Reese Harvey

Nonuniqueness of solutions to the Lagrangian mean curvature equation

We resolve a question posed by Harvey and Lawson concerning the uniqueness of continuous viscosity solutions to the Dirichlet problem for the Lagrangian mean curvature equation on the unit ball with continuous boundary data. For every dimension $n\geq 2$, we construct a continuous phase $\psi \colon \overline{B_1}\to(-n\frac{\pi}{2},n\frac{\pi}{2})$ and continuous boundary data $g \colon \partial B_1\to \mathbb{R}$ for which the Dirichlet problem \[ \sum_{i=1}^n \arctan\lambda_i(D^2u)=\psi(x)\quad\text{in }B_1, \qquad u=g\quad\text{on }\partial B_1, \] admits a continuum of distinct continuous viscosity solutions.