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Jeremy Wall

Publications and source records attributed to Jeremy Wall.

4 recordsLinked to original sources
arXiv2025

Doubling and the two-dimensional critical valued Lagrangian phase

In this paper, we establish interior Hessian and gradient estimates for the two-dimensional Lagrangian mean curvature equation when the phase changes signs, provided the gradient of the phase vanishes along its zero set. At the critical phase in two dimensions, the Jacobi inequality degenerates, preventing the use of higher-dimensional methods to obtain Hessian estimates. To address this difficulty, we introduce a modified doubling technique that applies to degenerate Jacobi inequalities and yields interior estimates.

math.AP↗
arXiv2025

Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase

We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., $|\Theta|\geq (n-2)\tfrac{\pi}{2}$, and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct $C^{\alpha}$ singular viscosity solutions to show that criticality of the phase is necessary, and that these conditions cannot be removed in dimension one. We also introduce a new method for proving $C^{2,\alpha}$ estimates by exponentiating the arctangent operator into a concave one when $|\Theta|\geq (n-2)\tfrac{\pi}{2}$ and $n>2$.

math.AP↗