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Manifold Derivative in the Laplace–Beltrami Equation

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Manifold Derivative in the Laplace–Beltrami Equation Journal of Functional Analysis FU3130 journal of functional analysis 151, 234269 (1997) article no. FU973130 Manifold Derivative in the LaplaceBeltrami Equation* F. R. Desaint Departement de Mathematiques, Universite de Montpellier II, Place EugeÁne Bataillon, F-34095 Montpellier, France and Jean-Paul Zolesio CNRS Institut Non Lineaire de Nice, 1361 route des Lucioles, 06904 Sophia Antipolis Cedex, France; and Centre de Mathematiques Appliquees (CMAÂMEIJE, INRIA), 2004 route des Lucioles, View metadata, citation and similar06902 papersSophia at Antipolis core.ac.uk Cedex, France brought to you by CORE Received March 11, 1997; accepted April 9, 1997 provided by Elsevier - Publisher Connector 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 solu- tion to the LaplaceBeltrami boundary value problem with homogeneous Dirichlet boundary conditions. The domain is an open subset | of a smooth compact manifold N 1 of R . The flow of a vector field V(t, } ) changes | in |t (and 1 in 1t). The relative boundary #t of |t in 1t is smooth enough and y(|t) is the solution in |t of the LaplaceDirichlet problem with zero boundary value on #t . The shape tangential derivative is characterized as being the solution of a similar non homo- geneous 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. 1997 Academic Press Contents. 1. Introduction. 1.1. Shape tangential derivative. 1.2. Material derivative. 1.3. Shape derivative. 2. Structure of the shape tangential derivative. 2.1. Tangential calculus. 2.2. Flow mapping. 2.3. Structure theorem. * The research of the first author has been partially supported by a European FBP Grant. 234 0022-1236Â97 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved. File: DISTIL 313001 . By:DS . Date:24:11:97 . Time:09:41 LOP8M. V8.0. Page 01:01 Codes: 4377 Signs: 2335 . Length: 50 pic 3 pts, 212 mm MANIFOLD DERIVATIVE 235 3. The oriented distance function and extensions. 3.1. Definition and main properties. 3.2. Shape boundary derivative; examples. 4. Material derivative. 4.1. Shape continuity. 4.2. Existence of the material derivative y* . 4.3. Equation satisfied by the transported element yt. 4.4. Boundedness of %t in H 1(1). 4.5. Smoothness result. 5. Shape tangential derivative. 5.1. The shape boundary derivative. 5.2. The shape tangential derivative. 1. INTRODUCTION 1.1. Shape Tangential Derivative We consider a domain 0 in RN and we denote by 1 its boundary which is a smooth compact manifold with empty boundary and is oriented by the unitary normal field n outgoing to 0. The solution y(1) to the elliptic problem associated with the LaplaceBeltrami operator on 1 belongs to 1 1 the Sobolev space H*(1) of functions in H (1) with zero mean on 1. For any subset | of 1, open in 1, we denote by # its relative boundary in 1 and by & the unitary normal field to # contained in the linear tangent space to 1 and outgoing to |. The solution z(|) to the Dirichlet Boundary value problem associated with the LaplaceBeltrami operator on | and Dirichlet 1 homogeneous conditions on # is in the Sobolev space H 0(|). We consider a family of perturbed manifolds 1t in the form 1t=Tt(1) where Tt is a N smooth one parameter family of transformations of R . We set |t=Tt(|) and we consider the perturbed solutions y(1t) and z(|t) associated with the perturbed boundary value problems. We emphasis the fact that any such family of transformations can be regarded as being the flow mapping &1 of its ``speed vector field'' V(t, x)=((Ât) Tt) o(Tt) and we chose that field V as being the deformation parameter. Following [13, 16], the shape boundary derivative of y(1)at1, in the direction V, is defined by y$1 (1, V) =((dÂdt)( y(1t) oTt(V)))|t=0&{1 y(1)}V. We extend that notion to the element z(|), introducing here the concept of shape tangential derivative of z(|)at|, in the direction of the field V through the shape boundary derivative of the extension by zero to the surface 1 of the element z(|). These concepts will be developed in the next section and we shall see in which sense these elements are derivatives. In both situations the shape tangential derivative exists and z$(|; V), y$(1; V) are linear on V. In [13] only the case where # is empty was considered (for different and easier problems). In that situation no tangential variation is involved in the problem (as there is no relative boundary) and the terminology was ``shape boundary derivative''. The present concept of shape tangential derivative contains as specific case that shape boundary derivative concept but is really a generalization of the shape derivative (developed for open domains File: DISTIL 313002 . By:DS . Date:24:11:97 . Time:09:41 LOP8M. V8.0. Page 01:01 Codes: 3376 Signs: 2755 . Length: 45 pic 0 pts, 190 mm 236 DESAINT AND ZOLESIO in RN, see subsection 1.3) as it characterizes the variation of the solution with respect to the boundary # on which a boundary condition is imposed. Following the structure theorem, cf. [16, 17], in the generalized form we establish in the next section, these expressions only depend on the normal component (V(0), n) on | of the vector field V(0) and on the tangential field V1=V&(V, n) n through its normal component (V1(0), &) on #. We make use of intrinsic geometry approach and without use of any local coordinates we give a complete analysis of these shape derivatives from which it appears that the shape sensitivity analysis of the solution in both cases depends on the deviatoric part of the second fundamental form of the surface 1. 1.2. Material Derivative For scalar functions depending on a moving domain we reserve classically (from control theory terminology) the name of shape functional. In the present context we are not concerned with scalar functions but with ``state variables'' (again following the control theory terminology), i.e. with elements such as z(|) ranging in a variable Hilbert space depending on the domain |. Concerning state variable shape sensitivity analysis, the study has been mostly concentrated on the material derivative, see many exam- ples in [13]. That material derivative is useful but cannot be considered as the ``definitive'' concept for several reasons, for example the fact that for some transformations which do not move the domain (``do not change the shape''), the material derivative is not zero. In that approach the deforma- tion mappings are ``smooth enough'', in [1] the author introduces a more general approach. 1.3. Shape Derivative 1.3.1. The Case without Curvature We have 1=RN. In the classical situation of a moving open domain | in RN the element z(|) is extended by zero to an element of some fixed 1 Hilbert space H 0(D) where D is any domain containing the moving family of open domains. For general boundary value problems it has been proved in [5], [16] and [17] that when the material derivative z* (|; V) exists, say in H1(|), then there exists elements Z # C 1([0, {[, H1(D)) such that Z(t,})| = |t z(|t). Moreover we have the property that (Ât) Z(0, })|| is independent on the element Z. That element is, by definition, the shape derivative z$(|; V) of the state variable z at the domain 0 in the direction V. The structure result for the shape boundary derivative states that if the material derivative exists and is linearly and continuously depending on V, then the File: DISTIL 313003 . By:DS . Date:24:11:97 . Time:09:41 LOP8M. V8.0. Page 01:01 Codes: 3174 Signs: 2552 . Length: 45 pic 0 pts, 190 mm MANIFOLD DERIVATIVE 237 shape boundary derivative exists and is given by (Ât) Z(0, })||=z* ||& [{z } V(0)] . || 1.3.2. For a Submanifold | For the shape boundary derivative that structure result is preserved in a generalized form as the derivative will depend, again on the vector field V1 (0) through its normal component (V1(0), &) on the relative boundary # but also, at points where the curvature is not zero, on the component (V(0), n). For technical reasons it is useful to consider that there exists a bounded open domain D containing the family (1t). The right hand side of the tangential problem on 1t will be associated with a given element F in Hs(D), s> 1. The LaplaceBeltrami operator on 1 will be denoted by 2 2 t 1t and y(1 )(ory) will denote the solution of the problem &2 y = f on t t 1t t t 1 where f is the trace of F on 1 : f =F| . Similarly, z(| )(orz) will t t t t 1t t t denote the solution of the problem &2 z = f in | and z # H 1(| ).
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