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

By Giovanni E. Comi, Giorgio Stefani. Journal of Functional Analysis, 2018

98 citations recorded by the source corpus.

Abstract We introduce the new space B V 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 ) ⊂ B V 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.

Subfields: Applied Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/73019301566f95a7dea917ecdbd6098fb67cf05e

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In Applied Mathematics

1. Javier Cueto
2. C. Mora‐Corral
3. José Carlos Bellido

In Mechanics of Materials

1. Javier Cueto
2. C. Mora‐Corral
3. José Carlos Bellido

Citations in the corpus

• Fractional Piola identity and polyconvexity in fractional spaces (2018, background): “In [8], it is addressed the space of functions u whose total fractional variation is finite, naturally leading to the definition of the space BV s(Rn) and to a s-fractional Caccioppoli perimeter concept.”