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.

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