Schedule for: 19w5065 - Nonlinear Geometric PDE's

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Let K be an elliptic (not necessarily linear) second order differential operator. Suppose that we want to minimise the L-infinity norm of K(u) for functions u satisfying suitable boundary conditions. Here K may represent, e.g., the curvature of a curve in the plane or the scalar curvature of a Riemannian manifold in a fixed conformal class, but the problem is not restricted to questions with a geometric background. If the operator and the boundary conditions are such that the equation K(u) = 0 has a solution, then the problem is of course trivial. But since this is a second order variational problem, it may be appropriate to prescribe u as well as its first derivative on the boundary of its domain, which in general rules out this situation. In the cases studied so far, the solution, while not trivial, still has a nice structure, and one feature is that |K(u)| is always constant. The sign of K(u) may jump, but we have a characterisation of the jump set in terms of a linear PDE. Furthermore, in some cases we have a unique solution, even though the underlying functional is not strictly convex. This talk is based on joint papers with H. Schwetlick and with N. Katzourakis and the PhD thesis of Z. Sakellaris.

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^2$ containing the origin. We are concerned with the following Liouville equation with Dirac mass measure $$ \left\{ \begin{aligned}&- \Delta u = \lambda e^u-4\pi N_\lambda {\mathcal \delta}_0& \hbox{ in }& \Omega,\\ & \ u=0 & \hbox{ on }& \partial \Omega. \end{aligned} \right. $$ Here $\lambda$ is a positive small parameter, ${ \mathbb \delta}_0$ denotes Dirac mass supported at $0$ and $N_\lambda $ is a positive number close to an integer $N$ ($N\geq 2$) from the right side. We assume that $\Omega$ is $(N+1)$-symmetric and the regular part of the Green's function satisfies a nondegeneracy condition (both assumptions are verified if $\Omega$ is the unit ball) and we provide an example of non-simple blow-up as $\lambda \to 0^+$ exhibiting a non-symmetric scenario. More precisely we construct a family of solutions split in a combination of $N+1$ bubbles concentrating at 0 arranged on a tiny polygonal configuration centered at $0$. This is a joint work with Juncheng Wei (University of British Columbia).

We construct solutions of the Keller-Segel system which blow-up in infinite time in the form of asymptotic aggregation in the critical mass case, with a method that does not rely on radial symmetry, and applies to establish estability of the phenomenon.

We construct a family of entire solutions of the 3D Ginzburg-Landau equation with vortex lines given by interacting helices, with degree one around each filament and total degree an arbitrary positive integer. Existence of these solutions was conjectured by del Pino and Kowalczyk (2008), and answers negatively a question of Brezis analogous to the the Gibbons conjecture for the Allen-Cahn equation. This is joint work with Manuel del Pino, María Medina and Remy Rodiac.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Motivated in part by problems arising in micromagnetics, we study several variational models of Ginzburg-Landau type, depending on a small parameter $\epsilon >0$, for (tangent) vector fields on a 2-dimensional compact Riemannian surface. As $\epsilon\to 0$, the vector fields tend to be of unit length and develop singular points of a (non-zero) index, called vortices. Our main result determines the interaction energy between these vortices as $\epsilon\to 0$, allowing us to characterize the asymptotic behaviour of minimizing sequence. This is joint work with Radu Ignat.

In this talk, we first recall how some monotonicity formulas can be derived along the level set flow of the capacitary potential associated with a given bounded domain $\Omega$. A careful analysis is required in order to preserve the monotonicity across the singular times, leading in turn to a new quantitative version of the Willmore inequality. Remarkably, such analysis can be carried out without any \emph{a priori} knowledge of the size of the singular set. Hence, the same order of ideas applies to the $p$-capacitary potential of $\Omega$, whose critical set, for $p\neq2$, is not necessarily negligible. In this context, a generalised version of the Minkowski inequality is deduced. Joint works with M. Fogagnolo and L. Mazzieri.

In this talk, we first recall how some monotonicity formulas can be derived along the level set flow of the capacitary potential associated with a given bounded domain $\Omega$. A careful analysis is required in order to preserve the monotonicity across the singular times, leading in turn to a new quantitative version of the Willmore inequality. Remarkably, such analysis can be carried out without any \emph{a priori} knowledge of the size of the singular set. Hence, the same order of ideas applies to the $p$-capacitary potential of $\Omega$, whose critical set, for $p\neq2$, is not necessarily negligible. In this context, a generalised version of the Minkowski inequality is deduced. Joint works with M. Fogagnolo and L. Mazzieri.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider a class of three-dimensional ``CR manifolds'' which are modelled on the Heisenberg group. We introduce a natural concept of ``mass'' and prove its positivity under the conditions that the Webster curvature is positive and in relation to their (holomorphic) embeddability properties. We apply this result to the CR Yamabe problem, and we discuss the properties of Sobolev-type quotients, giving some counterexamples to the existence of minimisers for ``Rossi spheres'', in sharp contrast to the Riemannian case. This is joint work with J.H.Cheng and P.Yang.

In this talk I will discuss recent work with P. Daskalopoulos on sufficient conditions to prove uniqueness of complete graphs evolving by mean curvature flow. It is interesting to remark that the behaviour of solutions to mean curvature flow differs from the heat equation, where non-uniqueness may occur even for smooth initial conditions if the behaviour at infinity is not prescribed for all times.

I will consider blowing-up solution for the Gelfand problem on planar domains. It is well known that blow up at a single point must occur at a critical point x of a ``reduced functional'' F, whereas uniqueness of blowing up families has been recently shown provided x is a non-degenerate critical point of F. We showed that, if x is a degenerate critical point of F and satisfies some additional generic condition, then one may have two solutions blowing up at the same point. Solutions are constructed using a Lyapunov-Schmidt reduction. This is a joint work with Luca Battaglia and Angela Pistoia.

We consider the periodic solutions of a nonlinear elliptic system derived from the Maxwell-Chern-Simons model on a flat torus $\Omega$: $$\left\{\begin{array}{l} \Delta u=\mu(\lambda e^u-N)+4\pi\sum_{i=1}^n m_{i}\delta_{p_i},\\ \Delta N=\mu (\mu+\lambda e^u)N-\lambda \mu(\lambda+\mu)e^u \end{array} \right. \mbox{ in }\Omega, $$ where $\lambda, \mu>0$ are positive parameters. We obtain a Brezis-Merle type classification result for this system when $\lambda, \mu \to \infty$ and $\lambda<<\mu$. We also construct blow up solutions to this system. This is a joint work with Youngae Lee and Kwon Ohsang.

In this talk I will discuss two new constructions of the multiple end solutions. In the case of the Allen-Cahn equation the ends are asymptotic to the Simons cone in $\mathbb R^8$. The case of the generalized second Painlev\'e equation in $\mathbb R^2$ is somehow different since there is no apparent underlying geometric problem. Yet we can interpret the behavior of the solution as being asymptotic along the axis to: two one dimensional Hastings-McLeod solution, the heteroclinic solution of the Allen-Cahn equation and the trivial solution.

I will discuss some results obtained in collaboration with Massimo Grossi, Angela Pistoia and Daisuke Naimen concerning the existence of nodal solutions for the problem $$ -\Delta u = \lambda u e^{u^2+|u|^p} \text{ in }\Omega, u = 0 \text{ on }\partial \Omega, $$ where $\Omega\subseteq \mathbb R^2$ is a bounded smooth domain and $p\to 1^+$. If $\Omega$ is ball, it is known that the case $p=1$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions with $\lambda$ close to $0$. In our work we construct a blowing-up family of nodal solutions to such problem as $p\to 1^+$, when $\Omega$ is an arbitrary domain and $\lambda$ is small enough. To our knowledge this is the first construction of sign-changing solutions for a Moser-Trudinger type critical equation on a non-symmetric domain.

In this talk, which is the first part of a joint seminar with G. Catino (Politecnico di Milano), I will introduce some generalization of certain canonical Riemannian metrics, presenting two possible approaches (curvature conditions with potential and critical metrics of Riemannian functionals). The main result is related to the existence of a new canonical metric, which generalizes the condition of harmonic Weyl curvature, on every 4-dimensional closed manifold.

In this talk, which is the second part of a joint seminar with P. Mastrolia (Università degli Studi di Milano), I will present some results concerning rigidity and existence of canonical metrics on closed (compact without boundary) four manifolds. In particular I will consider Einstein metrics, Harmonic Weyl metrics and some generalizations. These are joint works with P. Mastrolia (UniMI), D.D. Monticelli and F. Punzo (PoliMi).

We consider the equation $\triangle_g u+hu=|u|^{2^*-2}u$ in a closed Riemannian manifold $(M,g)$, where $h$ is some Holder continuous function in $M$ and $2^* = \frac{2n}{n-2}$, $n:=dim M$. We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is a priori bounded. We obtain this result under the conditions that $n \ge 7$ and $h<(n-2)/(4(n-1)) S_g$ in $M$, where $S_g$ is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function $h$. This is a joint work with J. V\'etois (McGill University, Montr\'eal)

Motivated by the the work of K. Uhlenbeck, we discuss minimal immersions of closed surfaces of genus larger than 1 on hyperbolic 3-manifold. In this respect we establish multiple existence for the Gauss-Codazzi equations and describe the asymptotic behaviour of the solutions in terms of the prescribed conformal structure and holomorphic quadratic differential whose real part identifies the corresponding second fundamental form. Joint work with Z. Huang and M. Lucia

In this talk I will discuss some recent constructions of blow-up solutions for a Fujita type problem for powers $p$ related to the critical Sobolev exponent. Both finite type blow-up (of type II) and infinite time blow-up are considered. This research program is in collaboration with C. Cortazar,M. del Pino and J. Wei.

We discuss the existence of critical points of the Moser-Trudinger functional in dimension 2 with arbitrarily prescribed Dirichlet energy using degree theory. If time permits, we will also sketch an approach on Riemann surfaces using a min-max method \'a la Djadli-Malchiodi. This talk is based on joint works (and a work in progress) with Francesca De Marchis, Olivier Druet, Andrea Malchiodi, Gabriele Mancini and Pierre-Damien Thizy.

Since Schoen raised the question of compactness of the full set of solutions of the Yamabe problem in the $C^0$ topology (in 1988), it had been generally expected that the solution set must be $C^0$-compact unless the underlying manifold is conformally equivalent to the standard sphere. In 2008-09, Khuri, Marques, Schoen himself and Brendle gave the surprising answer that the expectation holds whenever the dimension of the manifold is less than 25 (under the validity of the positive mass theorem whose proof is recently announced by Schoen and Yau) but does not if the dimension is 25 or greater. On the other hand, concerning the fractional Yamabe problem on a conformal infinity of an asymptotically hyperbolic manifold, Kim, Musso, and Wei considered an analogous question and constructed manifolds of high dimensions whose solution sets are $C^0$-noncompact (in 2017). In this talk, we show that the solution set is $C^0$-compact if the conformal infinity is non-umbilic and its dimension is 7 or greater. Our proof provides a general scheme toward other possible compactness theorems for the fractional Yamabe problem. This is joint work with Monica Musso (University of Bath, UK) and Juncheng Wei (University of British Columbia, Canada).

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) such that $0\in\partial \Omega$. In this talk, we consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, $$ \left\{ \begin{array}{llll} -\Delta u-\gamma \frac{u}{|x|^2}- h(x) u &=& \frac{|u|^{2^\star(s)-2}u}{|x|^s} \ \ &\text{in } \Omega,\\ \hfill u&=&0 &\text{on }\ \partial\Omega, \end{array} \right.\eqno{(E)} $$ where $0 (TCPL 201)

We study the quantization properties for a non-local mean-field equation and give a necessary and sufficient condition for the existence of solution for a ``Mean Field''-type equation in an interval with Dirichlet-type boundary condition. We restrict the study to the $1$-dimensional case and consider the fractional mean-field equation on the interval $I = (-1, 1)$ $$(-\Delta)^\frac{1}{2} u=\rho \frac{e^{u}}{\int_I e^{u}dx},$$ subject to Dirichlet boundary conditions. As in the $2-$dimensional case, it is expected that for a sequence of solutions to our equation, either we get a $\mathcal{C}^{\infty}$ limiting solution or, after a suitable rescaling, we obtain convergence to the Liouville equation. Then, we can reduce the problem to the study of the non-local Liouville's equation. One of the key points here is to find an appropriate Pohozaev identity. We prove that existence holds if and only if $\rho < 2\pi$. This requires the study of blowing-up sequences of solutions. In particular, we provide a series of tools which can be used (and extended) to higher-order mean field equations of non-local type. We provide a completely non-local method for this study, since we do not use the localization through the extension method. Instead, we use the study of blowing-up sequences of solutions. This is a work done in collaboration with with A. Hyder, Y. Sire and L. Martinazzi.

We prove uniqueness and non-degeneracy of solutions for the mean field equation blowing-up on a non-degenerate blow-up set. Analogous results are derived for the Gelfand equation. The argument is based on sharp estimates for bubbling solutions and suitably defined Pohozaev-type identities. This is joint project with D. Bartolucci, Y. Lee and W. Yang.

We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying a weighted Poincar\'e inequality outside a compact set. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green's function vanishing at infinity. On the source function we assume a sharp pointwise decay depending on the weight appearing in the Poincar\'e inequality and on the behavior of the Ricci curvature at infinity. We do not require any curvature or spectral assumptions on the manifold. This result is a joint work with G. Catino and F. Punzo (Politecnico di Milano).

Given an isoparametric function $f$ on the round sphere and considering the space of functions $w\circ f$, we reduce the Yamabe-type problem $$(1)\qquad -\Delta_{g_0}+\lambda u=\lambda |u|^{p-1}u\ \hbox{on}\ \mathbb S^n$$ with $\lambda>0$ and $p>1$, into a second order singular ODE of the form $$w\rq{}\rq{}+{h(r)\over \sin r} w\rq{}+\lambda \left(|w|^{p-1}w-w\right)=0,$$ with boundary conditions $w\rq{}(0)=0$ and $w\rq{}(\pi)=0$, and where $h$ is a monotone function with exactly one zero on $[0, \pi]$. Using a double shooting method, for any $k\in\mathbb N$, if $n_1\le n_2$ are the dimensions of the focal submanifolds determined by $f$ and if $p \in \left(1,\frac{n-n_1+2}{n-n_1-2}\right)$, this problem admits a nodal solution having at least $k$ zeroes. This yields a solution to problem $(1)$ having as nodal set a disjoint union of at least $k$ connected isoparametric hypersurfaces. As an application and using that the Hopf fibrations are Riemannian submersions with minimal fibers, we give a multiplicity result of nodal solutions to the Yamabe problem on $\mathbb C P^m$ and on $\mathbb HP^m,$ the complex and quaternionic projective spaces respectively, with $m $ odd. This is a joint work with Jimmy Petean and Oscar Palmas.

We consider an equation in divergence form with a singular/degenerate weight \[ -\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or}\; \textrm{div}(|y|^aF(x,y))\;, \] Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove H\"older continuity of solutions and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $C^{0,\alpha}$ and $C^{1,\alpha}$ a priori bounds for approximating problems in the form \[ -\mathrm{div}((\varepsilon^2+y^2)^a A(x,y)\nabla u)=(\varepsilon^2+y^2)^a f(x,y)\; \quad\textrm{or}\; \textrm{div}((\varepsilon^2+y^2)^aF(x,y)) \] as $\varepsilon\to 0$. Finally, we derive $C^{0,\alpha}$ and $C^{1,\alpha}$ bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.

In this talk, we will consider the question of existence of positive blowing-up solutions to a class of conformally invariant elliptic equations of second order on a closed Riemannian manifold. A result of Olivier Druet provides necessary conditions for the existence of blowing-up solutions whose energy is a priori bounded. Essentially, these conditions say that for such solutions to exist, the potential function in the limit equation must coincide, up to a constant factor, at least at one point, with the scalar curvature of the manifold and moreover, in low dimensions, the weak limit of the solutions must be identically zero. I will present new existence results showing the optimality of these conditions. I will discuss in particular the case of low dimensions and the important role of the mass of the Green's function in this case. This is a joint work with Fr\'ed\'eric Robert (Universit\'e de Lorraine).

We study radially symmetric solutions for a semilinear equation with fractional Laplacian. Contrary to the local case, where we can give a solution to an ODE by simply looking at its phase portrait, in the nonlocal case we develop several new methods. We will give some applications, in particular to the existence of solutions of the singular fractional Yamabe problem, and the uniqueness of steady states of aggregation-diffusion equations.

In this talk I will consider nonlinear elliptic equations involving the Laplace or the p-Laplace operator and nonlinearities with supercritical growth, from the viewpoint of the Sobolev embedding. I'll present some new nonexistence results in contractible and non starshaped domains. The domains that are considered can be arbitrarily close to non contractible domains and their geometry can be very complex. The results presented are developed in collaboration with Donato Passaseo.

I will report on a recent result, in collaboration with A. Malchiodi, concerning a four-dimensional PDE of Liouville type arising in the theory of log-determinants in conformal geometry. The differential operator combines a linear fourth-order part with a quasi-linear second-order one. Since both have the same scaling behavior, compactness issues are very delicate and even the ``linear theory'' is problematic. For the log-determinant of the conformal laplacian and of the spin laplacian we succeed to show existence and logarithmic behavior of fundamental solutions, quantization property for non-compact solutions and existence results via critical point theory.

I will consider a double mean field-type Liouville PDE on a compact surface with boundary, with a nonlinear Neumann condition. This equation is related to the problem of prescribing both the Gaussian curvature and the geodesic curvature on the boundary. I will discuss blow-up analysis, a sharp Moser-Trudinger inequality for the energy functional, existence of minmax solution when the energy functional is not coercive. The talk is based on a work in progress with Rafael Lopez-Soriano (Universitat de Valencia).

Let $(M,g,\sigma)$ be a compact Riemannian spin manifold of dimension $m \ge 2,$ let $\mathbb S(M)$ denote the spinor bundle on $M$, and let $D$ be the Atiyah-Singer Dirac operator acting on spinors $\Psi:M\to \mathbb S(M)$. We present recent results on the existence of solutions of the nonlinear Dirac equation with critical exponent $$D\Psi=\lambda \Psi+f(|\Psi|)\Psi+|\Psi| ^{2\over m-1}\Psi$$ where $\lambda\in\mathbb R$ and $f(|\Psi|)\Psi$ is a subcritical nonlinearity in the sense that $f(s)=o\left(s^{2\over m-1}\right)$ as $s\to\infty$. This is joint work with Tian Xu.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.