On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions

TL;DR

This paper extends Phillips's 2002 result by showing that certain elliptic systems in divergence form in two dimensions either lack non-trivial one-homogeneous solutions or only admit affine solutions, even when the system depends smoothly on spatial variables.

Contribution

It generalizes the non-existence result to systems with spatial variable dependence and includes solutions in Sobolev spaces, broadening the scope of previous findings.

Findings

Non-existence of non-trivial one-homogeneous solutions under certain conditions

Existence of a singular solution violating the main theorem's hypotheses

Extension of Phillips's result to systems with spatial dependence

Abstract

We extend the result of D. Phillips (On one-homogeneous solutions to elliptic systems in two dimensions. C. R. Math. Acad. Sci. Paris 335 (2002), no. 1, 39-42) by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to one-homogeneous solutions belonging to the Sobolev space H^{1}, from which his treatment of Lipschitz solutions follows as a special case. A singular one-homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method which has links to nonlinear elasticity.