On the well-posedness of global fully nonlinear first order elliptic systems

Abstract

In the very recent paper [15], the second author proved that for any f∈L2⁢(ℝn,ℝN){f\in L^{2}(\mathbb{R}^{n},\mathbb{R}^{N})}, the fully nonlinear first order system F⁢(⋅,D⁢u)=f{F(\,\cdot\,,\mathrm{D}u)=f} is well posed in the so-called J. L. Lions space and, moreover, the unique strong solution u:ℝn→ℝN{u\colon\mathbb{R}^{n}\rightarrow\mathbb{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 [15] by introducing a new strictly weaker ellipticity condition and by proving well-posedness in the same “energy” space.

Keywords

• cauchy–riemann equations
• fully nonlinear systems
• elliptic first order systems
• calculus of variations
• campanato’s near operators
• cordes’ condition
• compensated compactness
• baire category method
• convex integration
• 35j46
• 35j47
• 35j60
• 35d30
• 32a50
• 32w50