Critical Points and Level Sets of Grushin-Harmonic Functions in the Plane

This paper concerns the critical points and the level sets of solutions of the Grushin equation in the plane. After exactly establishing descriptions about the critical points of the homogeneous Gruhin-harmonic polynomials and investigating the local geometric properties of the level sets near these critical points, we prove that the critical points of solutions of the Grushin equation are isolated and each critical point has finite multiplicity. We further estimate the numbers of interior critical points of solutions of the Dirichlet boundary value problem.

Automated summary

This paper investigates the critical points and level sets of solutions of the Grushin equation in the plane. It proves that the critical points of solutions are isolated with finite multiplicity, and further estimates the number of interior critical points in the Dirichlet boundary value problem.