  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.
