The Yamabe problem on manifolds with boundary: Existence and compactness results
Let \((M,g)\) be a smooth, \(n\)-dimensional (\(n\geq 3\)) compact Riemannian manifold without boundary. The well known Yamabe problem asks whether there is a metric \(\overline g\) that is pointwise conformal to \(g\) and has constant scalar curvature. Analytically the problem reduces to finding a positive solution \(u\) of the equation \((*)\) \(-(\Delta_g - c(n)R_g)u = \overline R u^{(n+2)/(n-2)}\) in \(M\), where \(\Delta_g\) and \(R_g\) are the Laplace-Beltrami operator and the scalar curvature, respectively, of the metric \(g\), \(c(n)=(n-2)/4(n-1)\) and \(\overline R = 0\) or \(\pm n(n-2)\). The existence of a positive solution was established in certain cases by N.S.~Trudinger in 1968, in some other cases by T.~Aubin in 1976, and finally, in the remaining cases, by R.~Schoen in 1984. More recently, \textit{R. Schoen} [Pitman Monogr. Surv. Pure Appl. Math. 52, 311-320 (1991; Zbl 0733.53021)] proved the compactness in \(C^2(M)\) of the set of positive solutions of \((*)\) in the case that \((M,g)\) is locally conformally flat but not conformally equivalent to the standard sphere. Moreover, he showed that the total degree of all solutions is \(-1\). In the same paper, he also gave an indication of the proof of the same result for general manifolds. These results are much stronger than the existence of a positive solution of \((*)\). If one considers the Yamabe problem for compact manifolds with boundary, one needs to add a boundary condition to \((*)\). A suitable boundary condition is \((**)_c\) \(({{\partial}\over{\partial\nu}} + {{n-2}\over 2} h_g)u = c u^{n/(n-2)}\) on \(\partial M\) for some \(c\in{\mathbb R}\), where \(\nu\) is the outer unit normal to \(\partial M\) and \(h_g\) is the mean curvature of \(\partial M\) with respect to the inner normal. This means that \(\partial M\) has constant mean curvature \(c\) in the new metric \(\overline g = u^{4/(n-2)}g\). \textit{J. F. Escobar} [J. Differ. Geom. 35, 21-84 (1992; Zbl 0771.53017)] has shown that a large class of compact manifolds with boundary are conformally equivalent to one of constant scalar curvature and boundary of zero mean curvature. In further work [Indiana Univ. Math. J. 45, 917-943 (1996; Zbl 0881.53037)] he showed that there exist \(c^{+}>0\) and \(c^{-}<0\) such that \((*)\), \((**)_c\) has a positive solution for \(c=c^{+}\) or \(c^{-}\). The main result of this paper is an analogue for manifolds with boundary of the compactness result of Schoen mentioned above. More precisely, the authors establish a bound \({1\over C}\leq u(x) \leq C\), \(\|u\|_{C^2(M)} \leq C\), with \(C=C(M,g,\overline c)\), for a class of compact Riemannian manifolds with boundary of the set of positive solutions \(u\) of \((*)\), \((**)_c\) with \(|c|\leq \overline c\) for any given \(\overline c\). A slightly stronger result is also obtained: one can replace the exponents \((n+2)/(n-2)\) and \(n/(n-2)\) in \((*)\), \((**)_c\) by \(p\) and \((p+1)/2\) respectively, for \((n+2)/(n-2) -\delta_0 \leq p \leq (n+2)/(n-2)\) for some \(\delta_0>0\) depending on \(M,g\) and \(\overline c\). The proof proceeds by an analysis of the possible blow-up behaviour of solutions using a Liouville type theorem, coupled with the positive mass theorem of Schoen and Yau.
scientific article; zbMATH DE number 1425244
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