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The singular anisotropic Adams' type inequality in $\mathbb{R}^n$

Tao Zhang, Meixia Li, Fan Yang, Chunqin Zhou

TL;DR

This work establishes singular anisotropic Adams-type inequalities under the Finsler-Laplacian framework, proving best constants and exact growth in $\mathbb{R}^n$ and extending the results to bounded domains $\Omega$. The authors leverage anisotropic rearrangement, the Wulff shape, and a suite of sharp lemmas to control the exponential-type nonlinearity with a singular weight $F^o(x)^{-\beta}$ under $\|\Delta_F u\|_{\frac{n}{2}}\le1$, achieving the critical constant $\lambda_n\bigl(1-\frac{\beta}{n}\bigr)$. A key contribution is the introduction and use of an optimal descending growth mechanism in the anisotropic setting, which yields exact-growth inequalities and sharpness via carefully designed test functions. The results generalize Adams-type inequalities to anisotropic and singular contexts and lay groundwork for potential extremal analysis and applications in geometric PDEs.

Abstract

In this paper, using anisotropic rearrangement techniques, we first establish the best constants for the singular anisotropic Adams' type inequality with exact growth in $\mathbb{R}^n$. Furthermore, by the same trick, we also prove the singular anisotropic Adams' type inequality on bounded domain $Ω\subset \mathbb{R}^n$.

The singular anisotropic Adams' type inequality in $\mathbb{R}^n$

TL;DR

This work establishes singular anisotropic Adams-type inequalities under the Finsler-Laplacian framework, proving best constants and exact growth in and extending the results to bounded domains . The authors leverage anisotropic rearrangement, the Wulff shape, and a suite of sharp lemmas to control the exponential-type nonlinearity with a singular weight under , achieving the critical constant . A key contribution is the introduction and use of an optimal descending growth mechanism in the anisotropic setting, which yields exact-growth inequalities and sharpness via carefully designed test functions. The results generalize Adams-type inequalities to anisotropic and singular contexts and lay groundwork for potential extremal analysis and applications in geometric PDEs.

Abstract

In this paper, using anisotropic rearrangement techniques, we first establish the best constants for the singular anisotropic Adams' type inequality with exact growth in . Furthermore, by the same trick, we also prove the singular anisotropic Adams' type inequality on bounded domain .

Paper Structure

This paper contains 5 sections, 10 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Some crucial Lemmas
  4. Proof of Theorem \ref{['main-thm-1']}
  5. Proof of Theorem \ref{['main-thm-2']}

Key Result

Theorem 1.1

Let $n\geq3$, $0\leq\beta<n$ and $0<\lambda\leq\lambda_n(1-\frac{\beta}{n})$, where $\lambda_n=n^\frac{n}{n-2}(n-2)^\frac{n}{n-2}\kappa_{n}^\frac{2}{n-2}$. Then there exists a constant $C=C(\lambda,\beta,n)>0$ such that for all $u\in W^{2,\frac{n}{2}}(\mathbb{R}^n)$ with $\|\Delta_F u\|_\frac{n}{2}\ where $\Phi(t)=e^t-\sum\limits_{j=0}^{j_\frac{n}{2}-2}\frac{t^j}{j!}$, $j_\frac{n}{2}=\min\{j\in\ma

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • ...and 2 more