# Schedule for: 16w5065 - Asymptotic Patterns in Variational Problems: PDE and Geometric Aspects

Beginning on Sunday, September 25 and ending Friday September 30, 2016

All times in Oaxaca, Mexico time, CDT (UTC-5).

Sunday, September 25 | |
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19:30 - 22:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Monday, September 26 | |
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07:30 - 08:45 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 09:40 |
Zhi-Qiang Wang: Localized nodal solutions for semiclassical nonlinear Schroedinger equations ↓ We investigate the existence of localized sign-changing solutions for the semiclassical nonlinear Schr\"odinger equation $−\epsilon^2 \Delta v + V (x)v = |v|^{p-2} v, v \in H^1 (\mathbb{R}^N ) $ where $N \ge 2$, $2 < p < 2^*$, $\epsilon> 0$ is a small parameter, and V is assumed to be bounded and bounded away from zero. When V has a local minimum point P, as $\epsilon \to 0$, we construct an infinite sequence of localized sign-changing solutions clustered at P and these solutions are of higher topological type in the sense that they are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. Our method combines the Byeon and Wang’s penalization approach and minimax method via a variant of the classical symmetric mountain pass theorem, and is rather robust without
using any non-degeneracy conditions. (Conference Room San Felipe) |

09:45 - 10:25 |
Isabella Ianni: A Morse index formula for the Lane-Emden problem ↓ We consider the semilinear Lane-Emden problem
\begin{equation}
\label{problemAbstract}
\left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }B\\
u=0\qquad\qquad\qquad\mbox{ on }\partial B
\end{array}\right.\tag{$\mathcal E_p$}
\end{equation}
where $B$ is the unit ball of $\mathbb R^N$, $N\geq3$, centered at the origin and $p\in(1,p_S)$, $p_S=\frac{N+2}{N-2}$.
We compute the Morse index of any radial solution $u_p$ of \eqref{problemAbstract}, for $p$ sufficiently
close to $p_S$. The proof exploits the asymptotic behavior of $u_p$
as $p\rightarrow p_S$ and the analysis of a limit eigenvalue problem. The result is obtained in collaboration with F. De Marchis and F. Pacella. (Conference Room San Felipe) |

10:30 - 11:00 | Coffee Break (Conference Room San Felipe) |

11:00 - 11:40 |
Jaeyoung Byeon: Nonlinear Schroedinger systems with large interaction forces between different components ↓ For nonlinear Schrodinger systems, one of basic concerns is a
construction of least energy vector solutions whose components are all
positive. Especially, we are interested in cases that the interactions
between different components are very strong while the strength of
self-interactions are fixed. For the systems
with two components, there have been many studies and we have a
relatively good understanding about existence and asymptotic behavior of
least energy vector solutions for large interaction force. When we
consider systems with three components and two different types of
interaction, repulsion and attraction, are involved, the construction of
a least energy vector solution is more involved. In this talk, I would
like to introduce recent results on construction of least energy vector
solutions and their asymptotic behavior when their interaction forces
are large. (Conference Room San Felipe) |

11:45 - 12:25 |
Manon Nys: Properties of ground states of nonlinear Schroedinger equations under a weak constant magnetic field ↓ Joint work with Denis Bonheure and Jean Van Schaftingen.
In this talk, we consider the nonlinear Schr\"odinger equation in the
presence of an external magnetic field $$−(\nabla + iA)^2 u + u = |u|^{p-2}u, \quad\text{in } \mathbb{R}^N,$$
where the magnetic operator is defined as
$$-(\nabla + iA)^2 := -\Delta - i\nabla \cdot A - 2iA \cdot \nabla + |A|^2.$$
Here we choose $A = (A_i)_{1\leq i\leq N}$ to be a linear magnetic potential,
corresponding to a constant magnetic field $B = (B_{ij})_{1\leq i,j\leq N}$, where
$B_{ij} = \partial_i A_j - \partial_j A_i$.
In particular, we focus on the ground states of this equation for $|B|$
sufficiently small.
First, we consider uniqueness (up to some "`translations"' and multiplication
by a complex phase) and symmetry properties of such ground states.
To study this, we use the known properties of the limit equation
$$−\Delta u + u = |u|^{p-2}u,\quad \text{in } \mathbb{R}^N,$$
and an implicit function argument.
Then, we obtain an improved asymptotic decay at infinity (with
respect to the case without magnetic field).
Finally, we consider the dependence of the groundstate energy $\mathcal{E}(B)$
on the magnetic field B and in particular we succeed to prove its monotonicity
(and to give an exact expression). (Conference Room San Felipe) |

12:30 - 12:40 | Group Photo (Hotel Hacienda Los Laureles) |

13:00 - 14:30 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:00 - 15:40 |
Kazunaga Tanaka: Nonlinear elliptic problems in singularly perturbed cylindrical domains ↓ In this talk we consider singular perturbation problem
for a nonlinear elliptic problem in perturbed cylindrical domains:
$$ \left\{
\begin{aligned}
-&\Delta u = g(u) \quad \hbox{in}\ \Omega_\epsilon,\\
&u>0 \qquad\quad \hbox{on}\ \Omega_\epsilon,\\
&u=0 \quad\qquad \hbox{on}\ \partial\Omega_\epsilon.\
\end{aligned}
\right.
$$
Here $g(s):\, \mathbb{R}\to\mathbb{R}$ is a continuous function with subcritical growth and
$$ \Omega_\epsilon=\{ (x,y)\in \mathbb{R}^k\times \mathbb{R}^\ell; \, x\in \mathbb{R}^k,\,
y\in D_{\epsilon x}\}
$$
and $D_x\subset \mathbb{R}^\ell$ is a domain depending on $x\in\mathbb{R}^k$ smoothly.
We show the existence of solutions which concentrate at
a prescribed part of domain. Especially we consider the situation where
the prescribed part is corresponding to \lq\lq local maxima'' (Conference Room San Felipe) |

15:45 - 16:25 |
Gianmaria Verzini: Spiralling asymptotic profiles of competition-diffusion systems ↓ In this talk we consider solutions of the competitive elliptic system
$$
\begin{cases}
-\Delta u_i = - \beta \sum_{j \neq i} a_{ij} u_i u_j & \text{in $\Omega\subset
\mathbb{R}^2$} \\
u_i =g & \text{in $\partial\Omega$}
\end{cases} \qquad i=1,\dots,k,
$$
and their asymptotic profiles when $\beta\to+\infty$. We shall focus our attention on the asymmetric case: $a_{ij}\neq a_{ji}$. This is a joint result with A. Salort, S. Terracini, A. Zilio. (Conference Room San Felipe) |

16:30 - 16:50 | Coffee Break (Conference Room San Felipe) |

16:50 - 17:30 |
Julian Fernando Chagoya Saldana: Ground States for Irregular and Indefinite Superlinear Schroedinger Equations ↓ We consider the existence of a ground state for the
subcritical stationary semilinear Schrodinger equation $-\Delta u +
u=a(x)|{u}|^{p-2}u$ in $H^1$, where $a\in L^\infty(\mathbb{R}^N)$ may change
sign. Our focus is on the case where loss of compactness occurs at the
ground state energy. By providing a new variant of the Splitting Lemma
we do not need to assume the existence of a limit problem at infinity,
be it in the form of a pointwise limit for $a$ as $|{x}|\to\infty$ or
of asymptotic periodicity. That is, our problem may be \emph{irregular}
at infinity. In addition, we allow $a$ to change sign near infinity, a
case that has never been treated before. (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Tuesday, September 27 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 09:40 |
Massimo Grossi: Entire radial and nonradial solutions for systems with critical growtht, part 1 ↓ We consider a (nonvariational) system involving the critical
Sobolev exponent in the whole space. This can be seen as an extension of
the Toda system to higher dimension. Using the bifurcation theory we
will show the existence of a radial solution. (Conference Room San Felipe) |

09:45 - 10:25 |
Francesca Gladiali: Entire radial and nonradial solutions for systems with critical growtht, part 2 ↓ We continue the discussion about the system considered in part
1. Using some suitable symmetries properties of the spherical harmonics
we prove the existence of nonradial solutions. (Conference Room San Felipe) |

10:30 - 11:00 | Coffee Break (Conference Room San Felipe) |

11:00 - 11:40 |
Hugo Tavares: Paths to uniqueness of critical points and applications ↓ In this talk we address the question whether a certain special
subset of critical points of a Fr\'{e}chet differentiable functional is
necessarily a singleton, and we prove an abstract result that gathers
many interesting problems in its framework. We will provide several
applications and prove uniqueness of positive solutions of some elliptic
problems, covering both old and new results. Essentially, we build paths
to uncover the hidden convexity of the associate functionals.
This is based on a joint work with D. Bonheure, J. Foldes, E. Moreira dos
Santos and A. Salda\~na (Conference Room San Felipe) |

11:45 - 12:25 |
Jorge Faya: Concentrating solutions for a Hénon-type problem on general domains ↓ We consider the problem%
\begin{equation*}
\qquad\left\{
\begin{array}
[c]{ll}%
-\Delta u = \beta(x)|u|^{p-1-\epsilon }u & \text{in }\Omega,\\
u=0 & \text{on }\partial\Omega,
\end{array}
\right.
\end{equation*}
where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3,$ $p:=\frac{N+2}{N-2}$ is the Sobolev critical exponent, $\epsilon$ is a small positive parameter and the function $\beta\in C^{1}(\overline{\Omega})$ is strictly positive on $\overline{\Omega}$.
In this talk we shall present a recent result about the existence of positive and sign changing solutions whose asymptotic profile is a sum of $k$ bubbles which accumulate at a single point at the boundary as $\varepsilon$ tends to zero.
This is joint work with professors Juan D\'avila and Fethi Mahmoudi. (Conference Room San Felipe) |

13:00 - 14:30 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:00 - 15:40 |
Yannick Sire: Geometry of solutions for semi-linear equations in convex domains ↓ It was believed for a long time by the community that solutions of semi linear equations with reasonable nonlinearities in convex domains would be quasi-concave (i.e. their super level sets would be convex). We provide a counter-example to this conjecture in the plane. This is joint work with F. Hamel and N. Nadirashvili. (Conference Room San Felipe) |

15:45 - 16:25 |
Monica Musso: Existence, compactness and non-compactness results on the fractional Yamabe problem in large dimensions ↓ Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [h])$.
The fractional Yamabe problem addresses to solve \[P^{\gamma}[g^+,h] (u) = cu^{n+2\gamma \over n-2\gamma}, \quad u > 0 \quad \text{on } M\] where $c \in \mathbb{R}$ and $P^{\gamma}[g^+,h]$ is the fractional conformal Laplacian whose principal symbol is $(-\Delta)^{\gamma}$.
In this talk, I will present some recent results concerning existence of solutions to the fractional Yamabe problem, and also properties of compactness and non compactness of its solution set, in comparison with what is known in the classical case.
These results are in collaboration with Seunghyeok Kim and Juncheng Wei. (Conference Room San Felipe) |

16:30 - 17:50 | Coffee Break (Conference Room San Felipe) |

16:50 - 17:30 |
Juan Carlos Fernández: Multiplicity of Nodal Solutions for Yamabe Type Equations ↓ Given a compact Riemannian manifold $(M, g)$ without boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish existence and multiplicity of positive and sign changing solutions to the following Yamabe type equation
$$
div g(a\nabla u) + bu = c|u|^{2*-2} u \quad\text{on } M
$$
where $\div g$ denotes the divergence operator on $(M; g)$,$ a, b$ and $c$ are smooth functions with $a$ and $c$ positive, and $2*=\frac{2m}{m-2}$ denotes the critical Sobolev exponent. In particular, if $R_g$ denotes the scalar curvature, we give some examples where the Yamabe equation
$$
-\frac{4(m-1)}{m-2}\Delta_g u+R_g u = \kappa u^{2*-2}\quad\text{on } M.
$$
admits an infinite number of sign changing solutions. We also study the lack of compactness of these problems in a symmetric setting and how the symmetries restore it at some energy levels. This allows us to use a suitable variational principle to show the existence and multiplicity of such solutions. This is joint work with Monica Clapp. (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Wednesday, September 28 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 09:40 |
Changfeng Gui: The Sphere Covering Inequality and its application to a Moser-Trudinger type inequality and mean field equations ↓ In this talk, I will present a new inequality: the Sphere
Covering Inequality, which states that the total area of two {\it
distinct} surfaces with Gaussian curvature 1, which are also
conformal to the Euclidean unit disk with the same conformal factor on
the boundary, must be at least $4 \pi$. In other words, the areas of
these surfaces must cover the whole unit sphere after a proper
rearrangement. We apply the Sphere Covering Inequality to show the best
constant of a Moser-Trudinger type inequality conjectured by A. Chang
and P. Yang. Other applications of this inequality include the
classification of certain Onsager vortices
on the sphere, the radially symmetry of solutions to Gaussian curvature
equation on the plane, classification of solutions for mean field
equations on flat tori and the standard sphere, etc. The resolution of
several interesting problems in these areas will be presented. The
work is jointly done with Amir Moradifam from UC Riverside. (Conference Room San Felipe) |

09:45 - 10:25 |
Teresa D'Aprile: Existence results for the prescribed Gauss curvature problem on closed surfaces ↓ Let $(\Sigma, g)$ be a compact orientable surface without boundary and with metric $g$ and Gauss curvature $\kappa_g$.
Given points $p_i\in \Sigma$ and a Lipschitz function $K$ defined on $\Sigma$, a classical problem in differential geometry is the question on the existence of a metric $\tilde g$ conformal to $g$ in $\Sigma\setminus\{p_1,\ldots, p_m\}$, namely $$\tilde g = e^{u} g\hbox{ in }\Sigma\setminus\{p_1,\ldots, p_m\}$$ admitting conical singularities of orders $\alpha_i$'s (with $\alpha_i>-1$) at the points $p_i$'s and having $K$ as the associated Gaussian curvature in $\Sigma\setminus\{p_1,\ldots, p_m\}$. The question reduces to solving a singular Liouville-type equation on $\Sigma$
\begin{equation*}-\Delta_g u+2\kappa_g =2K e^u-4\pi \sum_{i=1}^m\alpha_i\delta_{p_i}\hbox{ in }\Sigma\end{equation*} where $\delta_p$ denotes Dirac mass supported at $p$.
By employing a min-max scheme jointly with a finite dimensional reduction method, we deduce new existence results in the perturbative regime when the quantity
$\chi(\Sigma)+\sum_{i=1}^m \alpha_i$ approaches a positive even integer, where $\chi(\Sigma)$ is the Euler characteristic of the surface $\Sigma$. (Conference Room San Felipe) |

10:30 - 11:00 | Coffee Break (Conference Room San Felipe) |

11:00 - 11:40 |
David Ruiz: Blowing-up solutions for the Toda system ↓ The Toda system appears naturally in the non abelian
Chern-Simons theory, and has been very much studied recently. It can
also be regarded as a natural generalization to systems of the Liouville
equation. In this talk we first survey some of the known results
obtained by different approaches: blow-up analysis, degree theory,
variational methods \dots
Secondly we present constructions of blowing-up solutions for the Toda
system. Those solutions have the common feature that one component is
not quantized, and some global mass is present. This is a phenomenon
with no analogue in the single equation case. This is joint work with
Teresa D'Aprile (Rome II) and Angela Pistoia (Rome I). (Conference Room San Felipe) |

11:45 - 12:25 |
Shusun Yan: Degree Counting Formula and Shadow System ↓ We will discuss the degree counting formula for the
solutions of a shadow system. To obtain such formula, we will use the
blow-up analysis, together with the Pohozaev identity, to calculate the
local mass of the
solutions at the blow-up points. The motivation for us to carry out
this analysis is to derive the degree counting formula for the Tada systems.
This is a joint work with C.-S. Lin. (Conference Room San Felipe) |

13:00 - 14:30 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:00 - 15:40 |
Lia Bronsard: Minimizers of the Landau-de Gennes energy around a spherical colloid particle ↓ We consider energy minimizing configurations of a nematic
liquid crystal around a spherical colloid particle, in the context of
the Landau-de~Gennes model. The nematic is assumed to occupy the
exterior of a ball, and satisfy homeotropic weak anchoring at the
surface of the colloid and approach a uniform uniaxial state far from
the colloid. We study the minimizers in two different limiting regimes:
for balls which are small compared to the characteristic length scale,
and for large balls. The relationship between the radius and the
anchoring strength is also relevant. For small balls we obtain a
limiting quadrupolar configuration, with a ``Saturn ring'' defect for
relatively strong anchoring, corresponding to an exchange of eigenvalues
of the Q-tensor. In the limit of very large balls we obtain an
axisymmetric minimizer of the Oseen—Frank energy, and a dipole
configuration with exactly one point defect is obtained. This is joint
work with Stan Alama and Xavier Lamy. (Conference Room San Felipe) |

15:45 - 16:25 |
Enrico Valdinoci: Fractional Laplacian of divergent functions ↓ We introduce a notion of fractional Laplacian
for functions which grow more than linearly at infinity. In such case,
the operator is not defined in the classical sense: nevertheless,
we can give an ad-hoc definition which can be
useful for applications in various fields,
such as blowup and free boundary problems.
In this setting, when the solution has a polynomial
growth at infinity, the right hand side of the equation
is not just a function, but an equivalence class of functions
modulo polynomials of a fixed order.
We also present a sharp version of the Schauder estimates in this framework and a Liouville Theorem.
The results presented have been recently obtained in collaboration
with Serena Dipierro and Ovidiu Savin. (Conference Room San Felipe) |

16:30 - 16:50 | Coffee Break (Oaxaca) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Thursday, September 29 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 09:40 |
Filomena Pacella: The singular Liouville problem in the plane as limit of Lane-Emden problems: asymptotics and Morse index ↓ We present recent results that show as the singular Liouville equation in the plane can be viewed as a limit of semilinear elliptic equations of Lane-Emden type. This is achieved through the study of the asymptotic behavior of families of symmetric sign changing solutions whose nodal line does not touch the boundary. In particular this phenomenum arises while studying radial sign changing solutions of Lane Emden problems. An accurate study of the spectrum of the linearized operator shows a relation between the Morse index of these solutions and that of a specific solution of the limit problem.
The results are contained in some papers in collaboration with F. De Marchis-I.Ianni and M.Grossi-C. Grumiau. (Conference Room San Felipe) |

09:45 - 10:25 |
Jean Dolbeault: Symmetry in interpolation inequalities ↓ In presence of radially symmetric weights in an Euclidean
space, it is well known that symmetry breaking may occur: the minimizing
functions of functionals which are invariant under rotation are, in some
cases, not radially symmetric. This usually follows from a linear
stability analysis of the minimizers. The goal of this lecture is to
investigate the reverse property and establish, in some interpolation
inequalities, when the local linear stability of radial optimal
functions means that the global optimal functions are in fact radially
symmetric. The main tool is a flow: spectral properties of the
linearized operator around radial optimizers can be interpreted in terms
of large time asymptotics of the solution to the evolution equation and
related with the optimal constant in the inequality. The symmetry range
depends on a parameter, which can be used to classify the solutions of
the Euler-Lagrange equations. A singular limit can be identified when
the parameter takes large values. The interpolation inequality has a
spectral counterpart for Schrödinger operators, which allows to quantify
the distance to a semi-classical regime. Various equivalent problems on
spheres and cylinders can also be considered. This is joint work with
various collaborators, among which Maria J. Esteban, Michal Kowalczyk,
Michael Loss, and Matteo Muratori. (Conference Room San Felipe) |

10:30 - 11:00 | Coffee Break (Conference Room San Felipe) |

11:00 - 11:40 |
Serena Dipierro: Boundary behavior and graph properties of nonlocal minimal surfaces ↓ We recall the notion of nonlocal minimal surfaces and we discuss their
qualitative and quantitative interior and boundary behavior. In
particular, we present some optimal examples in which the surfaces stick
at the boundary. This phenomenon is purely nonlocal, since classical
minimal surfaces do not stick at the boundary of convex domains. We also
discuss the graph properties of the nonlocal minimal surfaces. (Conference Room San Felipe) |

11:45 - 12:25 |
Mouhamed Moustapha Fall: Constant Nonlocal Mean Curvature hypersurfaces ↓ We present recent results on the existence of critical points
of the nonlocal (or fractional) perimeter functional under volume
constraints in periodic media. These critical points are called sets
with Constant Nonlocal Mean Curvature (CNMC). Since the only bounded
CNMC set is the ball, we will consider unbounded CNMC sets for this
talk. These sets bifurcate from parallel cuves, parallel planes,
cylinders and translation invariant lattices of spheres. The
construction of these objects amounts to study quasilinear type
fractional equations, and local inversion arguments have been the main
tool we used to solve these equations.
Joint works with X. Cabrè, J. Solà-Morales, T. Weth, E.A. Thiam and I.A.
Minlend. (Conference Room San Felipe) |

13:00 - 14:30 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:00 - 15:40 |
Juan Davila: Hölder estimates for solutions of a MEMS equation ↓ We prove sharp Hölder estimates for sequences of positive solutions
of a nonlinear elliptic problem with negative exponent. As a
consequence, we prove the existence of solutions with isolated
ruptures in a bounded convex domain in two dimensions.
This is joint work with Kelei Wang (Wuhan University) and Juncheng Wei
(University of British Columbia). (Conference Room San Felipe) |

15:45 - 16:25 |
Pavol Quittner: Threshold solutions of a semilinear parabolic equation ↓ If $p>1+2/n$ then the nonlinear heat equation $u_t-\Delta u=u^p, \quad x\in \mathbb{R}^n,\ t>0,$ possesses both positive global solutions
and positive solutions which blow up in finite time. We are interested
in the large-time behavior of radial positive solutions lying on the
borderline between global existence and blow-up.
Reference:\\
P. Quittner: Threshold and strong threshold solutions of a semilinear
parabolic equation, arXiv:1605.07388\\ (Conference Room San Felipe) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Friday, September 30 | |
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07:45 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

12:00 - 15:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |