Shape stability of a quadrature surface problem in infinite Riemannian manifolds

Abstract

In this paper, we revisit a quadradure surface problem in shape optimization. With tools from infinite-dimensional Riemannian geometry, we give simple control over how an optimal shape can be characterized. The framework of the infinite-dimensional Riemannian manifold is essential in the control of optimal geometric shape. The covariant derivative plays a key role in calculating and analyzing the qualitative properties of the shape hessian. Control only depends on the mean curvature of the domain, which is a minimum or a critical point. In the two-dimensional case, Gauss-Bonnet's theorem gives a control within the framework of the algorithm for the minimum.