# On Homogenization Problems with Oscillating Dirichlet Conditions in Space-Time Domains

## TL;DR

This paper establishes the homogenization of fully nonlinear parabolic equations with oscillating Dirichlet boundary conditions in space-time domains, highlighting differences from elliptic cases regarding boundary data continuity.

## Contribution

It proves the existence of continuous homogenized boundary data on certain boundary parts and reveals potential discontinuities even with symmetric operators.

## Key findings

- Homogenized boundary data exists and is continuous on flat moving boundaries.
- Discontinuities in homogenized boundary data can occur even with rotation/reflection invariant operators.
- The results extend homogenization theory to more general space-time domains for parabolic equations.

## Abstract

We prove the homogenization of fully nonlinear parabolic equations with periodic oscillating Dirichlet boundary conditions on certain general prescribed space-time domains. It was proved in [9,10] that for elliptic equations, the homogenized boundary data exists at boundary points with irrational normal directions, and it is generically discontinuous elsewhere. However for parabolic problems, on a flat moving part of the boundary, we prove the existence of continuous homogenized boundary data $\bar{g}$. We also show that, unlike the elliptic case, $\bar{g}$ can be discontinuous even if the operator is rotation/reflection invariant.