Manifold derivative in the Laplace-Beltrami equation

This paper is concerned with the derivative of the solution with respect to the manifold, more precisely with the shape tangential sensitivity analysis of the solution to the Laplace-Beltrami boundary value problem with homogeneous Dirichlet boundary conditions. The domain is an open subset omega of a smooth compact manifold Gamma of R(N). The flow of a vector field V(t,.) changes omega in omega(t) (and Gamma in Gamma(t)). The relative boundary gamma(t) of omega(t) in Gamma(t) is smooth enough and gamma(omega(t)) is the solution in omega(t) of the Laplace-Dirichlet problem with zero boundary value on gamma(t). The shape tangential derivative is characterized as being the solution of a similar non homogeneous boundary value problem; that element gamma(Gamma)'(omega; V) can be simply defined by the restriction to omega of <(gamma)over dot>-V(Gamma)gamma.V where <(gamma)over dot> is the material derivative of gamma and del(Gamma)gamma is the tangential gradient of gamma. The study splits in two parts whether the relative boundary gamma of omega is empty or not. In both cases the shape derivative depends on the deviatoric part of the second fundamental form of the surface, on the field V(0) through its normal component on omega and on the tangential field V(0)(Gamma) through its normal component on the relative boundary gamma. We extend the structure results for the shape tangential derivative making use of intrinsic geometry approach and intensive use of extension operators. (C) 1997 Academic Press.