Manifold Derivative in the Laplace–Beltrami Equation

Abstract

This paper is concerned with the derivative of the solution with respect to the manifold, more precisely with theshape tangentialsensitivity analysis of the solution to the Laplace–Beltrami boundary value problem with homogeneous Dirichlet boundary conditions. The domain is an open subsetωof a smooth compact manifoldΓof RN. The flow of a vector fieldV(t,·) changesωinωt(andΓinΓt). The relative boundaryγtofωtinΓtis smooth enough andy(ωt) is the solution inωtof the Laplace–Dirichlet problem with zero boundary value onγt. The shape tangential derivative is characterized as being the solution of a similar non homogeneous boundary value problem; that elementy′Γ(ω;V) can be simply defined by the restriction toωofy−∇Γy·Vwhereyis the material derivative ofyand ∇Γyis the tangential gradient ofy. The study splits in two parts whether the relative boundaryγofωis empty or not. In both cases the shape derivative depends on the deviatoric part of the second fundamental form of the surface, on the fieldV(0) through its normal component onωand on the tangential fieldV(0)Γthrough its normal component on the relative boundaryγ. We extend the structure results for theshape tangentialderivative making use ofintrinsic geometryapproach and intensive use of extension operators.