Weighted anisotropic Sobolev inequality with extremal and associated singular problems

2021/07/01 by Kaushik Bal, Bal, Kaushik, Prashanta Garain +1 · 2 citations
Engineering · Mathematics · #35A23 #35B65 #35J75 #35J92 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2107.00336

openalex publication_date 2021/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a given Finsler-Minkowski norm F in ℝ^N and a bounded smooth domain Ω⊂ℝ^N (N≥ 2), we establish the following weighted anisotropic Sobolev inequality S(∫_Ω|u|^q f dx)^(1)/(q)≤(∫_ΩF(∇ u)^p w dx)^(1)/(p), ∀ u∈ W₀^1,p(Ω,w)\leqno(P) where W₀^1,p(Ω,w) is the weighted Sobolev space under a class of p-admissible weights w, where f is some nonnegative integrable function in Ω. We discuss the case 0

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