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Fractional Sobolev-Poincare inequalities in irregular domains

Chang-Yu Guo

Abstract

This paper is devoted to the study of fractional (q,p)-Sobolev-Poincare inequalities in irregular domains. In particular, we establish (essentially) sharp fractional (q,p)-Sobolev-Poincare inequality in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tends to the results for the usual derivative. Furthermore, we verified that those domains that support the fractional (q,p)-Sobolev-Poincare inequality together with a separation property are s-diam John domains for certain s, depending only on the associated data. We also point out an inaccurate statement in [2].

Fractional Sobolev-Poincare inequalities in irregular domains

Abstract

This paper is devoted to the study of fractional (q,p)-Sobolev-Poincare inequalities in irregular domains. In particular, we establish (essentially) sharp fractional (q,p)-Sobolev-Poincare inequality in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tends to the results for the usual derivative. Furthermore, we verified that those domains that support the fractional (q,p)-Sobolev-Poincare inequality together with a separation property are s-diam John domains for certain s, depending only on the associated data. We also point out an inaccurate statement in [2].

Paper Structure

This paper contains 5 sections, 10 theorems, 85 equations.

Table of Contents

  1. Notations and definitions
  2. Auxiliary results
  3. Main proofs
  4. Fractional $(q,p)$-Sobolev-Poincaré inequalities in domains with quasihyperbolic boundary condition
  5. Necessary conditions for the fractional $(q,p)$-Sobolev-Poincaré domains

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^n$, $n\geq 2$, be a domain with finite Lebesgue measure and $1\leq p\leq q<\infty$. Then the following statements are equivalent:

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Lemma 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Remark 3.4
  • ...and 11 more