Home > Analysis, BV functions, Calculus of Variations, Optimization, Real Analysis, shape optimization > Relaxation of the Anisotropic Perimeter – Part 2

## Relaxation of the Anisotropic Perimeter – Part 2

(Read this previous post to see the context and the statement of the problem)

While it is usually difficult to construct a recovery sequence for an arbitrary ${u \in BV(\Omega,\{0,1\})}$ (or all ${u \in L^1(\Omega)}$), the construction is often much simpler if the target function ${u}$ has some special structure.

It is useful to consider a subset ${\mathcal{D}\subset L^1(\Omega)}$ which is dense in ${BV(\Omega,\{0,1\})}$ such that for every ${u \in BV(\Omega,\{0,1\})}$ there exists ${(u_j) \subset \mathcal{D}}$ such that ${u_j \rightarrow u}$ in ${L^1(\Omega)}$ and ${P_\varphi(u)=\lim\limits_{j \rightarrow \infty} P_\varphi(u_j)}$.

A common choice for ${\mathcal{D}}$ is the class of characteristic functions of smooth sets, which is dense in ${BV(\Omega,\{0,1\})}$ in the sense that for every finite perimeter set ${E\subset \Omega}$ there exists a sequence ${E_n}$ of smooth sets such that ${|E \Delta E_n| \rightarrow 0}$ and ${|D\chi_{E_n}|(\Omega) \rightarrow |D\chi_E|(\Omega)}$ as ${n \rightarrow \infty}$. Note that this is different than the BV-norm convergence, but it is the notion of convergence very well suited for BV functions. In Approximation of Free-Discontinuity Problems by A. Braides a proof of the limsup estimate is given using the dense class of smooth functions.

When dealing with the anisotropic perimeter, a more natural way of working would be choosing sets ${E}$ whose boundary is polyhedral (piecewise affine). The main reason for doing that is the fact that polyhedrons have constant normals corresponding to one face. To be able to choose ${\mathcal{D}}$ as the class of characteristic functions of polyhedral subsets of ${\Omega}$ (with finite perimeter, of course), we must have a density result similar to the smooth case.

G. Cortesani and R. Toader prove the following result in their article A density result in SBV with respect to non-isotropic energies:

First, let’s write some preliminaries. Let ${\Omega}$ be an open and bounded subset of ${\Bbb{R}^N}$ and denote ${\mathcal{W}(\Omega,\Bbb{R}^n)}$ the space of all functions ${w \in SBV(\Omega,\Bbb{R}^n)}$ which satisfy the following properties:

(i) ${S_w}$ is essentially closed (${\mathcal{H}^{n-1}(\overline{S_w}\setminus S_w)=0}$).
(ii) ${\overline{S_w}}$ is a polyhedral set (the intersection of ${\Omega}$ with the union of a finite number of ${(n-1)}$ simplexes.
(iii) ${w \in W^{k,\infty}(\Omega \setminus \overline{S_w},\Bbb{R}^n)}$ for every ${k \in \Bbb{N}}$.
Theorem Assume that ${\partial \Omega}$ is Lipschitz, and let ${u \in SBV^p(\Omega,\Bbb{R}^n)\cap L^\infty(\Omega,\Bbb{R}^n)}$. Then there exists a sequence ${(w_h) \subset \mathcal{W}(\Omega,\Bbb{R}^n)}$ such that

${w_h \rightarrow w}$ strongly in ${L^1(\Omega,\Bbb{R}^n)}$,
${\nabla w_h \rightarrow \nabla u}$ strongly in ${L^p(\Omega,M^{N \times n})}$
${\limsup_{h \rightarrow \infty} \|w_h\|_{\infty} \leq \|u\|_\infty}$
${\displaystyle \limsup_{h \rightarrow \infty} \int_{\overline A \cap S_{w_h}} \varphi(x,w_h^+,w_h^-,\nu_{w_h})d\mathcal{H}^{n-1} \leq \int_{\overline A \cap S_u} \varphi(x,u^+,u^-,\nu_u)d\mathcal{H}^{n-1} }$ for every ${A \subset \subset \Omega}$ and for every upper semicontinuous function ${\varphi: \Omega \times \Bbb{R}^n \times \Bbb{R}^n \times S^{N-1} \rightarrow [0,\infty)}$ such that ${\varphi(x,a,b,\nu)=\varphi(x,b,a,-\nu)}$ for every ${x \in \Omega,\ a,b \in \Bbb{R}^n}$ and ${\nu \in S^{N-1}}$.
This is not quite exactly what we want here, because note that the approximate sequence ${(w_h)}$ is not made of characteristic functions. Still, we can construct a sequence of characteristic functions starting from ${(w_h)}$ in the following way: choose ${v_h \in SBV(\Omega,\{0,1\})}$ such that ${v_h}$ and ${w_h}$ have the same jump set (which is polyhedral). Moreover, we choose the side of the jump set on which ${v_h}$ is ${1}$ such that ${\|u-v_h\|}$ is smaller (i.e. choose the side on which most of the set defined by ${u}$ lies).

What we really need in our problem is the ${L^1}$ convergence and the last part with the integral on the jump set. Note that since we do not use ${u^+,u^-}$ in our problem and the rest of the expressions in the fourth property above depend only on the jump set, it follows that the property remains true. As for the ${L^1(\Omega)}$ convergence, we have:

$\displaystyle \|u-v_h\|=|u^{-1}(1) \Delta v_h^{-1}(1)|$

which tends to zero, since the symmetric differente will appear only in a small neighbourhood of the jump set whose measure goes to ${0}$ ans ${h}$ goes to ${\infty}$.
The above argument allows us to consider ${\mathcal{D}}$ the subclass of ${SBV(\Omega,\{0,1\})}$ of functions with polyhedral jump set.

In order to prove the upper estimate for the ${\Gamma}$-convergence problem we can consider the optimal profile problem

$\displaystyle c_p= \inf \{ \int_{-\infty}^\infty \left(\frac{1}{p'}W(v) +\frac{1}{p}|v'|^p \right)dt:\ v(-\infty)=0, v(+\infty)=1\}.$

It is immediate to see that if we apply Young’s inequality we find that
$\displaystyle c_p \geq \int_{-\infty}^\infty (W(v))^{1/p'}v' dt=\int_0^1 (W(s))^{1/p'}ds$

and the equality holds if ${v}$ satisfies the differential equation ${v'=(W(v))^{1/p}}$. We will consider ${v}$ a solution of the problem
$\displaystyle \begin{cases} v'=(W(v))^{1/p} \\ v(0)=1/2. \end{cases}$

Note that truncating between ${0}$ and ${1}$ makes the functional smaller, which means that ${v}$ is contained between ${0}$ and ${1}$ (this is logical… to go from a value to another in an optimal way, you don’t go oscilating). This also proves that ${v'}$ is bounded and therefore ${v}$ is Lipschitz continuous.

Note that the optimal profile realizes the minimal cost of the functional while passing from ${0}$ to ${1}$ on the real line. We would like to get from ${0}$ to ${1}$ in finite time, and one way to do this is to consider slightly dilatated profiles:

$\displaystyle v^\eta = \max\{0,\min\{(1+2\eta)v-\eta,1\}\},\ \forall \eta>0$

Note that if we denote
$\displaystyle c_p^\eta = \int_{-\infty}^\infty \left(\frac{1}{p'}W(v^\eta) +\frac{1}{p}|(v^\eta)'|^p \right)dt$

then we have
$\displaystyle c_p^\eta \rightarrow c_p \text{ as } \eta \rightarrow 0.$

Now we assume that ${u}$ is the characteristic function of a polyhedral set ${E}$. Fix ${\eta>0}$ and denote ${M=\max\{ \varphi(\vec{n}) : \|\vec n \|=1\}}$. Choose ${T>0}$ such that ${v^\eta}$ passes from ${0}$ to ${1}$ in the interval ${[-TM,TM]}$. For each ${\varepsilon>0}$ we will consider rectangular neighborhoods of the ${(N-1)}$-dimensional facets of ${E}$ the points at distance of smaller than ${T\varepsilon}$, but we choose these rectangular boxes so that boxes corresponding to different facets do not intersect. It remains a zone ${N_\varepsilon}$ at distance smaller than ${T\varepsilon}$ of the jump set which is not covered by the rectangular boxes, and this region has a volume of order ${\varepsilon^2}$. We also consider the signed distance ${d(x)=\text{dist}(x,\Omega \setminus E)-\text{dist}(x,E)}$.

Define the recovery sequence on ${\Omega \setminus N_\varepsilon}$ by

$\displaystyle u_\varepsilon(x)=\begin{cases} \displaystyle v^\eta(\frac{d(x)}{\varepsilon \varphi(\nu_u(x))}) & \text{ if }|d(x)|\leq TM\varepsilon \\ 0 & \text{ otherwise in }\Omega \setminus E \\ 1 & \text{ otherwise in }E\end{cases}$

and extend it on the entire ${\Omega}$ with the same Lipschitz constant of order ${1/\varepsilon}$ using Kirszbraun’s Theorem.
Let’s estimate first the part of the integral on ${N_\varepsilon}$:

$\displaystyle \int_{N_\varepsilon} \frac{1}{\varepsilon p'}W(u_\varepsilon)+\frac{\varepsilon^{p-1}}{p}\varphi(\nabla u_\varepsilon)\leq |N_\varepsilon|\left(\frac{\max_{[0,1]}W}{p\varepsilon}+\frac{\varepsilon^{p-1}}{p}M \sup_\Bbb{R} (v^\eta)'\frac{K}{\varepsilon^p}\right)$

It is easy to see that because of the fact that ${|N_\varepsilon|=O(\varepsilon^2)}$ if we take the limsup above we get zero so ${N_\varepsilon}$ is negligible in the limsup estimate.
On ${\Omega \setminus N_\varepsilon}$ we have for ${\varepsilon}$ small enough:

$\displaystyle \int_{\Omega \setminus N_\varepsilon} \left( \frac{1}{\varepsilon p'} W(u_\varepsilon) +\frac{\varepsilon^{p-1}}{p} \varphi^p(\nabla u_\varepsilon) \right) dx =$

$\displaystyle =\int_{\Omega \setminus N_\varepsilon} \left( \frac{1}{\varepsilon p'} W(v^\eta (\frac{d(x)}{\varepsilon \varphi(\nu_u(x)})) +\frac{\varepsilon^{p-1}}{p} \varphi^p(\frac{\nabla u_\varepsilon}{|\nabla u_\varepsilon|})|\nabla u_\varepsilon|^p \right) dx \leq$

$\displaystyle \leq \int_{-TM\varepsilon}^{TM\varepsilon} \int_{{d(x)=t} \setminus N_\varepsilon} \left(\frac{1}{\varepsilon p'}W(v^\eta(\frac{t}{\varepsilon \varphi(\nu_u(x))}))+\frac{\varepsilon^{p-1}}{p}\varphi^p(\nu_u(x))\frac{1}{\varepsilon^p \varphi^p(\nu(x))}|(v^\eta)'(\frac{t}{\varepsilon \varphi(\nu_u(x))})|^p \right)d\mathcal{H}^{n-1}(x)dt =$

$\displaystyle = \int_{S(u)\setminus N_\varepsilon} \int_{-TM\varepsilon}^{TM\varepsilon} \left(\frac{1}{\varepsilon p'}W(v^\eta(\frac{t}{\varepsilon \varphi(\nu_u(x))}))+\frac{1}{\varepsilon p}|(v^\eta)'(\frac{t}{\varepsilon \varphi(\nu_u(x))})|^p \right)d\mathcal{H}^{n-1}(x)dt$

$\displaystyle = \int_{S(u)\setminus N_\varepsilon} \varphi(\nu_u(x))\int_{-TM/\varphi(\nu_u(x))}^{TM/\varphi(\nu_u(x))} \frac{1}{p'}W(v^\eta)+\frac{1}{p}|(v^\eta)'|^p dt d\mathcal{H}^{n-1}(x)\leq$

$\displaystyle \leq \int_{S(u)} c_p^\eta \varphi(\nu(x))d\mathcal{H}^{n-1}(x).$

Taking limsup as ${\varepsilon \rightarrow 0}$, and then making ${\eta \rightarrow 0}$ gives us the desired conclusion.