# Research Interests

I’m interested in Partial Differential Equations (PDEs). Most of my research has been in free boundary and obstacle problems. I’m trying to keep this webpage as accessible as possible, so if you don’t know what a free boundary problem is, or a PDE for that matter, just click on the page listed above and find out.

Speaking technically, my work has focused on the regularity of free boundaries to a Bernoulli-type free boundary problem with a parabolic variable coefficient operator. My main results for this problem are the optimal Lipschitz regularity of the solution, and that Lipschitz free boundaries are $C^{1,\alpha}$. If you’re really interested in this, you can look up my papers listed below.

I also have an ongoing project involving a thin obstacle-type problem as well as a problem best classified as a shape optimization problem with a parabolic state equation.

If you’re interested in anything I’ve done, feel free to contact me. My contact info is on the ‘About Me’ page.

### Papers:

Regularity of Solutions to a Bernoulli-type Parabolic Free Boundary
Nov 2016 Problem with Variable Coefficients (Backing, T.) Nonlinear Analysis:
Theory, Methods & Applications, Volume 146, November 2016.

Regularity of the Free Boundary for a Bernoulli-Type Parabolic Problem
Nov 2016 with Variable Coefficients (Backing, T.) Nonlinear Analysis: Theory,
Methods & Applications, Volume 146, November 2016.

### Preprints:

Existence, Uniqueness and Regularity of Solutions for a Penalized
Boundary Obstacle Problem (Backing, T., Danielli, D.)