Existence and Uniqueness of Global Strong Solutions to Fully Nonlinear Second Order Elliptic Systems

TL;DR

This paper establishes the existence and uniqueness of strong global solutions for a class of fully nonlinear second order elliptic systems on \\mathbb{R}^n, introducing a weaker ellipticity condition and employing Fourier transform and perturbation methods.

Contribution

It introduces a new weaker ellipticity condition for fully nonlinear elliptic systems and proves global existence and uniqueness of solutions in a tailored Sobolev space.

Findings

Proved solvability of the nonlinear PDE system in a Sobolev energy space.

Established a uniqueness estimate for solutions.

Provided counterexamples and stability results for the systems.

Abstract

We consider the problem of existence and uniqueness of strong a.e. solutions $u: \mathbb{R}^n \longrightarrow \mathbb{R}^N$ to the fully nonlinear PDE system \[\label{1} \tag{1} F(\cdot,D^2u ) \,=\, f, \ \ \text{ a.e. on }\mathbb{R}^n, \] when $ f\in L^2(\mathbb{R}^n)^N$ and $F$ is a Carath\'eodory map. \eqref{1} has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato's ellipticity condition on $F$. By introducing a new much weaker notion of ellipticity, we prove solvability of \eqref{1} in a tailored Sobolev "energy" space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a "perturbation device" which allows to use Campanato's near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form \eqref{1}.