulas in distributional fractional spaces

本文基于分布方法，在分数阶函数空间中证明了新的莱布尼茨法则，并将其应用于一般分数阶椭圆算子边值问题的适定性证明。

分数阶Sobolev空间与分数阶变分的一种分布方法：渐近性I

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I

本文研究分数阶变分的渐近行为，证明分数阶α-变分在α趋于1时收敛于标准De Giorgi变分，且β-变分在β趋于α时收敛于α-变分，均逐点及Γ-极限意义下成立。

Distributional Fractional Gradients and a Bourgain-Brezis-type Estimate

In this paper, we extend the definition of fractional gradients found in Mazowiecka-Schikorra to tempered distributions on $\R^n$, introduce associated regularisation procedures and establish some first regularity results for distributional fractional gradients in $L^{1}_{od}$. The key feature is the introduction of a suitable space of off-diagonal Schwarz functions $\mathcal{S}_{od}(\R^{2n})$, allowing for a dual definition of the fractional gradient on an appropriate space of distributions $\mathcal{S}^{\prime}_{od}(\R^{2n})$ by means of fractional divergences defined on $\mathcal{S}_{od}(\R…