I study elliptic PDE and elliptic systems, primarily using viscosity techniques, geometric analysis, geometric measure theory, and the calculus of variations. Lately, I’ve been thinking about the regularity & existence theory for the minimal surface system, which is the Euler-Lagrange system for the area functional in high codimensions. When the codimension is one, the minimal surface system reduces to the minimal surface equation.
Papers
- Dimler, B. Partial regularity for Lipschitz solutions to the minimal surface system. Calc. Var. Partial Differential Equations 60, 260 (2023). [.pdf]
M.S. Thesis
While in the M.S. program in mathematics at UTSA, I studied a conjecture first posed by Enrico De Giorgi for a class of nonlinear PDE which includes the Allen-Cahn equation. In particular, I studied recent results pertaining to the conjecture for both the classical and fractional Laplacians. You can find my thesis here.