# Holder Regularity up to the Boundary for Critical SQG on Bounded Domains

## TL;DR

This paper proves that solutions to the critical SQG equation in bounded domains are globally Holder continuous up to the boundary, overcoming boundary-related challenges using De Giorgi methods.

## Contribution

It establishes boundary regularity for the critical SQG equation in bounded domains, a problem previously unresolved due to boundary complications.

## Key findings

- Global Holder regularity up to the boundary is achieved.
- De Giorgi techniques are effective in boundary regularity analysis.
- The boundary introduces specific challenges that are addressed in the proof.

## Abstract

We consider the dissipative SQG equation in bounded domains, first introduced by Constantin and Ignatova in 2016. We show global Holder regularity up to the boundary of the solution, with a method based on the De Giorgi techniques. The boundary introduces several difficulties. In particular, the Dirichlet Laplacian is not translation invariant near the boundary, which leads to complications involving the Riesz transform.