Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on non-local Sobolev spaces and non-local perimeters

Konstantinos Bessas, Giuseppe Cosma Brusca

Abstract

We investigate the space of non-local Sobolev functions associated with an integral kernel. We prove an extension result, Sobolev and Poincaré inequalities and an isoperimetric inequality for the non-local perimeter restricted to a set. Finally, we remark on non-local isoperimetric problems, even when the underlying kernel is not necessarily radially symmetric.

A note on non-local Sobolev spaces and non-local perimeters

Abstract

We investigate the space of non-local Sobolev functions associated with an integral kernel. We prove an extension result, Sobolev and Poincaré inequalities and an isoperimetric inequality for the non-local perimeter restricted to a set. Finally, we remark on non-local isoperimetric problems, even when the underlying kernel is not necessarily radially symmetric.

Paper Structure

This paper contains 10 sections, 28 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Non-local geometric variational problems
  3. Plan of the paper
  4. Notation and preliminary results
  5. Properties of non-local Sobolev functions
  6. An extension result
  7. Sobolev and Poincaré inequalities
  8. Remarks on the non-local isoperimetric problem
  9. Non-local perimeter of balls
  10. Existence of non-local isoperimetric sets

Key Result

Theorem 1

lemmaO o 1

Theorems & Definitions (63)

  • Theorem 1
  • Theorem 2: 1
  • Lemma 3
  • Lemma 4: 1
  • Proposition 5
  • Proposition 6: 1
  • Corollary 7
  • Corollary 8: 1
  • Definition 9
  • Definition 10: 1
  • ...and 53 more