I’m trying to understand the intuition behind shape optimization using Hadamard’s method. Please consider the following simple example:

Let $lambda$ denote the Lebesgue measure on $mathcal B(mathbb R)$, $dinmathbb N$, $Usubseteqmathbb R^d$ be open, $$mathcal F(Omega):=lambda^{otimes d}(Omega);;;text{for }Omegainmathcal B(U)$$ and $$mathcal G(Omega):=sigma_{partialOmega}(partialOmega);;;text{for }Omegainmathcal A,$$ where $sigma_{partialOmega}$ denotes the surface measure on $mathcal B(partialOmega)$ for $Omegainmathcal A$ and $$mathcal A:={Omegasubseteqmathbb R^d:Omegatext{ is bounded and open},overlineOmegasubseteq Utext{ and }partialOmegatext{ is of class }C^1}.$$

Consider the simple problems of minimizing $mathcal F$ and $mathcal G$ over their respective domains.

If I got it right, the problem is that there is no canonical notion of a “derivative” of $mathcal F$ or $mathcal G$. So, the idea is take a look at what happens to those functionals if their argument is subject to a small perturbation. The question is what we can infer from that, but let me be precise:

Let $tau>0$ and $T_t$ be a $C^1$-diffeomorphism from $U$ onto $U$ for $tin(0,tau)$ with $T_0=operatorname{id}_U$. Assume $$(0,tau)ni T_t(x)tag1$$ is continuously differentiable for all $xin U$. Moreover, assume that $$(0,tau)times Uni(t,x)mapsto{rm D}T_t(x)tag2$$ is continuous in the first argument uniformly with respect to the second so that we may assume that $$det{rm D}T_t(x)>0;;;text{for all }(t,x)in(0,tau)times Utag3$$ by taking $tau$ sufficiently small. (I guess, intuitively, this ensures that the transformations are orientation-preserving; maybe someone can comment on this.) Let $$v_t(x):=fracpartial{partial t}T_t(T_t^{-1}(x));;;text{for }(t,x)in(0,tau)times U.$$

Now let $Omega^{(1)}inmathcal B(U)$, $Omega^{(2)}inmathcal A$ and $$Omega^{(i)}_t:=T_tleft(Omega^{(i)}right);;;text{for }tin(0,tau).$$ We can show that $$frac{mathcal Fleft(Omega^{(1)}_tright)-mathcal Fleft(Omega^{(1)}_0right)}txrightarrow{tto0+}int_{Omega^{(1)}}nablacdot v_0:{rm d}lambda^{otimes d}tag4$$ and if $Omega^{(1)}inmathcal A$, then the right-hand side is equal to $$int_{Omega^{(1)}}nablacdot v_0:{rm d}lambda^{otimes d}=intlangle v_0,nu_{partialOmega^{(1)}}rangle:{rm d}sigma_{partialOmega^{(1)}}tag5,$$ where $nu_{partialOmega^{(1)}}$ denotes the unit outer normal field on $partialOmega^{(1)}$. Moreover, $$frac{mathcal Gleft(Omega^{(2)}_tright)-mathcal Gleft(Omega^{(2)}_0right)}txrightarrow{tto0+}intnabla_{partialOmega^{(2)}}cdot v_0:{rm d}sigma_{partialOmega^{(2)}}tag6,$$ where $nabla_{partialOmega^{(2)}}cdot v_0$ denotes the tangential divergence of $v_0$.

The question is: What can we infer from that? How do $(4)$, $(5)$ and $(6)$ help us to find a (local) minimum of $mathcal F$ or $mathcal G$?

Here is what my guess is: I think the implications are less helpful in deriving a theoretical (local) minimum, but they help to come up with a numerical “gradient descent”-like algorithm. It’s trivial to observe that if $mathcal H$ is any functional on a system $mathcal B$ of sets contained in $2^U$ and $Omega_t:=T_t(Omega_0)inmathcal B$ for $tin(0,tau)$, then $$frac{mathcal H(Omega_t)-mathcal H(Omega_0)}txrightarrow{tto0+}atag7$$ for some $a<0$ tells us that $$mathcal H(Omega_t)<mathcal H(Omega_0);;;text{for all }tin(0,delta)tag8$$ for some $delta>0$.

Now, we know that the family $v$ of “velocity” fields and the family $(T_t)_{tin(0,:tau)}$ of transformations are in a one-to-one relationship. As the results infer (and please correct me, if this is a bad conclusion) the derivatives in above only depend on $v_0$ and hence we could show a family of transformations corresponding to a “time-independent” velocity field.

The results above tell us that if we should choose $v_0$ such that the right-hand side of $(4)$, $(5)$ or $(6)$, respectively, is negative.

But is there a “steepest descent” direction? I don’t know whether we can show this, but intuitively, it seems like $v_0=-nu_{partialOmega}$ would yield the “steepest descent” for $mathcal F$. This not only yields a negative right-hand side of $(5)$, but it is even somehow plausible that we shrink the volume of the shape by squeezing it at every point in the opossite direction of the normals. Can this be made rigorous? It would obviously end in the emptyset which is clearly a global minimum of $mathcal F$ …