On the Well-Posedness of Global Fully Nonlinear First Order Elliptic Systems

2015/11/09 by Hussien Abugirda, Abugirda, Hussien, Nikos Katzourakis +1
Engineering · Mathematics · #32A50 #32W50 #35J47 #35J60 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Primary 35J46 #Secondary 35D30 #Stability and Controllability of Differential Equations #math.AP #msc:32A50 #msc:32W50 #msc:35D30 #msc:35J46 #msc:35J47 #msc:35J60

paper · pdf · doi:10.48550/arxiv.1511.02809

Journal: Advances in Nonlinear Analysis, 13 pages

openalex publication_date 2015/11/09 · arxiv created 2016/04/07 · arxiv updated 2016/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the very recent paper [K1], the second author proved that for any f∈ L²(ℝⁿ,ℝ^N), the fully nonlinear first order system F(⋅,D u) =f is well posed in the so-called J.L. Lions space and moreover the unique strong solution u:ℝⁿ\longrightarrow ℝ^N to the problem satisfies a quantitative estimate. A central ingredient in the proof was the introduction of an appropriate notion of ellipticity for F inspired by Campanato's classical work in the 2nd order case. Herein we extend the results of [K1] by introducing a new strictly weaker ellipticity condition and by proving well posedness in the same "energy" space.

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