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.

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