Ill-posedness of the Dirichlet problem for 2D Lagrangian mean curvature equation

We investigate the Dirichlet problem of the two dimensional Lagrangian mean curvature equation in a bounded domain. Infinitely many $C^{1,\alpha }(\alpha \in (0,\frac{1}{5}))$ very weak solutions are built through Nash-Kuiper construction. Moreover, we note there are infinitely many $C^{1,\alpha }$ very weak solutions that can not be improved to be $C^{2,\alpha }$.