Shape Derivative in the Wave Equation with Dirichlet Boundary Conditions

✍ Scribed by John Cagnol; Jean-Paul Zolésio

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 223 KB
• Volume: 158
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1999.3643

✦ Synopsis

The aim of this paper is to give a full analysis of the the shape differentiability for the solution to the second order hyperbolic equation with Dirichlet boundary conditions. The implicit function theorem does not work to solve the problem of weak regularity of the data; nevertheless by a more technical approach we prove an analogous result. We will first prove the theorem under strong regularity of the right hand side, then using the hidden regularity we will prove the shape derivative continues to exist under weak condition of regularity. We end up with a second order shape derivative for this problem. 1999 Academic Press Contents. 1. Introduction. 1.1. Shape and material derivatives. 1.2. The generalized wave problem. 1.3. The results. 2. Transformation of domains. 2.1. The shape difference quotient. 2.2. Properties of the flow mapping. 2.3. Limits as the perturbation parameter tends to 0. 3. Homogeneous Dirichlet B.C. with strong regularity of the data. 3.1. Material derivative. 3.2. Shape derivative with strong regularity of the data. 3.3. Proof of Theorem 1. 4. Homogeneous Dirichlet B.C. with weak regularity of the data. 4.1. Absolute continuity. 4.2. Differentiability. 4.3. Enhancement of the result with special property of the data. 4.4. The case of non-integer Sobolev spaces. 5. Shape derivative with non-homogeneous Dirichlet B.C. 5.1. Shape differentiability. 5.2. Application to the second order shape derivative.