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Optimal weighted Wente's inequality

Matilde Gianocca

Abstract

We prove $L^\infty$ and $W^{1,2}$ weighted Wente's inequalities. We prove in particular the critical case: for the $|x|^2$ weighted Wente's estimate the optimal weight is $|x|^2\log|x|$.

Optimal weighted Wente's inequality

Abstract

We prove and weighted Wente's inequalities. We prove in particular the critical case: for the weighted Wente's estimate the optimal weight is .
Paper Structure (3 sections, 6 theorems, 99 equations)

This paper contains 3 sections, 6 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Proof of weighted Wente's inequality
  3. The case $\alpha=1$

Key Result

Theorem 1

Let $\varphi\in W^{1,2}_0(B_1)$ be the unique solution of for $a,b\in W^{1,2}(B_1)$, where $B_1\subset\mathbb{R}^2$ is the two-dimensional open disk. Then for any $0<\alpha<1$ there exists a constant $C_\alpha>0$, s.t. Moreover there exists a universal constant $C>0$ such that

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof