Blow-up analysis of large conformal metrics with prescribed Gaussian and geodesic curvatures (Q6644833)

This paper discusses the following problem: given a compact Riemannian surface \((M, g)\) with a nonempty boundary and having negative Euler characteristic and given two smooth non-constant functions \(f\in C^\infty(M)\) and \(h\in C^\infty(\partial M)\) with \(\max f = \max h = 0\), what are the qualitative properties of \[-\Delta_g+K_g=fe^{2u}\] on \(M\) and \[\frac{\partial u}{\partial \nu_g}+\kappa_g=h e^u\] on \(\partial M\), where \(\nu_g\) is the outward unit normal to \(\partial M\) and \(\kappa_g\) is the geodesic curvature of the boundary \(\partial M\) with respect to \(g\). The authors obtain that for small enough \(\lambda, \mu>0\) there exist two distinct conformal metrics \(g_{\lambda, \mu}=e^{2u_{\lambda,\mu}}g\) and \(g^{\lambda, \mu}=e^{2u^{\lambda,\mu}}g\) with prescribed sign-changing Gaussian and geodesic curvature equal to \(f + \mu\) and \(h + \lambda\). The asymptotic behaviour for \(\lambda, \mu\to 0\) is also investigated.