#!/usr/bin/env python
# coding: utf-8

# 
# # Recent developments in the study of self-dual Chern-Simons vortices
# 
# > #### [Gabriella Tarantello](http://www.mat.uniroma2.it/~tarantel/), Università di Roma "Tor Vergata"
# 
# #### [Ascona Winter School 2016](http://www.math.uzh.ch/pde16/index-Ascona2016.html), [(alternate link)](http://www.monteverita.org/en/90/default.aspx?idEvent=295&archive=)
# 
# Book: [Selfdual Gauge Field Vortices, Birkhauser 2008](http://www.springer.com/gb/book/9780817643102)
# 
# 1. [Lecture 1](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/tarantello-lecture1.ipynb)
# 2. [Lecture 2](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/tarantello-lecture2.ipynb)
# 3. [Lecture 3](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/tarantello-lecture3.ipynb)
# 4. [Lecture 4](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/tarantello-lecture4.ipynb)
# 
# [source](https://github.com/colliand/ascona2016)

# ...since we've been talking about bubbling, I'm going to show you how bubbling also occurs in the theory of vortices.
# 
# This morning, we focused on planar Chern-Simons vortices (topological type). This amounted to proving the following...
# 
# **Theorem:** The problem:
# $$ -\Delta u = \frac{4}{k^2} e^u (1 - e^u) - 4 \pi \sum_{j=1}^N \delta_{p_j}$$
# with boundary condition $1 -e^u \in L^1 (\mathbb{R}^2) \iff u \rightarrow 0$ as $|x| \rightarrow +\infty$  
# admits a unique solution for fixed $\{ p_1, \dots, p_N \}$ repeated with multiplicity $\implies$ vortex configuration given as a minimizer of the energy with (Higgs field) $\phi$ with fixed degree = $N$. 
# 
# We also have in this case that the total energy $\frac{4}{k^2} \int_{\mathbb{R}^2} e^u (1 - e^u) = 4 \pi N$, so quantized. 

# ### Nontopological vortices:
# 
# $$ -\Delta u = ... $$
# with the (different) boundary condition $e^u \in L^1 (\mathbb{R}^2) \implies u \rightarrow - \infty \implies e^u \rightarrow 0$ which in turn tells me that $|\phi|$ admits a zero at $\infty$. 
# 
# $\implies$ total energy $ = \frac{4}{k^2} \int_{\mathbb{R}^2} e^u (1 - e^u)$ is no longer quantized by the parameter $N$. We typically call this integral $\beta$ and we investigate for which $beta$ we can perform an analysis. We have $^u \rightarrow 0$ but the convergence to zero is like $|x|^{2N - \frac{\beta}{2\pi}}$. The decay of $u$ is connected to the value of $\beta$. 
# 
# We have a nice ansatz: $u = v + \sum_{j=1}^N \ln |x - p_j|^2.$ We can determine that $\beta > 4 \pi (N+1)$. In fact, this is not sharp and the right size is $\beta > 8 \pi (N+1)$. The range of $\beta$ is big. For each such beta, we know there is a solution.

# Tomorrow will focus on the nontopological vortex case. Today, we will focus our efforts on the topological case. 

# ***
# 
# $$ -\Delta u = \frac{4}{k^2} e^u (1 - e^u) - 4 \pi \sum_{j=1}^N \delta_{p_j}$$ 
# in $(M,g)$ where $M$ is a closed surface. There is again an analogous situation with two different boundary conditions leading to "toplogical" and "nontopological" solutions.
# 
# * $u = w+c$
# * $\int w d \sigma_g = 0$
# * $C = \frac{1}{|M|} \int_M u$

# ...moving fast..discussing constraint equations...I'm not keeping up with the typing...reverse Hölder inequality....leads to a necessary condition for existence:
# $$ \frac{4 \pi N k^2}{|M|} \leq 1.$$

# **Theorem:** There exists a critical $k_c >0$ (depending upon various parameter) such that the Chern-Simons Vortex equation admits a solution on $M  \iff  0 < k \leq k_c$. Furthermore, $0<k\leq k_c$ the problem admits two solutions $u_{\pm}$ such that
# 
# 1. $u_+$ is local minimum where $u_+ = w_+ + c_+$ with $c_+$ solving the constraint condition with the + sign. In the limit $k \rightarrow 0^+, C_+ \rightarrow 0$ and $w_+ \rightarrow w_0$ pointwise.
# 2. (The other solutions is a mountain pass.) $u_- = w_- + c_-$ where $c_-$ satisfies the constraint with the minus sign... [K. Choi 2008]

# ...I got tired of typing...