Fractional Laplacians, perimeters and heat semigroups in Carnot groups

Abstract

We define and study the fractional Laplacian and the fractional perimeter of a set in Carnot groups and we compare the perimeter with the asymptotic behaviour of the fractional heat semigroup.

Fractional Sobolev norms and BV functions on manifolds

“…Independently, the authors of [7] and [26] showed, that the variation |Df |(M ) of a BV function on a manifold can be approximated by evolutions of the function under the heat semigroup. In [13] a characterization of perimeters in Carnot groups is provided via heat semigroup techniques. The authors raise the question if a characterization of perimeters can also be attained on Riemannian manifolds.…”

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

“…Different approaches have been developed for stating the definition or restriction of the fractional Laplacian in the bounded domains. [12][13][14][15][16][17] In this work, we use the definition based on the spectral decomposition technique 12,18,19 which offers an interesting way to introduce and exploit the fractional Laplacian in a bounded domain. In order to present the used definition of (− Δ) s , we denote by { k , k } k ≥ 1 the eigenpairs of the usual Laplacian operator on the bounded domain …”

“…However, this equivalence cannot be preserved when this type of operators is considered on bounded domains. Different approaches have been developed for stating the definition or restriction of the fractional Laplacian in the bounded domains 12‐17 …”

Some Nonlocal Operators in the First Heisenberg Group

“…In analogy to this, the choice of the constant c in the case of (−L H 1 ) = 0. For an application of the fractional Laplace operator in the Heisenberg group to the geometric measure theory in this noncommutative framework, see [9].…”

“…We are interested in a construction of nonlocal operators in the first Heisenberg group H 1 and possibly compare them with the fractional operator of the sub-Laplacian already known in the literature (see [7][8][9]). …”