(63) |

where with and the following shape optimization problem,

(64) |

where . We derive the necessary conditions for this problem, and obtain a formula for the gradient for this functional subject to the FP equation (63).

As a result of changes in the shape , the potential changes to
and into
.
Moreover,

(65) |

and the equation governing is

(66) |

The functional variation with respect to
(see equation (62)) can be written as

(67) |

From the boundary condition some terms are simplified, in particular . We also have the relation

(68) |

where stands for the tangential gradient. Substituting these into and using integration by parts for the terms gives

(69) |

This expression depends on which is to be eliminated using the same idea as before. To this end we use the identity

(70) |

which follows from (66) and holds for an arbitrary . The relation and integration by parts of each of the terms in the above integral give

(71) |

(72) |

where in the first integral we used the relation on as well as and on . Thus,

(73) |

for an arbitrary smooth function . This is the analog of equation (10) of section (2.2).

Before we add this term to the functional we need to express certain
terms in the boundary.
The wall boundary condition for on

(74) |

will be transfered to . It is easy to see that

(75) |

by considering one dimension at a time. Therefore

(76) |

Using the boundary condition we get

(77) |

The expression for the change in the functional as given in (69)
depends on as
well as on . To eliminate the dependence on we add
the left hand side of (73).
We then collect terms involving separately from
terms involving
, and use the boundary condition for
on , giving

(78) |

Now we choose such that it satisfies

(79) |

and then the variation of the functional simplifies to

(80) |

The gradient of the functional is given by

(81) |