gianluca orlando
Variational analyisis of the $N$-clock model
Gianluca Orlando 
Joint works with:
M. Cicalese (TU München), M. Ruf (JLU Gießen)
On the Bridge: Geometric Measure Theory and Calculus of Variations
Trento, 8 September 2026
Ising model
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \{+ e_2, - e_2\}$
Energy: $\displaystyle H_\varepsilon(u) = \sum_{i,j \text{ n.n.}} - \varepsilon^2 u(\varepsilon i) \cdot u(\varepsilon j)$
XY model
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathbb{S}^1$
Energy: $\displaystyle H_\varepsilon(u) = \sum_{i,j \text{ n.n.}} - \varepsilon^2 u(\varepsilon i) \cdot u(\varepsilon j)$
Ising model in Statistical Mechanics
Sampling: $\mathbb{P}(u) \propto e^{-\beta H_\varepsilon(u)}$
Energy: $\displaystyle H_\varepsilon(u) = \sum_{i,j \text{ n.n.}} - \varepsilon^2 u(\varepsilon i) \cdot u(\varepsilon j)$
$XY$ model in Statistical Mechanics
Sampling: $\mathbb{P}(u) \propto e^{-\beta H_\varepsilon(u)}$
Energy: $\displaystyle H_\varepsilon(u) = \sum_{i,j \text{ n.n.}} - \varepsilon^2 u(\varepsilon i) \cdot u(\varepsilon j)$
–
Physics literature
- Berezinskii. Sov. Phys. JETP (1971)
- Kosterlitz
. J. Phys. C, (1973) -
Kosterlitz
, Thouless. J. Phys. C (1973)
$N$-clock model
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathcal{S}_N$
Energy: $\displaystyle H_\varepsilon(u) = \sum_{i,j \text{ n.n.}} - \varepsilon^2 u(\varepsilon i) \cdot u(\varepsilon j)$
Discrete-to-continuum variational analysis (Ising)
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \{+ e_2, - e_2\}$
Energy: $ \displaystyle \sum_{i,j \text{ n.n.}} - \varepsilon^2 \big( u(\varepsilon i) \cdot u(\varepsilon j) - 1 \big) = \frac{1}{2} \sum_{i,j \text{ n.n.}} \varepsilon^2 | u(\varepsilon i) - u(\varepsilon j) |^2 =: E_\varepsilon(u) $
Low-energy configurations: $E_\varepsilon(u_\varepsilon) \lesssim \varepsilon \to 0$ as $\varepsilon \to 0$
$\Gamma$-limit
\[
\displaystyle \frac{1}{\varepsilon} E_\varepsilon(u_\varepsilon) \stackrel{\Gamma}{\to} c \int_{\partial^* \{u=\uparrow \}} |\nu_u|_1 \, \mathrm{d} \mathcal{H}^1
\]
Alicandro, Braides, Cicalese. Netw. Heterog. Media (2006)
The $XY$ model does not see interfaces
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathbb{S}^1$
$u^-$
$u^+$
$\theta$
\[ \hspace{-1.3em} E_\varepsilon(u_\varepsilon) = \frac{1}{2} \sum_{i,j \text{ n.n.}} \varepsilon^2 | u(\varepsilon i) - u(\varepsilon j) |^2 \approx \underbrace{\frac{1}{\varepsilon}\frac{\mathrm{d}_{\mathbb{S}^1}(u^-,u^+)}{\theta}}_{\text{\# \text{interactions}}} \varepsilon^2 \underbrace{\theta^2}_{\text{unit cost}} \approx \mathrm{d}_{\mathbb{S}^1}(u^-,u^+) \varepsilon \theta \]
The $N$-clock model sees interfaces
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathcal{S}_N$
$u^-$
$u^+$
$\theta$
\[ \hspace{-1.3em} E_\varepsilon(u_\varepsilon) = \frac{1}{2} \sum_{i,j \text{ n.n.}} \varepsilon^2 | u(\varepsilon i) - u(\varepsilon j) |^2 \approx \underbrace{\frac{1}{\varepsilon}\frac{\mathrm{d}_{\mathbb{S}^1}(u^-,u^+)}{\theta}}_{\text{\# \text{interazioni}}} \varepsilon^2 \underbrace{\theta^2}_{\text{interaction}} \approx \mathrm{d}_{\mathbb{S}^1}(u^-,u^+) \varepsilon \theta \]
$\Gamma$-limit as $\varepsilon \to 0$, $N$ fixed
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathcal{S}_N$, with $N$ fixed and minimal angle $\theta$
Theorem [Cicalese, O., Ruf - Interfaces and Free Boundaries (2021)]
\[ \frac{1}{\varepsilon \theta} E_\varepsilon(u_\varepsilon) \stackrel{\Gamma}{\to} \frac{\sin^2\big(\frac{\theta}{2}\big)}{\big(\frac{\theta}{2}\big)^2} \int_{J_u} \mathrm{d}_{\mathbb{S}^1}(u^-,u^+) |\nu_u|_1 \, \mathrm{d} \mathcal{H}^1 \]
Proof.
\[ |u(\varepsilon i) - u(\varepsilon j) |^2 = 4 \sin^2\Big( \frac{k \theta}{2} \Big) \geq 4 k \sin^2\Big(\frac{\theta}{2}\Big) = 4 \frac{\mathrm{d}_{\mathbb{S}^1}\big(u(\varepsilon i), u(\varepsilon j) \big)}{\theta} \sin^2\Big(\frac{\theta}{2}\Big) \]
$\Gamma$-limit as $N \to +\infty$
| Set $ | A | _{2,1} = $ sum of Euclidean norms of the columns of $A$. |
Theorem [Cicalese, O., Ruf - Interfaces and Free Boundaries (2021)]
As $N \to +\infty$, i.e., $\theta \to 0$,
\[
\begin{split}
& \frac{\sin^2\big(\frac{\theta}{2}\big)}{\big(\frac{\theta}{2}\big)^2} \int_{J_{u_N}} \mathrm{d}{\mathbb{S}^1}(u^-_N,u^+_N) |\nu{u_N}|1 \, \mathrm{d} \mathcal{H}^1 \stackrel{\Gamma}{\to} \
& \hspace{4em} \stackrel{\Gamma}{\to} \int{\Omega} |\nabla u|{2,1} \, \mathrm{d} x + |\mathrm{D}^{(c)} u|{2,1}(\Omega) + \int_{J_u} \mathrm{d}_{\mathbb{S}^1}(u^-,u^+) |\nu_u|_1\, \mathrm{d} \mathcal{H}^1,
\end{split}
\]
finite for $u \in BV(\Omega;\mathbb{S}^1)$.
Educated guess: The same $\Gamma$-limit is obtained if $N_\varepsilon \to +\infty$ "slowly", i.e., $\theta_\varepsilon \to 0$ "slowly" as $\varepsilon \to 0$.
Question: Can we quantify the threshold rate $\text{?} \ll \theta_\varepsilon \ll 1$ as $\varepsilon \to 0$?
$N_\varepsilon \to +\infty$, i.e., $\theta_\varepsilon \to 0$, as $\varepsilon \to 0$
Educated guess: The limit depends on the rate of convergence $\theta_\varepsilon \to 0$
$XY \text{?}$
$\text{?}$
$BV \text{?}$
fast
slow
$\theta_\varepsilon \to 0$ “slowly”
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathcal{S}{N\varepsilon}$, with $N_\varepsilon \to +\infty$, i.e., $\theta_\varepsilon \to 0$
Theorem [Cicalese, O., Ruf - ARMA (2022)]
Assume $\text{\color{red}?} \ll \theta_\varepsilon \ll 1$. Then \[ \frac{1}{\varepsilon \theta_\varepsilon} E_\varepsilon(u_\varepsilon) \stackrel{\Gamma}{\to} \int_{\Omega} |\nabla u|{2,1} \, \mathrm{d} x + |\mathrm{D}^{(c)} u|{2,1}(\Omega) + \int_{J_u} \mathrm{d}_{\mathbb{S}^1}(u^-,u^+) |\nu_u|_1 \, \mathrm{d} \mathcal{H}^1 \]
$XY$ vortices
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathbb{S}^1$
\(\Omega\)
\[ E_\varepsilon(\text{vortex}) = \frac{1}{2} \sum_{i.j. \text{n.n.}} \varepsilon^2 |u(\varepsilon i) - u(\varepsilon j)|^2 \approx \varepsilon^2 \int_{B \setminus B_\varepsilon} \Big| \nabla \Big( \frac{x}{|x|} \Big) \Big|^2 \approx 2 \pi \varepsilon^2 |\log \varepsilon| \]
\[ \frac{1}{\varepsilon^2 |\log \varepsilon|} E_\varepsilon(u_\varepsilon) \stackrel{\Gamma}{\to} 2 \pi |\mu|(\Omega) \]
–
Literature on Ginzburg-Landau and $XY$
- Bethuel, Brezis, Hélein. (1994)
- Sandier, Serfaty. J. Funct. Anal. (1998)
- Jerrard, Soner. Calc. Var. PDEs (2002)
- Alberti, Baldo, Orlandi. Indiana Univ. Math. J. (2005)
- Alicandro, Cicalese. ARMA (2009)
- Alicandro, Cicalese, Ponsiglione. Indiana Univ. Math. J. (2011)
- Alicandro, Ponsiglione. J. Funct. Anal. (2014)
- Alicandro, De Luca, Garroni, Ponsiglione. ARMA (2014)
[Jump forward](#/theorem-with-vortices)
---
## $N$-clock vortices
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathcal{S}_{N_\varepsilon}$, with $N_\varepsilon \to +\infty$, i.e., $\theta_\varepsilon \to 0$
\\[
\frac{1}{\varepsilon \theta_\varepsilon} E_\varepsilon(u_\varepsilon) \leq C \implies \frac{1}{\varepsilon^2 |\log \varepsilon|} E_\varepsilon(u_\varepsilon) \leq C \frac{\theta_\varepsilon}{\varepsilon |\log \varepsilon|}
\\]
- $\displaystyle \frac{\theta_\varepsilon}{\varepsilon |\log \varepsilon|} \to +\infty \implies$ no information on $\displaystyle \frac{1}{\varepsilon^2 |\log \varepsilon|} E_\varepsilon(u_\varepsilon)$
- $\displaystyle \frac{\theta_\varepsilon}{\varepsilon |\log \varepsilon|} \to 0 \implies \frac{1}{\varepsilon^2 |\log \varepsilon|} E_\varepsilon(u_\varepsilon) \to 0 \implies $ no vortices
Conclusion: The $BV$-limit occurs if $\varepsilon |\log \varepsilon| \ll \theta_\varepsilon \ll 1$
---
## No vortices regime
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathcal{S}_{N_\varepsilon}$, with $N_\varepsilon \to +\infty$, i.e., $\theta_\varepsilon \to 0$
Constraint: $\displaystyle \theta_\varepsilon \ll \varepsilon |\log \varepsilon| \implies \frac{1}{\varepsilon^2 |\log \varepsilon|} E_\varepsilon(u_\varepsilon) \to 0 \implies $ no vortices
A Lavrentiev gap?
For $u_\varepsilon \to u$ in $L^1(\Omega;\mathbb{S}^1)$, do we expect this strict inequality?
\\[
\liminf_{\varepsilon \to 0} \frac{1}{\varepsilon \theta_\varepsilon} E_\varepsilon(u_\varepsilon) > \int_\{\Omega\} |\nabla u|_\{2,1\} \\, \mathrm{d} x + |\mathrm{D}^\{(c)\} u|_\{2,1\}(\Omega) + \int_\{J_u\} \mathrm{d}_\{\mathbb{S}^1\}(u^-,u^+) |\nu_u|_1 \\, \mathrm{d} \mathcal{H}^1
\\]
--
## Back to the original problem
Spoiler alert: [Yes](#/back-to-original-problem)
---
## An analogous problem in the smooth setting
Problem: Relax
\\[
\int_{\Omega} |\nabla u| \\, \mathrm{d} x \\, , \quad u \in C^\infty(\Omega;\mathbb{S}^1)
\\]
- Giaquinta, Modica, Souček. Calc. Var. PDEs (1993)
$XY \text{?}$
$\text{vortices?}$
$BV$
fast
slow
Facts:
- The relaxation domain is $BV(\Omega;\mathbb{S}^1)$
- $\displaystyle \frac{x}{|x|} \in W^{1,p}(B_1;\mathbb{S}^1)$ for $p \in [1,2)$, hence it is in the relaxation domain
- $\displaystyle \mathrm{deg}\Big( \frac{x}{|x|}, \partial B_\rho \Big) = 1$, for every $\rho \in (0,1)$
- $u \in C^\infty(B_1;\mathbb{S}^1) \implies \mathrm{deg}\big( u, \partial B_\rho \big) = 0$, for every $\rho \in (0,1)$
- Topological obstruction: $\nexists \\, u_\varepsilon \in C^{\infty}(B_1;\mathbb{S}^1)$ such that $u_\varepsilon \stackrel{\text{stricly-}BV}{\longrightarrow} \frac{x}{|x|}$
---
## Approximating singularities with smooth maps
Aim: Approximate $\displaystyle \frac{x}{|x|}$ via $C^{\infty}(\Omega;\mathbb{S}^1)$ maps and minimal energy $\displaystyle \int_\Omega |\nabla u| \mathrm{d}x$
$u = e^{\iota 2 \pi \varphi}$
$\varphi$
Fact 1: If $u \in C^\infty(\Omega;\mathbb{S}^1)$, then $G_u$ has no boundary
Fact 2: The graph $G_{\frac{x}{|x|}}$ has boundary $-\delta_0 \times \llbracket \mathbb{S}^1 \rrbracket$
---
## Currents induced by graphs
Graph current: $u \in C^\infty(\Omega;\mathbb{S}^1) \leadsto G_u := (\mathrm{id}, u)_\\# \llbracket \Omega \rrbracket \in \mathcal{D}_2(\Omega \times \mathbb{R}^2)$
Duality: $\displaystyle \langle G_u, \omega \rangle := \int_{\Omega \times \mathbb{R}^2} \langle \omega , \vec G_\{u_\varepsilon\} \rangle \\, \mathrm{d} \mathcal{H}^2 {\LARGE \llcorner} \text{graph}(u)$
$\hphantom{\text{Duality:}\langle G_u, \omega \rangle} \displaystyle = \int_{\Omega} \langle \omega(x,u(x)), M(\nabla u(x)) \rangle \\, \mathrm{d} x \\,, \quad \omega \in \mathcal{D}^2(\Omega \times \mathbb{R}^2)$
Properties:
- $G_u$ is integer multiplicity rectifiable
- $\partial G_u |_{\Omega \times \mathbb{R}^2} = 0$ in $\mathcal{D}_1(\Omega \times \mathbb{R}^2)$
- $\pi^\Omega_\\# G_u = \llbracket \Omega \rrbracket$
- $\langle G_u, \phi(x,y) \mathrm{d} x \rangle \geq 0$ for every $\phi \in \mathcal{D}(\Omega \times \mathbb{R}^2)$
- $\mathbb{M}(G_u) < +\infty$
- $\displaystyle \\| G_u \\|_1 = \sup\_{\phi \in C^\infty_c(\Omega \times \mathbb{R}^2)} \langle G_u, \phi(x,y)|y| \mathrm{d} x \rangle < + \infty$
- $\text{supp}(G_u) \subset \overline \Omega \times \mathbb{S}^1$
---
## Cartesian currents
Cartesian current: $T \in \mathrm{cart}(\Omega \times \mathbb{S}^1)$ if $T \in \mathcal{D}_2(\Omega \times \mathbb{R}^2)$ and satisfies the following properties
Properties:
- $T$ is integer multiplicity rectifiable
- $\partial T |_{\Omega \times \mathbb{R}^2} = 0$ in $\mathcal{D}_1(\Omega \times \mathbb{R}^2)$
- $\pi^\Omega_\\# T = \llbracket \Omega \rrbracket$
- $\langle T, \phi(x,y) \mathrm{d} x \rangle \geq 0$ for every $\phi \in \mathcal{D}(\Omega \times \mathbb{R}^2)$
- $\mathbb{M}(T) < +\infty$
- $\displaystyle \\| T \\|_1 = \sup\_{\phi \in C^\infty_c(\Omega \times \mathbb{R}^2)} \langle T, \phi(x,y)|y| \mathrm{d} x \rangle < + \infty$
- $\text{supp}(T) \subset \overline \Omega \times \mathbb{S}^1$
---
## Limits of smooth functions
Fact: Let $u_\varepsilon \in C^{\infty}(\Omega;\mathbb{S}^1)$ and assume that $\displaystyle \int_\Omega |\nabla u_\varepsilon| \\, \mathrm{d} x \leq C$
Then $\displaystyle \mathbb{M}(G_{u_\varepsilon}) = \mathcal{H}^2(\text{graph}(u_\varepsilon)) = \int_\Omega \sqrt{\mathrm{det}( \mathrm{Id} + (\nabla u_\varepsilon)^T \nabla u_\varepsilon)} \\, \mathrm{d} x$
$\displaystyle \hphantom{\text{Then} \mathbb{M}(G_{u_\varepsilon})} \hspace{-2px} = \int_\Omega \sqrt{1 + |\nabla u_\varepsilon|^2 + \mathrm{det}(\nabla u_\varepsilon)^2} \\, \mathrm{d} x = \int_\Omega \sqrt{ 1 + |\nabla u_\varepsilon|^2} \\, \mathrm{d} x$
(Compactness): $\hspace{2em} G_{u_\varepsilon} \rightharpoonup T$ and $T \in \mathrm{cart}(\Omega \times \mathbb{S}^1)$
($\Gamma$-liminf): $\hspace{4.2em} \displaystyle \liminf_{\varepsilon \to 0}\int_\Omega |\nabla u_\varepsilon| \\, \mathrm{d} x \geq \int_\{\Omega \times \mathbb{R}^2\} \Phi(\vec T)\\, \mathrm{d} |T|$
($\Gamma$-limsup): $\hspace{3.5em}$ Given $T \in \mathrm{cart}(\Omega \times \mathbb{S}^1)$, $\exists \\, u_\varepsilon$ such that $G_{u_\varepsilon} \rightharpoonup T$ and $\hspace{8.3em} \displaystyle \lim_{\varepsilon \to 0}\int_\Omega |\nabla u_\varepsilon| \\, \mathrm{d} x = \int_\{\Omega \times \mathbb{R}^2\} \Phi(\vec T)\\, \mathrm{d} |T|$
$\hspace{8em}$ Giaquinta, Modica, Souček. Calc. Var. PDEs (1993)
Note: If $u_\varepsilon \stackrel{\*}{\rightharpoonup} u$, then, in general, $T \neq G_u$
---
## Structure theorem
Theorem [Giaquinta, Modica, Souček - Calc. Var. PDEs (1993)]
1. Let $T \in \mathrm{cart}(\Omega \times \mathbb{S}^1)$. Then there exist:
- a unique $u_T \in BV(\Omega;\mathbb{S}^1)$;
- an i.m. rectifiable 1-current $L \in \mathcal{D}_1(\Omega)$ such that $T = G_\{u_T\} + L \times \llbracket \mathbb{S}^1 \rrbracket$.
2. Given $u \in BV(\Omega;\mathbb{S}^1)$, there exists $T \in \mathrm{cart}(\Omega \times \mathbb{S}^1)$ such that $u = u_T$.
--
## Lifting in $BV$
Strictly related to the problem of lifting $BV$ functions with values in $\mathbb{S}^1$
- Dávila, Ignat. C. R. Acad. Sci. Paris, Ser. I (2003)
- Ignat. Ann. Inst. H. Poincaré Anal. Non Linéaire (2005)
- Canevari, Orlandi. J. Funct. Anal. (2020)
---
## An analogous problem in the smooth setting
Theorem [Giaquinta, Modica, Souček - Calc. Var. PDEs (1993)]
$\displaystyle \hspace{1.5em} \int_\Omega |\nabla u| \\, \mathrm{d} x \hspace{0em} \stackrel{\Gamma}{\rightarrow} \hspace{0em} \int_{\Omega \times \mathbb{R}^2} \Phi(\vec T) \\, \mathrm{d} |T| = \int_\Omega |\nabla u| \\, \mathrm{d} x + |\mathrm{D}^{\text{(c)}}u|(\Omega) + \mathcal{J}(u;\Omega)$
$\displaystyle \hspace{0em} \big(u \in C^\infty(\Omega;\mathbb{S}^1)\big) \hspace{12.5em} \big( u \in BV(\Omega;\mathbb{S}^1)\big)$
where $\displaystyle \mathcal{J}(u;\Omega) = \inf_{T \in \mathrm{cart}(\Omega \times \mathbb{S}^1)} \Big\\{ \int_{J_T} \ell_T(x) \mathrm{d} \mathcal{H}^1(x) : T = G_u + L \times \llbracket \mathbb{S}^1 \rrbracket \Big\\}$.
--
## Not subadditive
$\displaystyle u \colon \varepsilon \mathbb{Z}^2 \to \mathcal{S}_{N_\varepsilon} \quad \leadsto \quad G_\{u_\varepsilon\} \in \mathcal{D}_2(\Omega \times \mathbb{R}^2)$ with $\partial G_\{u_\varepsilon\} = - \mu_\{u_\varepsilon} \times \llbracket \mathbb{S}^1 \rrbracket$
---
## Theorem for $\varepsilon \ll \theta_\varepsilon \ll \varepsilon |\log \varepsilon|$ with no vortices
Recall:
- $\displaystyle \frac{1}{\varepsilon \theta_\varepsilon} E_\varepsilon(u_\varepsilon) \leq C \hspace{-0.3em} \implies \hspace{-0.3em} \frac{1}{\varepsilon^2 |\log \varepsilon|} E_\varepsilon(u_\varepsilon) \to 0 \hspace{-0.3em} \implies \hspace{-0.3em} \mu_{u_\varepsilon} \stackrel{\mathrm{f}}{\to} 0 \hspace{-0.3em} \implies \hspace{-0.3em} \partial G_{u_\varepsilon} \rightharpoonup 0$
- $\displaystyle \frac{1}{\varepsilon \theta_\varepsilon} E_\varepsilon(u_\varepsilon) \approx \int_{J_{u_\varepsilon}} \mathrm{d}_{\mathbb{S}^1}(u_\varepsilon^-, u_\varepsilon^+) |\nu_\{u_\varepsilon\}|_1 \mathrm{d} \mathcal{H}^1 = \int_\{\Omega \times \mathbb{R}^2\} \Phi_\{2,1\}(\vec G_\{u_\varepsilon\}) \mathrm{d}| G_\{u_\varepsilon\}|$
Theorem [Cicalese, O., Ruf - *CPAM* (2022)]
([Compactness](#/compactness)): $\hspace{0em} G_\{u_\varepsilon\} \rightharpoonup T = G_u + L \times \llbracket \mathbb{S}^1 \rrbracket \in \mathrm{cart}(\Omega \times \mathbb{S}^1)$
($\Gamma$-liminf): $\hspace{2.4em} \displaystyle \liminf_{\varepsilon \to 0} \frac{1}{\varepsilon \theta_{\varepsilon}} E_\varepsilon(u_\varepsilon) \geq \int_\{\Omega \times \mathbb{R}^2\} \Phi_\{2,1\}(\vec T) \mathrm{d}|T|$
$\hspace{12.8em} \displaystyle = \int_{\Omega} |\nabla u|_\{2,1\} \\, \mathrm{d} x + |\mathrm{D}^{(c)} u|_\{2,1\}(\Omega) + \mathcal{J}_\{2,1\}(u;\Omega)$
([$\Gamma$-limsup](#/limsup)): $\hspace{1.7em}$ Given $u \in BV(\Omega;\mathbb{S}^1)$ there exists $u_\varepsilon \to u$ such that
$\hspace{6.5em} \displaystyle \lim_{\varepsilon \to 0} \frac{1}{\varepsilon \theta_{\varepsilon}} E_\varepsilon(u_\varepsilon) = \int_{\Omega} |\nabla u|_\{2,1\} \\, \mathrm{d} x + |\mathrm{D}^{(c)} u|_\{2,1\}(\Omega) + \mathcal{J}_\{2,1\}(u;\Omega)$
--
## Compactness
[Back to Theorem](#/main-thm)
Note: $\quad \partial G_\{u_\varepsilon\} = - \mu_\{u_\varepsilon\} \times \llbracket \mathbb{S}^1 \rrbracket \quad$ and $\quad \mu_\{u_\varepsilon\} \stackrel{\text{flat}}{\to} 0 \quad \nRightarrow \quad \mathbb{M}(\partial G_{u_\varepsilon}) \leq C$
Solution: Modification lemma based on ball construction
--
## $\Gamma$-limsup
[Back to Theorem](#/main-thm)
Recipe:
1. Start with $u \in BV(\Omega;\mathbb{S}^1)$
2. Build the energy-optimal $T = G_u + L \times \llbracket \mathbb{S}^1 \rrbracket$
3. Approximate $G_{u_j} \rightharpoonup T$ with $u_j \in C^{\infty}(\Omega;\mathbb{S}^1)$
4. Discretize $u_j$ with piecewise constant function (with "close" values)
5. Build recovery sequence for piecewise constant function
6. [?](#/limsup-vortices)
---
## Theorem for $\varepsilon \ll \theta_\varepsilon \ll \varepsilon |\log \varepsilon|$ with vortices
Recall: [Cost of one vortex](#/vortex-cost)
$\displaystyle \frac{1}{\varepsilon \theta_\varepsilon} E_\varepsilon(u_\varepsilon) \approx \frac{1}{\varepsilon \theta_\varepsilon} 2 \pi \varepsilon^2 |\log \varepsilon| = 2 \pi \frac{\varepsilon |\log \varepsilon|}{\theta_\varepsilon} \to +\infty$
Theorem [Cicalese, O., Ruf - *CPAM* (2022)]
Let $\varepsilon \ll \theta_\varepsilon \ll \varepsilon |\log \varepsilon|$
Assume that $\displaystyle \frac{1}{\varepsilon \theta_\varepsilon} E_\varepsilon(u_\varepsilon) - 2 \pi M \frac{\varepsilon |\log \varepsilon|}{\theta_\varepsilon} \leq C$
$\displaystyle \hspace{6em} \Big( \implies \frac{1}{\varepsilon^2 |\log \varepsilon|} E_\varepsilon(u_\varepsilon) \leq 2\pi M + C\frac{\theta_\varepsilon}{\varepsilon |\log \varepsilon|}\Big)$
Then we have that:
$\hspace{2em}$ ([Compactness](#/compactness-vortices)): $\hspace{1em} \ldots$
$\hspace{2em}$ ([$\Gamma$-liminf](#/liminf-vortices)): $\hspace{3.1em} \ldots$
$\hspace{2em}$ ([$\Gamma$-limsup](#/limsup-vortices)): $\hspace{2.7em} \ldots$
--
[Back to Theorem](#/theorem-with-vortices)
## Compactness
(Compactness): $\hspace{0.5em} \mu_\{u_\varepsilon\} \stackrel{\text{flat}}{\to} \mu = \sum_{i=1}^N d_i \delta_{x_i}$ with $|\mu|(\Omega) \leq M$
$\hspace{6.5em}$ Moreover, if $|\mu|(\Omega)=M$, then $G_\{u_\varepsilon\} \rightharpoonup T$ in $\mathcal{D}_2(\Omega \times \mathbb{R}^2)$
$\hspace{6.5em}$ with $T \in \mathrm{cart}\big((\Omega \setminus \mathrm{supp}\\, \mu ) \times \mathbb{S}^1\big)$
$\hspace{6.5em}$ and $\partial T = - \mu \times \llbracket \mathbb{S}^1 \rrbracket$ in $\Omega \times \mathbb{R}^2$
--
## $\Gamma$-liminf
[Back to Theorem](#/theorem-with-vortices)
($\Gamma$-liminf): $\hspace{1em}$ If $u_\varepsilon \to u$ and $\mu_\{u_\varepsilon\} \stackrel{\text{flat}}{\to} \mu$ with $|\mu|(\Omega) = M$, then
$\displaystyle \hspace{5.35em} \liminf_{\varepsilon \to 0} \Big( \frac{1}{\varepsilon \theta_\varepsilon} E_\varepsilon(u_\varepsilon) - 2 \pi M \frac{\varepsilon |\log \varepsilon|}{\theta_\varepsilon} \Big)$
$\displaystyle \hspace{11em} \geq \int_{\Omega} |\nabla u|_\{2,1\} \\, \mathrm{d} x + |\mathrm{D}^{(c)}u|_\{2,1\}(\Omega) + \mathcal{J}_\{2,1\}(u,\mu;\Omega)$
$\hspace{5.35em}$ where $\displaystyle \mathcal{J}_\{2,1\}(u,\mu;\Omega) = \inf_\{ \stackrel{ T \in \mathrm{cart}((\Omega \setminus \mathrm{supp} \\, \mu) \times \mathbb{S}^1 ) }{\partial T = - \mu \times \llbracket \mathbb{S}^1 \rrbracket} \} \Big\\{ \int_\{J_T\} \ell_T |\nu_T|_1 \\, \mathrm{d} \mathcal{H}^1 \Big\\}$
--
## $\Gamma$-limsup
($\Gamma$-limsup): $\hspace{1em}$ Given $\mu = \sum_{i=1}^N d_i \delta_\{x_i\}$ with $|\mu|(\Omega) = M$ and $u \in BV(\Omega;\mathbb{S}^1)$
$\hspace{5.8em}$ there exists $u_\varepsilon \to u$ such that $\mu_{u_\varepsilon} \stackrel{\text{flat}}{\to} \mu$ and
$\hspace{5.8em} \displaystyle \lim_{\varepsilon \to 0} \Big( \frac{1}{\varepsilon \theta_\varepsilon} E_\varepsilon(u_\varepsilon) - 2 \pi M \frac{\varepsilon |\log \varepsilon|}{\theta_\varepsilon} \Big)$
$\displaystyle \hspace{8em} = \int_{\Omega} |\nabla u|_\{2,1\} \\, \mathrm{d} x + |\mathrm{D}^{(c)} u|_\{2,1\}(\Omega) + \mathcal{J}_\{2,1\}(u,\mu;\Omega)$
Follow the previous [**recipe**](#/limsup) far from vortices + dyadic interpolation
---
## A formal expansion (and Theorem for $\theta_\varepsilon \ll \varepsilon$)
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathcal{S}_{N_\varepsilon}$, with $N_\varepsilon \to +\infty$, i.e., $\theta_\varepsilon \to 0$
$\hspace{2em} \displaystyle \frac{1}{\varepsilon^2} E_\varepsilon(u_\varepsilon) \approx 2 \pi |\mu|(\Omega) |\log \varepsilon|$
$\hspace{8em} \displaystyle + \frac{\theta_\varepsilon}{\varepsilon} \Big( \int_{\Omega} |\nabla u|_\{2,1\} \\, \mathrm{d} x + |\mathrm{D}^{(c)} u|_\{2,1\}(\Omega) + \mathcal{J}_\{2,1\}(u,\mu;\Omega) \Big)$
Recall, in particular, Alicandro, De Luca, Garroni, Ponsiglione. *ARMA* (2014)
Spin field: $u \colon \varepsilon \mathbb{Z}^2 \to \mathbb{S}^1$
$\hspace{2em} \displaystyle \frac{1}{\varepsilon^2} E_\varepsilon(u_\varepsilon) \approx 2 \pi |\mu|(\Omega) |\log \varepsilon|$
$\hspace{8em} + \mathbb{W}(\mu) + \gamma |\mu|(\Omega)$
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# Thank you for the attention
$\displaystyle E\Big(\frac{x}{|x|};B_{R}\Big) = \int_{B_{R}} \Big| \nabla \frac{x}{|x|} \Big| \, \mathrm{d} x + 2 \pi R = 2 \pi R + 2 \pi R = 4 \pi R$
$\displaystyle E\Big(\frac{x}{|x|};B_1 \setminus \overline B_{r}\Big) = \int_{B_1 \setminus \overline B_{r}} \Big| \nabla \frac{x}{|x|} \Big| \, \mathrm{d} x = 2 \pi (1 - r)$
$\displaystyle E\Big(\frac{x}{|x|};B_1\Big) - E\Big(\frac{x}{|x|};B_{R}\Big) - E\Big(\frac{x}{|x|};B_1 \setminus \overline B_{r}\Big) = 4 \pi - 4 \pi R - 2 \pi (1 - r)$
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## Back to the original problem
Let us recall the [**discrete setting**](#/no-vortices-regime)
$XY$ $\text{?}$
$\text{vortices + cart}$
$BV$
fast
slow