2024/12/19 by Peter Lewintan, Lewintan, Peter, Paul Stephan +1
Mathematics · #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Advanced Harmonic Analysis Research
paper · pdf · doi:10.48550/arxiv.2412.14866
We focus on Korn-Maxwell-Sobolev inequalities for operators of reduced constant rank. These inequalities take the form ‖P - Π_\mathbbB Π_ker\mathscrA P‖_Wk-1, p^*(ℝn) ≤ c (‖\mathscrA[P]‖_Wk-1, p^*(ℝn) + ‖\mathbbB P‖Lp(ℝn)) for all P ∈ Cc^∞(ℝn; V) , where V is a finite-dimensional vector space, \mathscrA is a linear mapping, and \mathbbB is a constant coefficient homogeneous differential operator of order k . In particular, we can treat the combination (p,\mathscrA,\mathbbB,k)=(1,tr,Curl,1). Our results generalize the techniques from Gmeineder et al. (Math.Mod.Met.Appl.Sci,34:03,2024; arXiv:2405.10349), which exclusively dealt with reduced elliptic operators. In contrast to the reduced ellipticity case, however, the reduced constant rank case necessitates to introduce a correction, namely the projection Π_\mathbbB on the left-hand side of the inequality.