Shape stability of a quadrature surface problem in infinite Riemannian manifolds

Abstract

In this paper, we give a simple control on how an optimal shape can be characterized. The framework of Riemannian manifold of infinite dimension is essential. And the covariant derivative plays a key role in the computation and in the analysis of qualitative properties from the shape hessian. The control depends only on the mean curvature of the domain which is a minimum or a critical point. In the two dimensional case, the Gauss Bonnet theorem gives a control in the setting of the algorithm for the minimum. Mathematical classification subject: 49Q10, 53B20