Large conformal metrics with prescribed Gaussian and geodesic curvatures

Abstract

We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk. This is equivalent to solving the following P.D.E. $$\begin{aligned} {\left\{ \begin{array}{ll}-\Delta u=2K(z)e^u&{}\hbox {in}\;\mathbb {D}^2,\\ \partial _\nu u+2=2h(z)e^\frac{u}{2}&{}\hbox {on}\;\partial \mathbb {D}^2,\end{array}\right. } \end{aligned}$$ where K, h are the prescribed curvatures. We construct a family of conformal metrics with curvatures $$K_\varepsilon ,h_\varepsilon $$ converging to K, h respectively as $$\varepsilon $$ goes to 0, which blows up at one boundary point under some generic assumptions.