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Sobolev improvements on sharp Rellich inequalities

Gerassimos Barbatis, Achilles Tertikas

Abstract

There are two Rellich inequalities for the bilaplacian, that is for $\int (Δu)^2dx$, the one involving $|\nabla u|$ and the other involving $|u|$ at the RHS. In this article we consider these inequalities with sharp constants and obtain sharp Sobolev-type improvements. More precisely, in our first result we improve the Rellich inequality with $|\nabla u|$ obtained recently by Cazacu in dimensions $n=3,4$ by a sharp Sobolev term thus complementing existing results for the case $n\geq 5$. In the second theorem the sharp constant of the Sobolev improvement for the Rellich inequality with $|u|$ is obtained.

Sobolev improvements on sharp Rellich inequalities

Abstract

There are two Rellich inequalities for the bilaplacian, that is for , the one involving and the other involving at the RHS. In this article we consider these inequalities with sharp constants and obtain sharp Sobolev-type improvements. More precisely, in our first result we improve the Rellich inequality with obtained recently by Cazacu in dimensions by a sharp Sobolev term thus complementing existing results for the case . In the second theorem the sharp constant of the Sobolev improvement for the Rellich inequality with is obtained.
Paper Structure (2 sections, 12 theorems, 111 equations)

This paper contains 2 sections, 12 theorems, 111 equations.

Table of Contents

  1. Rellich-Sobolev inequality I
  2. Rellich-Sobolev inequality II

Key Result

Theorem 1

Let $\Omega\subset{\mathbb{R}}^n$, $n=3$ or $n=4$, be a bounded domain and let $D= \sup_{x \in \Omega} |x|$. There exists $C>0$ such that: $({\rm i})$ If $n=3$ then $({\rm ii})$ If $n=4$ then Moreover the power $X^4$ in case $n=3$ is the best possible.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 2 more