Critical points of solutions of degenerate elliptic equations in the plane

TL;DR

This paper investigates the critical points of solutions to degenerate elliptic equations in the plane, showing they cannot be isolated and deriving an Euler-Lagrange equation using approximation and viscosity solutions.

Contribution

It establishes the non-isolation of critical points for minimizers of certain convex functionals and derives associated Euler-Lagrange equations via approximation schemes.

Findings

Critical points of smooth minimizers are not isolated.

An Euler-Lagrange equation for the minimizer is derived.

A pairing with a stream function is constructed through approximation.

Abstract

We study the minimizer u of a convex functional in the plane which is not G\^ateaux-differentiable. Namely, we show that the set of critical points of any C^1-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler-Lagrange equation that u must satisfy. By applying the same approximating scheme, we can pair u with a function v which may be regarded as the stream function of u in a suitable generalized sense.