Special Session 47: Singular limits in elliptic and parabolic PDEs

Clustering phenomena in low dimensions for a boundary Yamabe problem

Giusi Vaira
University of Bari Aldo Moro
Co-Author(s):    Sergio Cruz-Blazquez and Angela Pistoia
We consider the classical geometric problem of prescribing scalar and boundary mean curvature via conformal deformation of the metric on a n-dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if n=3 all the blow-up points are isolated and simple. In this work we prove that this is not true anymore in low dimensions (that is n=4, 5, 6, 7). In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and minimizes the norm of the trace-free second fundamental form of the boundary.