Beni Bogoşel's blog

Relaxation of the Anisotropic Perimeter – Part 1

April 24, 2013 beni22sof Leave a comment

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 ${\Gamma-\lim_{\varepsilon \rightarrow 0^+}F_\varepsilon(u)=P_\varphi(u)}$ .

Numerical Approximation using Relaxed Formulation

April 23, 2013 beni22sof 1 comment

Sometimes it is easier to replace an optimization problem with a sequence of relaxed problems whose solutions approximate the solution to the initial problem.

This kind of procedure can be useful when we need to approximate numerically discontinuous functions (in particular the characteristic function). Modica Mortola theorem states that the functionals

$\displaystyle F_\varepsilon (u) = \begin{cases} \varepsilon \int_D |\nabla u|^2+\frac{1}{\varepsilon} \int_D W(u) & u \in H^1(D) \\ \infty & \text{ otherwise} \end{cases}$

${\Gamma}$ -converges to the functional

$\displaystyle F(u) = \begin{cases} \text{Per}(\{u=1\}) & \text{ if }\{u=1\} \text{ has finite perimeter} \\ \infty & \text{ otherwise} \end{cases}.$

(Recall that ${W}$ is a real function which is positive except for ${0}$ and ${1}$ where it is zero.)

SEEMOUS 2013 + Solutions

March 23, 2013 beni22sof 9 comments

Here are some of the problems of SEEMOUS 2013. Update: the 4th problem has arrived; it is number 3 below.

1. Let ${f:[1,8] \rightarrow \Bbb{R}}$ be a continuous mapping, such that

$\displaystyle \int_1^2 f^2(t^3)dt+2\int_1^2f(t^3)dt=\frac{2}{3}\int_1^8 f(t)dt-\int_1^2 (t^2-1)^2 dt.$

Find the form of the map ${f}$ .

Solution: Change the variable from ${t}$ to ${t^3}$ in the RHS integral and DO NOT calculate the last integral in the RHS. Get all the terms in the left and find that it is in fact the integral of a square equal to zero.

2. Let ${M,N \in \mathcal{M}_2(\Bbb{C})}$ be nonzero matrices such that ${M^2=N^2=0}$ and ${MN+NM=I_2}$ . Prove that there is an invertible matrix ${A \in \mathcal{M}_2(\Bbb{C})}$ such that ${M=A\begin{pmatrix} 0&1\\ 0&0\end{pmatrix}A^{-1}}$ and ${N=A\begin{pmatrix} 0&0 \\ 1&0\end{pmatrix}A^{-1}}$ .

Solution: One solution can be given using the fact that ${M,N}$ can be written in that form, but for different matrices ${A}$ .

Another way to do it is to consider applications ${f,g: \Bbb{C}^2 \rightarrow \Bbb{C}^2,\ f(x)=Mx,\ g(x)=Nx}$ . We get at once ${f^2=0,g^2=0,fg+gf=Id}$ and from these we deduce that ${(fg)^2=fg}$ and ${(gf)^2=gf}$ . First note that ${fg}$ is not the zero application. Then there exists ${u \in Im(fg) \setminus\{0\}}$ , i.e. there exists ${w (\neq 0)}$ such that ${f(g(w))=v}$ . We have ${fg(u)=(fg)^2(w)=fg(w)=u}$ . Consider ${v=g(u)}$ .

Then ${u,v}$ are not collinear, ${f(u)=0,f(v)=u, g(u)=v,g(v)=0}$ . Consider now the basis formed by ${u,v}$ and take ${A}$ to be the change of base matrix from the canonical base to ${\{u,v\}}$ .

3. Find the maximum possible value of

$\displaystyle \int_0^1 |f'(x)|^2|f(x)|\frac{1}{\sqrt{x}}dx$

over all continuously differentiable functions ${f:[0,1] \rightarrow \Bbb{R}}$ with ${f(0)=0}$ and ${\int_0^1|f'(x)|^2 dx\leq 1}$ .

4. Let ${A \in \mathcal{M}_2(\Bbb{Q})}$ such that there is ${n \in \Bbb{N},\ n\neq 0}$ , with ${A^n=-I_2}$ . Prove that either ${A^2=-I_2}$ or ${A^3=-I_2}$ .

Solution: Consider ${p \in \Bbb{Q}[X]}$ the minimal polynomial of ${A}$ , which has degree at most ${2}$ . The eigenvalues of ${A}$ satisfy ${\lambda_1^n=\lambda_2^n=-1}$ . We have two cases: either the eigenvalues are real and therefore they are both equal to ${-1}$ either they are complex and conjugate of modulus one. In both cases the determinant of ${A}$ is equal to ${1}$ . Therefore, by Cayley Hamiltoh theorem ${A}$ satisfies an equation of the type ${A^2-qA+I_2=0}$ .

By hypothesis the minimal polynomial ${p}$ divides ${X^n+1}$ . If ${p}$ has degree one then ${A=\lambda I_2}$ and ${\lambda \in \Bbb{Q},\lambda^n=-1}$ so ${A=-I_2}$ .

If not, then the minimal polynomial is ${X^2-qX+1}$ and we must have

$\displaystyle X^2-qX+1 | X^n+1.$

It can be proved that ${q}$ is in fact an integer. (using the fact that the product of two primitive polynomials is primitive)

Since ${q=2\cos \theta}$ it follows that ${q \in \{0,\pm 1,\pm 2\}}$ . The rest is just casework.

Categories: Algebra, Analysis, Undergraduate Tags: SEEMOUS

SEEMOUS 2013 (unofficial sources)

March 23, 2013 beni22sof Leave a comment

Here are some of the problems of SEEMOUS 2013.

1. Let ${f:[1,8] \rightarrow \Bbb{R}}$ be a continuous mapping, such that

$\displaystyle \int_1^2 f^2(t^3)dt+2\int_1^2f(t^3)dt=\frac{2}{3}\int_1^8 f(t)dt-\int_1^2 (t^2-1)^2 dt.$

Find the form of the map ${f}$ .

2. Let ${M,N \in \mathcal{M}_2(\Bbb{C})}$ be nonzero matrices such that ${M^2=N^2=0}$ and ${MN+NM=I_2}$ . Prove that there is an invertible matrix ${A \in \mathcal{M}_2(\Bbb{C})}$ such that ${M=A\begin{pmatrix} 0&1\&0\end{pmatrix}A^{-1}}$ and ${N=A\begin{pmatrix} 0&0 \\ 1&0\end{pmatrix}A^{-1}}$ .

3. Let ${A \in \mathcal{M}_2(\Bbb{Q})}$ such that there is ${n \in \Bbb{N},\ n\neq 0}$ , with ${A^n=-I_2}$ . Prove that either ${A^2=-I_2}$ or ${A^3=-I_2}$ .

Categories: Uncategorized

Torricelli point and angles of 120 degrees

March 4, 2013 beni22sof Leave a comment

Denote ${ABC}$ a triangle with angles smaller than ${120^\circ}$ . The point ${T}$ which minimizes the sum ${TA+TB+TC}$ is called the Torricelli point of the triangle ${ABC}$ . One interesting property of the Toricelli point, besides the fact that it minimizes the above sum is that all the angles formed around ${T}$ are equal and have ${120^\circ}$ .

I will prove here that the fact that the angles around ${T}$ are of ${120^\circ}$ can be derived without any geometric considerations, just from the fact that ${T}$ is the solution of the problem

$\displaystyle \min_{T \in \Delta ABC } TA+TB+TC.$

Categories: Calculus of Variations, Geometry Tags: Geometry, shape optimization

The Basic properties of Gamma Convergence

March 3, 2013 beni22sof 1 comment

Let ${X}$ be a metric space, and for ${\varepsilon >0}$ let be given ${F_\varepsilon : X \rightarrow [0,\infty]}$ . We say that ${F_\varepsilon}$ ${\Gamma}$ -converges to ${F}$ on ${X}$ as ${\varepsilon \rightarrow 0}$ , and we write ${\Gamma-\lim F_\varepsilon =F}$ , if the following conditions hold:

(LI) For every ${u \in X}$ and every sequence ${(u_\varepsilon)}$ such that ${u_\varepsilon \rightarrow u}$ in ${X}$ we have

$\displaystyle \liminf_{\varepsilon \rightarrow 0}F_\varepsilon(u_\varepsilon)\geq F(u)$

(LS) For every ${u \in X}$ there exists a sequence ${(u_\varepsilon)}$ such that ${u_\varepsilon \rightarrow u}$ in ${X}$ and

$\displaystyle \limsup_{\varepsilon \rightarrow 0}F_\varepsilon(u_\varepsilon)\leq F(u).$

The ${\Gamma}$ -convergence has the following properties:

1. The ${\Gamma}$ -limit ${F}$ is always lower semicontinuous on ${X}$ ;

2. ${\Gamma}$ -convergence is stable under continuous perturbations: if ${F_\varepsilon \stackrel{\Gamma}{\longrightarrow} F}$ and ${G}$ is continuous, then

$\displaystyle F_\varepsilon + G \stackrel{\Gamma}{\longrightarrow} F+G$

3. If ${F_\varepsilon \stackrel{\Gamma}{\longrightarrow} F}$ and ${v_\varepsilon}$ minimizes ${F_\varepsilon}$ over ${X}$ , then every limit point of ${(v_\varepsilon)}$ minimizes ${F}$ over ${X}$ .

I have seen these properties stated in many places, but the proofs are usually left to the reader. I will try and give the proofs below.

Categories: Calculus of Variations Tags: gamma-convergence

Master 7

March 3, 2013 beni22sof Leave a comment

(For the context and pervious posts look on the Shape Optimization page for the links)

As a result of the theorem proved in the previous post and of the fact that every ${\Gamma}$ -limit is lower semicontinuous we can see that the functional

$\displaystyle F_0(u)=\sum_{i,j=1}^k d(\alpha_i,\alpha_j) \mathcal{H}^{N-1}(\partial^* S_i \cap \partial^* S_j \cap \Omega)$

is lower semicontinuous on the space ${X=\{ u \in L^1(\Omega;\Bbb{R}^N) : u=\sum_{i=1}^k \alpha_i \chi_{S_i},S_i \subset \Omega,\text{Per}_\Omega(S_i)<\infty,|\Omega\setminus (S_1 \cup ... \cup S_k)|=0, \text{ and } \sum_{i=1}^k |S_i|\alpha_i=m \}}$ .

Definition 1 We say that a set of ${\sigma_{ij},\ 1\leq i,j \leq k}$ , ${i \neq j}$ is an admissible configuration if there exist linearly independent vectors ${\alpha_1,..,\alpha_k \in \Bbb{R}_+^n}$ and a continuous function ${W : \Bbb{R}^n_+ \rightarrow [0,\infty)}$ with zeros precisely at points ${\alpha_i,\ i=1..k}$ such that we have ${d(\alpha_i,\alpha_j)=\sigma_{ij},\ i,j=1..n}$ .

Using the above result we can see that if ${\sigma_{ij},\ i,j=1..k}$ is an admissible configuration then the functional ${\mathcal{F} :\mathcal{K} \rightarrow [0,\infty)}$ defined by

$\displaystyle \mathcal{F}(S_1,..,S_k)=\sum_{1 \leq i <j \leq k} \sigma_{ij} \mathcal{H}^{N-1}(\partial^* S_i \cap \partial^* S_j \cap \Omega)$

is lower semicontinuous on

$\displaystyle \mathcal{K}=\{ (S_1,..,S_k) : S_i \subset \Omega, \text{Per}_\Omega(S_i)<\infty, |S_i|=c_i>0,\ c_1+...+c_k=|\Omega|\},$

where ${(c_i)}$ is the unique solution of the system

$\displaystyle \sum_{i=1}^k c_i\alpha_i=m,$

where ${m}$ is chosen such that all ${c_i}$ are positive.

Older Entries

Beni Bogoşel's blog

Relaxation of the Anisotropic Perimeter – Part 1

Numerical Approximation using Relaxed Formulation

SEEMOUS 2013 + Solutions

SEEMOUS 2013 (unofficial sources)

Torricelli point and angles of 120 degrees

The Basic properties of Gamma Convergence

Master 7

Beni Bogoşel

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