A sharp Adams inequality in dimension four and its extremal functions

2017/01/28 by Van Hoang Nguyen, Nguyen, Van Hoang · 1 citation
Computer Science · Mathematics · #46E35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1701.08249

openalex publication_date 2017/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be a smooth oriented bounded domain in \mathbb R⁴, H₀²(Ω) be the Sobolev space, and λ₁(Ω)= inf \‖Δu‖₂² : u∈ H₀²(Ω), ‖u‖₂ =1\ be the first eigenvalue of the bi-Laplacian operator Δ² on Ω. For α∈ [0,λ₁(Ω)), we define ‖u‖_2,α² = ‖Δu‖₂² - α‖u‖₂², for u ∈ H₀²(Ω). In this paper, we will prove the following inequality sup_u∈ H₀²(Ω), ‖u‖_2,α ≤ 1 ∫_Ω e^{32 π2 u(x)2} dx lt; ∞. This strengthens a recent result of Lu and Yang \citeLuYang. We also show that there exists a function u^*∈ H₀²(Ω)∩ C⁴(Ω) such that ‖u^*‖_2,α =1 and the supremum above is attained by u^*. Our proofs are based on the blow-up analysis method.

Citations

• Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn
• Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space

Cited by

• The sharp Adams type inequalities in the hyperbolic spaces under the Lorentz-Sobolev norms

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