2024/05/16 by Franz Gmeineder, Gmeineder, Franz, Peter Lewintan +3
Engineering · Mathematics · #26D10 #35A23 #35Q74/35Q75 #46E35 #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2405.10349
openalex publication_date 2024/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give sharp conditions for the limiting Korn-Maxwell-Sobolev inequalities ‖ P‖_Wk-1,(n)/(n-1)(ℝn)≤ c(‖\mathscrA[P]‖_Wk-1,(n)/(n-1)(ℝn)+‖\mathbbBP‖L1(ℝn)) to hold for all P∈ Cc∞(ℝn;V), where \mathscrA is a linear map between finite dimensional vector spaces and \mathbbB is a k-th order, linear and homogeneous constant-coefficient differential operator. By the appearance of the L1-norm of the differential expression \mathbbBP on the right-hand side, such inequalities generalise previously known estimates to the borderline case p=1, and thereby answer an open problem due to Müller, Neff and the second author (Calc. Var. PDE, 2021) in the affirmative.