An improvement for the sharp Adams inequalities in bounded domains and whole space ℝⁿ

2016/04/26 by Van Hoang Nguyen, Nguyen, Van Hoang
Computer Science · Mathematics · #26D10 #35B33 #46E30 #46E35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1604.07526

openalex publication_date 2016/04/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We prove an improvement for the sharp Adams inequality in W^{m,\frac nm}₀(Ω) where Ω is a bounded domain in ℝⁿ inspired by Lions Concentration--Compactness principle for the sharp Moser--Trudinger inequality. Our method gives an alternative approach to a Concentration--Compactness principle in W^{m,\frac nm}₀(Ω) recently established by do Ó and Macedo. Moreover, when m is odd, we obtain an improvement for their result by finding the best exponent in this principle. Our approach also is successfully applied to whole space ℝⁿ to establish an improvement for the sharp Adams inequalities in W^{m,\frac nm}(ℝⁿ) due to Ruf, Sani, Lam, Lu, Fontana and Morpurgo. This type of improvement is still unknown, in general, except the special case m=1 due to do Ó, de Souza, de Medeiros and Severo. Our method is a further development for the method of \check\rm Cerny, Cianchi and Hencl combining with some estimates for the decreasing rearrangement of a function in terms of the one of its higher order derivatives.

Citations

• Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn

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