We introduce the new space BV alpha(R-n) of functions with bounded fractional variation in R-n of order a is an element of (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 alpha-perimeter and we define its fractional reduced boundary (FE)-E-alpha. We are able to show that W-alpha,W-1 (R-n) subset of BV alpha (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. Our main result partially extends 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. (C) 2019 Elsevier Inc. All rights reserved.

We introduce the new space BV α (R n ) of functions with bounded fractional variation in R n of order α∈(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 F α E. We are able to show that W α,1 (R n )⊂BV α (R n ) continuously and, similarly, that sets with (locally) finite standard fractional α-perimeter have (locally) finite fractional Caccioppoli α-perimeter, so that our theory provides a natural extension of the known fractional framework. Our main result partially extends De Giorgi's Blow-up Theorem to sets of locally finite fractional Caccioppoli α-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets.

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

Comi, Giovanni E.
;
Stefani, Giorgio
2019

Abstract

We introduce the new space BV alpha(R-n) of functions with bounded fractional variation in R-n of order a is an element of (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 alpha-perimeter and we define its fractional reduced boundary (FE)-E-alpha. We are able to show that W-alpha,W-1 (R-n) subset of BV alpha (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. Our main result partially extends 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. (C) 2019 Elsevier Inc. All rights reserved.
2019
Settore MAT/05 - Analisi Matematica
Fractional gradient; Fractional perimeter; Fractional Sobolev spaces; Function with bounded fractional variation; Analysis
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/79116
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