A sharp trace Adams inequality in ℝ4 and existence of the extremals

Chen, Lu; Lu, Guozhen; Zhu, Maochun

doi:10.2140/apde.2026.19.413

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A sharp trace Adams inequality in $\mathbb{R}^{4}$ and existence of the extremals

Lu Chen, Guozhen Lu and Maochun Zhu

Vol. 19 (2026), No. 3, 413–448

DOI: 10.2140/apde.2026.19.413

Abstract

Let $Ω \subseteq ℝ^{4}$ be a bounded domain with smooth boundary $\partial Ω$ . In this paper, we establish the following sharp form of the trace Adams inequality in $W^{2, 2} (Ω)$ with zero mean value and zero Neumann boundary condition:

S (α) = \sup_{\begin{array}{c} u \in W^{2, 2} (Ω) ∖ {0}, ∥ Δ u ∥_{2} \leq 1 \\ \int_{Ω} u d ‘ x = 0, \frac{\partial u}{\partial ν} |_{\partial Ω} = 0 \end{array}} \int_{\partial Ω} e^{α u^{2}} d σ < \infty

holds if and only if $α \leq 12 π^{2}$ .

Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of biharmonic equations on the half-space $ℝ_{+}^{4}$ . With this classification result, we can show that $S (12 π^{2})$ is attained by using the blow-up analysis and capacitary estimate. As an application, we prove a sharp trace Adams–Onofri-type inequality in general four-dimensional bounded domains with smooth boundary.

Keywords

Adams trace inequality, blow-up analysis, extremal function, capacity estimate

Mathematical Subject Classification

Primary: 46E30

Secondary: 35B33, 35J60

Milestones

Received: 10 October 2023

Revised: 28 November 2024

Accepted: 14 April 2025

Published: 11 March 2026

Authors

Lu Chen
School of Mathematics and Statistics Beijing Institute of Technology Beijing China

Guozhen Lu
Department of Mathematics University of Connecticut Storrs, CT United States

Maochun Zhu
School of Mathematics and Statistics Nanjing University of Science and Technology Nanjing China

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Vol. 19, No. 3, 2026