An Introduction to the Topological Derivative Method by Antonio André Novotny & Jan Sokołowski

Author:Antonio André Novotny & Jan Sokołowski
Language: eng
Format: epub
ISBN: 9783030369156
Publisher: Springer International Publishing

(4.36)

(4.37)

We need to show that has an asymptotic expansion with respect to ε whose remainder term is uniformly bounded. More precisely, the following result has to be justified:

Lemma 4.1

The energy admits and asymptotic expansion for ε > 0, ε small enough, of the form

(4.38)

with

(4.39)

uniformly for any fixed compact set in , i.e., C depends on this set only.

Proof

By taking into account that all compact set can be covered by a finite number of balls, it is sufficient to show the result for one single fixed ball B R. Therefore, we can assume that (4.35) holds true. The proof consists in deriving the explicit formulas for w and w ε written in terms of Fourier series, similar to [88]. Thus, the associated energies can be evaluated explicitly and the properties of the remainder term immediately deduced. By constructing w from an expansion in Fourier series of the boundary condition on ∂B R, we have

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.