Ph.D. Thesis
Inserted: 22 may 2020
Last Updated: 22 may 2020
Pages: 158
Year: 2020
Abstract:
In this thesis, we present the distributional approach to fractional Sobolev spaces and fractional variation developed in three papers, two in collaboration with G. E. Comi (see A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up and A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I) and one in collaboration with E. Bruè, M. Calzi and G. E. Comi (in preparation). The new space $BV^{\alpha}(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha\in(0,1)$ is distributionally defined by 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 $\alpha$-perimeter and we define its fractional reduced boundary $\mathscr{F}^{\alpha} E$. We are able to show that $W^{\alpha,1}(\mathbb{R}^n)\subset BV^\alpha(\mathbb{R}^n)$ continuously and, similarly, that sets with (locally) finite standard fractional $\alpha$-perimeter have (locally) finite fractional Caccioppoli $\alpha$-perimeter, so that our theory provides a natural extension of the known fractional framework. We first extend De Giorgi's Blow-up Theorem to sets of locally finite fractional Caccioppoli $\alpha$-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets. We then prove that the fractional $\alpha$-variation converges to the standard De Giorgi's variation both pointwise and in the $\Gamma$-limit sense as $\alpha\to1^-$ and, similarly, that the fractional $\beta$-variation converges to the fractional $\alpha$-variation both pointwise and in the $\Gamma$-limit sense as $\beta\to\alpha^-$ for any given $\alpha\in(0,1)$. Finally, by exploiting some new interpolation inequalities on the fractional operators involved, we prove that the fractional $\alpha$-gradient converges to the Riesz transform as $\alpha\to0^+$ in $L^p$ for $p\in(1,+\infty)$ and in the Hardy space and that the $\alpha$-rescaled fractional $\alpha$-variation converges to the integral mean of the function as $\alpha\to0^+$.
Download: