CRITICAL POINTS OF SOLUTIONS FOR THE MEAN CURVATURE EQUATION IN STRICTLY CONVEX AND NONCONVEX DOMAINS

In this paper, we mainly investigate the set of critical points associated to solutions of the mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce that the mean curvature equation has exactly one nondegenerate critical point in a smooth, bounded and strictly convex domain of R-n (n >= 2). Secondly, we study the geometric structure about the critical set K of solutions u for the constant mean curvature equation in a concentric (respectively an eccentric) spherical annulus domain of R-n (n >= 3), and deduce that K consists (respectively does not consist) of a rotationally symmetric critical closed surface S. In fact, in an eccentric spherical annulus domain, K is made up of finitely many isolated critical points (p(1), p(2), ..., p(l)) on an axis and finitely many rotationally symmetric critical Jordan curves (C-1, C-2, ...,C-k) with respect to an axis.