The prescribed curvature flow on the disc

For given functions $f$ and $j$ on the disc $B$ and its boundary $\mathrm{\partial }B=S^1$, we study the existence of conformal metrics $g=e^{2u}{g}_0$ with prescribed Gauss curvature $K_g=f$ and boundary geodesic curvature $k_g=j$. Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz (2018), we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work Struwe (2005) on the prescribed curvature problem on the sphere, we are able to exhibit a $2$-dimensional shadow flow for the center of mass of the evolving metrics from which we obtain existence results complementing the results recently obtained by Ruiz (2021) by degree-theory.