Home > Analysis, Calculus of Variations, Measure Theory, Optimization, Real Analysis > Relaxation of the Anisotropic Perimeter – Part 1

## Relaxation of the Anisotropic Perimeter – Part 1

I have discussed in a previous post how Modica-Mortola theorem can provide a good framework for relaxing the perimeter functional in the single and multi-phase cases. The ideas can be extended further to a more generalized notion of perimeter, the anisotropic perimeter. (anisotropic = directionally dependent)

The main idea is that the anisotropic perimeter doesn’t count every part of the boundary in the same way; some directions are more favorized than others. The anisotropic perimeter associated to a norm ${\varphi}$ is defined by

$\displaystyle \text{Per}_\varphi(\Omega)=\int_{\partial \Omega} \varphi(\vec{n})d\mathcal{H}^{n-1}.$

There are variants of Modica-Mortola theorem for the anisotropic perimeter. Here is one of them:

Theorem – Relaxation of the Anisotropic Perimeter

Let ${\Omega}$ be a bounded open set with Lipschitz boundary. Let ${p>1}$, let ${W : \Bbb{R} \rightarrow [0,\infty)}$ be a continuous function such that ${W(z)=0}$ if and only if ${z \in \{0,1\}}$ and let ${\varphi : \Bbb{R}^n \rightarrow [0,\infty)}$ be a norm on ${\Bbb{R}^n}$. Let ${F_\varepsilon : L^1(\Omega) \rightarrow [0,\infty]}$ be defined by

$\displaystyle F_\varepsilon(u) = \begin{cases} \displaystyle \frac{1}{\varepsilon p'}\int_\Omega W(u)dx +\frac{1}{p}\varepsilon^p \int_\Omega \varphi^p(\nabla u)dx & \text{ if } u \in W^{1,p}(\Omega) \\ +\infty & \text{ otherwise} \end{cases}$

and let ${P_\varphi : L^1(\Omega) \rightarrow [0,\infty]}$ be defined by

$\displaystyle P_\varphi(u) =\begin{cases}\displaystyle c_p \int_{S(u)} \varphi(\nu_u)d\mathcal{H}^{n-1}& \text{ if }u \in SBV(\Omega) \text{ and } u \in \{0,1\} \text{ a.e.}\\ +\infty & \text{ otherwise} \end{cases}$

where ${c_p =\int_0^1 (W(s))^{1/p'}ds}$. Then ${\Gamma-\lim_{\varepsilon \rightarrow 0^+}F_\varepsilon(u)=P_\varphi(u)}$.

Proof: (the ${\liminf}$ estimate) As usual, for the ${\Gamma}$ convergence proofs we have two parts. First we prove that for every ${(u_\varepsilon) \rightarrow u}$ as ${\varepsilon \rightarrow 0^+}$ we have

$\displaystyle P_\varphi(u) \leq \liminf_{\varepsilon \rightarrow 0^+} F_\varepsilon(u_\varepsilon).$

Consider the function ${\phi(t)=\displaystyle \int_0^t (W(s))^{1/p'} ds}$ and let’s show that for ${u \in BV(\Omega)}$ with ${u \in \{0,1\}\ a.e}$ we have ${P_\varphi(u)= \int_\Omega \varphi(D(\phi \circ u)) }$, where we use the notation

$\displaystyle \int_\Omega \varphi(\mu) = \int_\Omega \varphi\left(\frac{d \mu}{d|\mu|}\right)d|\mu|$

for every measure ${\mu \in \mathcal{M}(\Omega,\Bbb{R}^N)}$.

First note that if ${u \in \{0,1\}}$ a.e then using the definition of the variation of a ${BV(\Omega)}$ function we can see that ${D(\phi \circ u)=\phi(1) Du}$. Moreover, if we have a function ${u \in BV(\Omega)}$ whose image contains only two real values then the absolutely continuous part and the Cantor part of ${Du}$ are zero, while the jump part is

$\displaystyle D^ju(B)=\int_{B \cap S(u)} (u^+-u^-)\nu_u d \mathcal{H}^{n-1}$

where ${\nu_u}$ is the normal to the jump set ${S(u)}$ defined by ${Du=\nu_u|Du|}$. Having these in mind and using the fact that ${\varphi}$ is homogeneous of degree one, we obtain

$\displaystyle c_p \int_{S(u)} \varphi(\nu_u)d\mathcal{H}^{n-1}=c_p \int_{\Omega}\varphi\left(\frac{d Du}{d |Du|}\right)d\mathcal{H}^{n-1}\llcorner S(u)=$

$\displaystyle =\phi(1) \int_\Omega\varphi \left(\frac{d Du}{d |Du|}\right)d |Du|=\int_\Omega\varphi \left(\frac{dD(\phi \circ u)}{d|D(\phi \circ u)|} \right)d |D(\phi \circ u)|=$

$\displaystyle = \int_\Omega \varphi(D(\phi \circ u)).$

In order to prove the first property of the ${\Gamma}$-convergence we take two cases. First, assume that ${u \notin \{0,1\}}$ a.e. Then for ${u_\varepsilon \rightarrow u}$ in ${L^1(\Omega)}$ we have

$\displaystyle \liminf_{\varepsilon \rightarrow 0} F_\varepsilon(u_\varepsilon) \geq \liminf \frac{1}{\varepsilon}W(u_\varepsilon)dx=+\infty=P_\varphi(u).$

If ${u \in \{0,1\}}$ a.e. then applying Young’s inequality we get

$\displaystyle \liminf_{\varepsilon \rightarrow 0} F_\varepsilon(u_\varepsilon) \geq \liminf_{\varepsilon \rightarrow 0}\int_\Omega \varphi(\nabla u_\varepsilon)(W(u_\varepsilon))^{1/p'} dx=$

$\displaystyle = \liminf_{\varepsilon \rightarrow 0}\int_\Omega \varphi(\nabla (\psi \circ u_\varepsilon))dx \geq \int_\Omega \varphi(D(\phi \circ u))=P_\varphi(u)$

where in the last inequality we have used Reshetnyak’s Theorem for the measures ${\nabla (\phi\circ u_\varepsilon) \stackrel{*}{\rightharpoonup}D(\phi \circ u)}$. This is one point where we use the convexity of the function ${\varphi}$. This finishes the first part of the proof.

This first part of the ${\Gamma}$-convergence proof can also be proved using a slicing technique and a reduction to the one dimensional case. For more details see Approximation of Free Discontinuity Problems by Andrea Braides.

I’ll come back with the ${\limsup}$ estimate in a following post.