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

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.

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.

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.

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).

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''

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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 $\Sigma$.

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).

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.

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.

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.

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.

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.

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.

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.

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).

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\\