ABSTRACT
Abstract . This paper is concerned 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 UJ of a smooth compact manifold of IRN. The Speed Method approach is used: the flow transformation of a vector field V(t,.) changes that open subset in ujt; in general that perturbed set is no more a subset of T. The relative boundary 7 of UJ is smooth enough and y(uJt) is the solution in ujt of the Dirichlet problem with zero boundary value on 7 t. The shape tangential derivative is characterized as being the solution of a similar boundary value problem; that element y'r{uJ\ V) can be simply defined by the restriction to UJ of y — Vr?/. V where y is the material derivative of y. The study splits in two parts whether the relative boundary 7 of UJ is empty or not. In both cases the shape derivative depends on the deviatone part of the second fundamental form of the surface, on the field V(0) through its normal component on UJ and on the tangential field 1^(0)r through its normal component on 7.
