On the prescribed scalar curvature problem on compact manifolds with boundary (Q2708677)

Let \((M,g)\), \(n\geq 3\), be a compact Riemannian manifold with boundary. In previous work [Ann. Math. (2) 136, 1-50 (1992; Zbl 0766.53033); ibid. 139, 749-750 (1994; Zbl 0861.53036)] the author studied the conformal deformation of \(g\) to a scalar flat metric with constant mean curvature on \(\partial M\). In particular, he pointed out that the sign of the first eigenvalue, \(\nu_1\), of a certain Neumann problem is a conformal invariant. It follows that if \(g_1=\varphi_1^{4/(n-2)}g\), where \(\varphi_1\) is the first eigenfunction of this problem (with \(\nu_1\) finite), then for any given function \(K\) on \(M\) there is a metric conformally equivalent to \(g_1\) (and hence also to \(g\)) with scalar curvature \(K\), provided there is a smooth positive solution \(u\) of the equation NEWLINE\[NEWLINE \Delta_{g_1} u + {{(n-2)}\over{4(n-1)}} K u^{{n+2}\over{n-2}} = 0 \quad\text{in}\quad M. \tag \(*\) NEWLINE\]NEWLINE Under suitable assumptions on \(K\), the author proves the existence of a positive solution of \((*)\) with either positive Dirichlet data on \(\partial M\), or with a mixed Dirichlet-Neumann boundary condition. In the first case the result extends work of \textit{L. Caffarelli} and \textit{J. Spruck} [Indiana Univ. Math. J. 39, 1-18 (1990; Zbl 0717.35028)] on domains in \({\mathbb R}^n\) with \(K=1\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00042].