Manifold Derivative in the Laplace–Beltrami Equation

✍ Scribed by F.R. Desaint; Jean-Paul Zolésio

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 558 KB
• Volume: 151
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3130

✦ Synopsis

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 | of a smooth compact manifold 1 of R N . The flow of a vector field V(t, } ) changes | in | t (and 1 in 1 t ). The relative boundary # t of | t in 1 t is smooth enough and y(| t ) is the solution in | t of the Laplace Dirichlet problem with zero boundary value on # t . The shape tangential derivative is characterized as being the solution of a similar non homogeneous boundary value problem; that element y$ 1 (|; V ) can be simply defined by the restriction to | of y* &{ 1 y } V where y* is the material derivative of y and { 1 y is the tangential gradient of y. The study splits in two parts whether the relative boundary # of | 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 | and on the tangential field V(0) 1 through its normal component on the relative boundary #. We extend the structure results for the shape tangential derivative making use of intrinsic geometry approach and intensive use of extension operators.