On local solvability of nonlinear elliptic partial differential systems of principle type: the second order

TL;DR

This paper establishes local solvability results for second-order nonlinear elliptic PDE systems of principal type, demonstrating the existence of non-radial solutions and applications to harmonic maps and geometric objects.

Contribution

It provides new existence theorems for nonlinear elliptic systems and their applications in differential geometry, especially for harmonic maps and Beltrami-Laplace structures.

Findings

Existence of non-radial solutions for nonlinear PDE systems

Local existence of harmonic maps with prescribed tangent planes

Local existence of geometric objects defined by Beltrami-Laplace

Abstract

We prove results on solvability of nonlinear elliptic partial differential systems of principle type of second order. They are consequences of existence of non-radial solutions for nonlinear partial differential systems of Poisson type. As applications to geometry, we prove the exsitence of local harmonic maps with given tangent plane at a point between any Riemannan manifolds. More generally geometric objects defined by Beltrami-Laplace always exist locally.