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Not an answer, but too long for a comment. The general idea with this stuff seems to be to pair your family $\mathscr{D}$ of admissible domains with a(ny) suitable normed-space $\mathscr{V}$ of vector fields and then insist that the 'shape derivative' be the element of $\mathscr{V}^*$ such that

$$ J(\Omega+V) = J(\Omega) + J'(\Omega)V + o(\Vert V\Vert) $$

as $\Vert V \Vert\to 0$ in $\mathscr{V}$ (where $\Omega+V:=\{x+V(x):x\in \Omega\}$). This seems like the minimal property which a 'derivative' should satisfy in an affine setting (domains are 'points', vector fields are 'vectors').

As for what 'suitable' means in this context will - I think - generally depends on what sort of regularity you want for the associated flow. It's common to choose $\mathscr{V}$ so that its elements are Lipschitz continuous because then you can apply the Picard–Lindelöf theorem to associate a unique $C^1$ path germ with every point of $\Omega$.

The one-parameter flow seems like a bit of a distraction in all this - choose $\mathscr{V}$ right and you'll get the properties you want from $(T_t)_{t>0}$ from an appropriate ODE existence theorem.

Note: I've gone for the 'full' (Frechet-like) shape derivative above, you could also work just in terms of directional derivatives $\nabla_VJ(.):\mathscr{D}\to \mathbb{R}$ defined by requiring that

$$ J(\Omega+tV) =J(\Omega)+t(\nabla_VJ)(\Omega)+o(t) \;\;\mbox{as $t\to 0$}, $$

Either way, I think it's best to choose $\mathscr{D}$ and $\mathscr{V}$ based on where you want to go, and define 'derivatives' in terms of their essential property of being the linear bit of a first-order Taylor expansion (and not get too hung up on the setup used in any particular book).

This is all just my opinion of course :)

Not an answer, but too long for a comment. The general idea with this stuff seems to be to pair your family $\mathscr{D}$ of admissible domains with a(ny) suitable normed-space $\mathscr{V}$ of vector fields and then insist that the 'shape derivative' be the element of $\mathscr{V}^*$ such that

$$ J(\Omega+V) = J(\Omega) + J'(\Omega)V + o(\Vert V\Vert) $$

as $\Vert V \Vert\to 0$ in $\mathscr{V}$ (where $\Omega+V$ is either $\{x+V(x):x\in \Omega\}$ or something similar). This seems like the minimal property which a 'derivative' should satisfy in an affine setting (domains are 'points', vector fields are 'vectors').

As for what 'suitable' means in this context will - I think - generally depends on what sort of regularity you want for the associated flow. It's common to choose $\mathscr{V}$ so that its elements are Lipschitz continuous because then you can apply the Picard–Lindelöf theorem to associate a unique $C^1$ path germ with every point of $\Omega$.

The one-parameter flow seems like a bit of a distraction in all this - choose $\mathscr{V}$ right and you'll get the properties you want from $(T_t)_{t>0}$ from an appropriate ODE existence theorem.

Note: I've gone for the 'full' (Frechet-like) shape derivative above, you could also work just in terms of directional derivatives $\nabla_VJ(.):\mathscr{D}\to \mathbb{R}$ defined by requiring that

$$ J(\Omega+tV) =J(\Omega)+t(\nabla_VJ)(\Omega)+o(t) \;\;\mbox{as $t\to 0$}, $$

Either way, I think it's best to choose $\mathscr{D}$ and $\mathscr{V}$ based on where you want to go, and define 'derivatives' in terms of their essential property of being the linear bit of a first-order Taylor expansion (and not get too hung up on the setup used in any particular book).

This is all just my opinion of course :)

Not an answer, but too long for a comment. The general idea with this stuff seems to be to pair your family $\mathscr{D}$ of admissible domains with a(ny) suitable normed-space $\mathscr{V}$ of vector fields and then insist that the 'shape derivative' be the element of $\mathscr{V}^*$ such that

$$ J(\Omega) = J(\Omega_0) + J'(\Omega_0)V + o(\Vert V\Vert) $$

as $\Vert V \Vert\to 0$ in $\mathscr{V}$. This seems like the minimal property which a 'derivative' should satisfy in an affine setting (domains are 'points', vector fields are 'vectors').

As for what 'suitable' means in this context will - I think - generally depends on what sort of regularity you want for the associated flow. It's common to choose $\mathscr{V}$ so that its elements are Lipschitz continuous because then you can apply the Picard–Lindelöf theorem to associate a unique $C^1$ path germ with every point of $\Omega_0$.

The one-parameter flow seems like a bit of a distraction in all this - choose $\mathscr{V}$ right and you'll get the properties you want from $(T_t)_{t>0}$ from an appropriate ODE existence theorem.

Note: I've gone for the 'full' (Frechet-like) shape derivative above, you could also work just in terms of directional derivatives $\nabla_VJ(.):\mathscr{D}\to \mathbb{R}$ defined by requiring that

$$ J(\Omega) =J(\Omega_0)+t(\nabla_VJ)(\Omega_0)+o(t) \;\;\mbox{as $t\to 0$}, $$

Either way, I think it's best to choose $\mathscr{D}$ and $\mathscr{V}$ based on where you want to go, and define 'derivatives' in terms of their essential property of being the linear bit of a first-order Taylor expansion (and not get too hung up on the setup used in any particular book).

This is all just my opinion of course :)

Not an answer, but too long for a comment. The general idea with this stuff seems to be to pair your family $\mathscr{D}$ of admissible domains with a(ny) suitable normed-space $\mathscr{V}$ of vector fields and then insist that the 'shape derivative' be the element of $\mathscr{V}^*$ such that

$$ J(\Omega+V) = J(\Omega) + J'(\Omega)V + o(\Vert V\Vert) $$

as $\Vert V \Vert\to 0$ in $\mathscr{V}$ (where $\Omega+V:=\{x+V(x):x\in \Omega\}$). This seems like the minimal property which a 'derivative' should satisfy in an affine setting (domains are 'points', vector fields are 'vectors').

As for what 'suitable' means in this context will - I think - generally depends on what sort of regularity you want for the associated flow. It's common to choose $\mathscr{V}$ so that its elements are Lipschitz continuous because then you can apply the Picard–Lindelöf theorem to associate a unique $C^1$ path germ with every point of $\Omega$.

The one-parameter flow seems like a bit of a distraction in all this - choose $\mathscr{V}$ right and you'll get the properties you want from $(T_t)_{t>0}$ from an appropriate ODE existence theorem.

Note: I've gone for the 'full' (Frechet-like) shape derivative above, you could also work just in terms of directional derivatives $\nabla_VJ(.):\mathscr{D}\to \mathbb{R}$ defined by requiring that

$$ J(\Omega+tV) =J(\Omega)+t(\nabla_VJ)(\Omega)+o(t) \;\;\mbox{as $t\to 0$}, $$

Either way, I think it's best to choose $\mathscr{D}$ and $\mathscr{V}$ based on where you want to go, and define 'derivatives' in terms of their essential property of being the linear bit of a first-order Taylor expansion (and not get too hung up on the setup used in any particular book).

This is all just my opinion of course :)

Not an answer, but too long for a comment. The general idea with this stuff seems to be to pair your family $\mathscr{D}$ of admissible domains with a(ny) suitable normed-space $\mathscr{V}$ of vector fields and then insist that the 'shape derivative' be the element of $\mathscr{V}^*$ such that

$$ J(\Omega) = J(\Omega_0) + J'(\Omega_0)V + o(\Vert V\Vert) $$

as $\Vert V \Vert\to 0$ in $\mathscr{V}$. This seems like the minimal property which a 'derivative' should satisfy in an affine setting (domains are 'points', vector fields are 'vectors').

As for what 'suitable' means in this context will - I think - generally depends on what sort of regularity you want for the associated flow. It's common to choose $\mathscr{V}$ so that its elements satisfy the hypothesis of the Picard–Lindelöf (ODE existence and uniqueness) theorem, because then you have a canonical $C^1$ path (germ) emanating from each point, at which point you can start to consider things the other way round (i.e. starting from the flow and obtaining $V$ as a limit of difference quotients).

The one-parameter flow seems like a bit of a distraction in all this - choose $\mathscr{V}$ right and you'll get the properties you want from $(T_t)_{t>0}$ from an appropriate ODE existence theorem.

Note: I've gone for the 'full' (Frechet-like) shape derivative above, you could also work just in terms of directional derivatives $\nabla_VJ(.):\mathscr{D}\to \mathbb{R}$ defined by requiring that

$$ J(\Omega) =J(\Omega_0)+t(\nabla_VJ)(\Omega_0)+o(t) \;\;\mbox{as $t\to 0$}, $$

Either way, I think it's best to choose $\mathscr{D}$ and $\mathscr{V}$ based on where you want to go, and define 'derivatives' in terms of their essential property of being the linear bit of a first-order Taylor expansion (and not get too hung up on the setup used in any particular book).

This is all just my opinion of course :)

Not an answer, but too long for a comment. The general idea with this stuff seems to be to pair your family $\mathscr{D}$ of admissible domains with a(ny) suitable normed-space $\mathscr{V}$ of vector fields and then insist that the 'shape derivative' be the element of $\mathscr{V}^*$ such that

$$ J(\Omega) = J(\Omega_0) + J'(\Omega_0)V + o(\Vert V\Vert) $$

as $\Vert V \Vert\to 0$ in $\mathscr{V}$. This seems like the minimal property which a 'derivative' should satisfy in an affine setting (domains are 'points', vector fields are 'vectors').

As for what 'suitable' means in this context will - I think - generally depends on what sort of regularity you want for the associated flow. It's common to choose $\mathscr{V}$ so that its elements are Lipschitz continuous because then you can apply the Picard–Lindelöf theorem to associate a unique $C^1$ path germ with every point of $\Omega_0$.

The one-parameter flow seems like a bit of a distraction in all this - choose $\mathscr{V}$ right and you'll get the properties you want from $(T_t)_{t>0}$ from an appropriate ODE existence theorem.

Note: I've gone for the 'full' (Frechet-like) shape derivative above, you could also work just in terms of directional derivatives $\nabla_VJ(.):\mathscr{D}\to \mathbb{R}$ defined by requiring that

$$ J(\Omega) =J(\Omega_0)+t(\nabla_VJ)(\Omega_0)+o(t) \;\;\mbox{as $t\to 0$}, $$

Either way, I think it's best to choose $\mathscr{D}$ and $\mathscr{V}$ based on where you want to go, and define 'derivatives' in terms of their essential property of being the linear bit of a first-order Taylor expansion (and not get too hung up on the setup used in any particular book).

This is all just my opinion of course :)