A SHARP TRACE ADAMS INEQUALITY IN R4AND EXISTENCE OF THE EXTREMALS

L. U. Chen; Guozhen Lu; Maochun Zhu

doi:10.2140/apde.2026.19.413

A SHARP TRACE ADAMS INEQUALITY IN R⁴AND EXISTENCE OF THE EXTREMALS

L. U. Chen
, Guozhen Lu
, Maochun Zhu

Beijing Institute of Technology
University of Connecticut
Nanjing University of Science and Technology

Research output: Contribution to journal › Article › peer-review

Abstract

Let R⁴ beaboundeddomainwithsmooth boundary ∂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: (formula presented) holds if and only if α ≤ 12π². Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of biharmonic equations on the half-space R⁴₊. With this classification result, we can show that S(12π²) 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.

Original language	English
Pages (from-to)	413-448
Number of pages	36
Journal	Analysis and PDE
Volume	19
Issue number	3
DOIs	https://doi.org/10.2140/apde.2026.19.413
Publication status	Published - 2026
Externally published	Yes

Keywords

Adams trace inequality
blow-up analysis
capacity estimate
extremal function

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10.2140/apde.2026.19.413

Cite this

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title = "A SHARP TRACE ADAMS INEQUALITY IN R4AND EXISTENCE OF THE EXTREMALS",

abstract = "Let R4 beaboundeddomainwithsmooth boundary ∂Ω . In this paper, we establish the following sharp form of the trace Adams inequality in W2,2(Ω) with zero mean value and zero Neumann boundary condition: (formula presented) holds if and only if α ≤ 12π2. Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of biharmonic equations on the half-space R4+. With this classification result, we can show that S(12π2) is attained by using the blow-up analysis and capacitary estimate. As an application, we prove a sharp trace Adams{\textendash}Onofri-type inequality in general four-dimensional bounded domains with smooth boundary.",

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AU - Chen, L. U.

AU - Lu, Guozhen

AU - Zhu, Maochun

N1 - Publisher Copyright: © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). https://creativecommons.org/licenses/by/4.0/. Open Access made possible by subscribing institutions via Subscribe to Open.

PY - 2026

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N2 - Let R4 beaboundeddomainwithsmooth boundary ∂Ω . In this paper, we establish the following sharp form of the trace Adams inequality in W2,2(Ω) with zero mean value and zero Neumann boundary condition: (formula presented) holds if and only if α ≤ 12π2. Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of biharmonic equations on the half-space R4+. With this classification result, we can show that S(12π2) 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.

AB - Let R4 beaboundeddomainwithsmooth boundary ∂Ω . In this paper, we establish the following sharp form of the trace Adams inequality in W2,2(Ω) with zero mean value and zero Neumann boundary condition: (formula presented) holds if and only if α ≤ 12π2. Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of biharmonic equations on the half-space R4+. With this classification result, we can show that S(12π2) 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.

KW - Adams trace inequality

KW - blow-up analysis

KW - capacity estimate

KW - extremal function

UR - https://www.scopus.com/pages/publications/105033043566

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DO - 10.2140/apde.2026.19.413

M3 - Article

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EP - 448

JO - Analysis and PDE

JF - Analysis and PDE

IS - 3

ER -

A SHARP TRACE ADAMS INEQUALITY IN R⁴AND EXISTENCE OF THE EXTREMALS

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Keywords

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