Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)



First Advisor

Donatella Danielli

Committee Chair

Donatella Danielli

Committee Member 1

Daniel Phillips

Committee Member 2

Monica Torres

Committee Member 3

Aaron Yip


In this work the regularity of solutions and of the free boundary for a type of parabolic free boundary problem with variable coefficients is proved. After introducing the problem and its history in the introduction, we proceed in Chapter 2 to prove the optimal Lipschitz regularity of viscosity solutions under the main assumption that the free boundary is Lipschitz. In Chapter 3, we prove that Lipschitz free boundaries possess a classical normal in both space and time at each point and that this normal varies with a Hölder modulus of continuity. As a consequence, the viscosity solution is in fact a classical solution to the problem.