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Extremal function for a sharp Moser-Trudinger type inequality on the upper half space

Yubo Ni

TL;DR

This work proves a sharp weighted Moser-Trudinger inequality in the 2D upper half-space with monomial weights $t^{\alpha}$ and $t^{\beta}$ and identifies the best constant $a_{\alpha,\beta}$ alongside the critical exponent $b_{\alpha}$. It develops a concentration-compactness framework to handle maximizing sequences in the weighted setting and demonstrates the sharpness of the inequality via a Moser-type sequence, with a weighted Onofri inequality derived as a corollary. The authors then establish the existence of an extremal function under dynamic changes of the unit ball using radial variational methods, culminating in a nonlinear Dirichlet problem on the upper half-ball. Overall, the results extend sharp Moser-Trudinger theory to weighted upper-half-space domains and provide tools for related geometric and PDE problems under dynamic constraints.

Abstract

Sharp Moser-Trudinger type inequalities and their extremal functions play an important role in studying nonlinear PDEs and geometry. We establish a new sharp Moser-Trudinger type inequality in the upper half space in two dimensions and prove the existence of extremal functions for a sharp Moser-Trudinger type inequality under dynamic changes in the unit ball.

Extremal function for a sharp Moser-Trudinger type inequality on the upper half space

TL;DR

This work proves a sharp weighted Moser-Trudinger inequality in the 2D upper half-space with monomial weights and and identifies the best constant alongside the critical exponent . It develops a concentration-compactness framework to handle maximizing sequences in the weighted setting and demonstrates the sharpness of the inequality via a Moser-type sequence, with a weighted Onofri inequality derived as a corollary. The authors then establish the existence of an extremal function under dynamic changes of the unit ball using radial variational methods, culminating in a nonlinear Dirichlet problem on the upper half-ball. Overall, the results extend sharp Moser-Trudinger theory to weighted upper-half-space domains and provide tools for related geometric and PDE problems under dynamic constraints.

Abstract

Sharp Moser-Trudinger type inequalities and their extremal functions play an important role in studying nonlinear PDEs and geometry. We establish a new sharp Moser-Trudinger type inequality in the upper half space in two dimensions and prove the existence of extremal functions for a sharp Moser-Trudinger type inequality under dynamic changes in the unit ball.
Paper Structure (4 sections, 13 theorems, 131 equations)

This paper contains 4 sections, 13 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Weighted sharp Moser-Trudinger inequalities
  3. Concentration-compactness principle
  4. Extremal function exists under dynamic change

Key Result

Theorem 1.1

Let $\alpha,\beta>-1$ and let $\Omega\subset \mathbb{\mathbb{\mathbb{R}}}_+^2$ be a bounded domain. Then there exists a constant $c_{0}=c_{0}(\alpha,\beta)$ such that for each $u\in C_c^\infty(\Omega)$ with ${\int_{\Omega}|\nabla u|^{2+\alpha}t^{\alpha}}$dxdt $\leq1$, where is the best constant.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Adams 1988 A1988
  • Lemma 2.2: Lam 2017 L2017
  • proof : Proof of theorem \ref{['theorem1.1']}
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Remark 2.5
  • ...and 15 more