Compactness for conformal scalar-flat metrics on umbilic boundary manifolds (Q2201739)

The main goal of the authors is to describe the compactness property of positive solutions of the following boundary value problem \[ \begin{cases} L_gu=0 &\text{in}~~~ M\\ B_gu+(n-2)u^p=0&\text{on}~~~\partial M \end{cases}\tag{1} \] where \(M\) is a smooth, \(n\)-dimensional Riemannian manifold of positive type with regular umbilic boundary \(\partial M\), \(L_gu=\Delta_gu-\frac{n-2}{4(n-1)}R_gu, B_gu=-\frac{\partial}{\partial\nu}u-\frac{n-2}{2}h_gu\) are respectively the conformal Laplacian and the conformal boundary operator, \(R_g\) is the scalar curvature of the manifold, \(h_g\) is the mean curvature of the \(\partial M\) and \(\nu\) is the outer normal with respect to \(\partial M\). They consider the case \(n\geq 8\) with non-vanishing Weyl tensor \(W_g\). Under these assumptions, given \(\bar p>1\), they prove that there is a positive universal constant \(C\) such that for any \(p\in[\bar p,\frac{n}{n-2}]\) and any positive solution \(u\) of problem (1) the following inequalities hold \[ C^{-1}\leq u\leq C,~~~\Vert u\Vert_{C^{2,\alpha}(M)}\leq C, \] for some \(0<\alpha<1\). The authors also give some survey remarks. The technical equipment of the paper is not simple, there are a lot of auxiliary results.