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

We consider the problem of existence and uniqueness of strong a.e. solutions $u:{\mathbb{R}}^n⟶{\mathbb{R}}^N$ to the fully nonlinear PDE system $$\begin{array}{cccc}& \text{label}1F(\cdot ,D^{2}u)=f,\text{ a.e. on }{\mathbb{R}}^n,& & \text{(1)}\end{array}$$ 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}.