Schedule

The plenary and session talks will take palce in the Solvay amphitheater.

The workshop will start on Monday at 09:20 am and end on Friday at 01:00 pm.

There will be a welcome beer party on monday at 05:00 pm. Here is the booklet. You will receive a printed copy the first day of the meeting.
Mon 7Tue 8Wed 9 Thu 10Fri 11
8:45Registration
9:20 Wei Dolbeault Loss Souplet Carrillo
10:10 Winkler Tanaka Dancer Felli Maris
10:40Coffee breakCoffee break Coffee breakCoffee break Coffee break
11:10 Hamel del Pino Quittner Pistoia Van Schaftingen
12:00 Reichel Fila Le Coz de Figueiredo Terracini
12:30 Sandwich buffet LunchLunch Lunch
13:00 Sandwich buffet LunchLunch Lunch Complimentary
14:00 Contributed talks Grossi Gazzola Farina closing lunch
14:50 6 parallel sessions Sirakov Ianni Stinner
15:30 Coffee break Coffee break Coffee break Van Phan
16:00 Soave d'Avenia Grunau Coffee break
16:30 Berchio Colorado Adami
17:00 Welcome party 16:50 Secchi Mederski

Schedule of contributed talks

Forum EForum FForum G Forum HOF 2070OF 2072
14:00 Hernández Barile Román Balbus Pimenta Choi
14:15 Pomponio Maia Loiudice Salazar Majda Kalli
14:30 Zographopoulos Cozzi Ikoma Duzgun Benmakhlouf Yoo
14:45 Tentarelli Picon Azzollini Hasegawa Zappale Cieślak
15:00 Özsari Kulikov Coelho Buoso Fourti Gallouët
15:15 Siciliano Figueiredo

Abstracts

▸ Click on the talk titles to see the corresponding abstracts.

\( \newcommand{\IR}{\mathbb{R}} \newcommand{\intd}{\,\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\I}{\mathbf{i}} \renewcommand{\div}{\mathrm{div}} \)

Plenary speakers

José Antonio Carrillo, Imperial College London
Minimizing Interaction Energies — Slides

I will start by reviewing some recent results on qualitative properties of local minimizers of the interaction energy to motivate the main topic of my talk: to discuss global minimizers. We will show the existence of compactly supported global miminizers under quite mild assumptions on the potential in the complementary set of classical H-stability in statistical mechanics. A strong connection with the classical obstacle problem appears very useful when the singularity is strong enough at zero. An approach from discrete to continuum is also quite nice under convexity assumptions on the potential. This is based on three works, one together with F. Patacchini, J.A. Caniizo, another one with M. Delgadino and A. Mellet, and finally with M. Chipot and Y. Huang.

Manuel del Pino, Universidad de Chile
Bubbling in the critical heat equation: the role of Green's function — Slides

We investigate the point-wise, infinite-time bubbling phenomenon for positive solutions of the semilinear heat equation at the critical exponent in a bounded domain. We build an invariant manifold for the flow which ends at \(k\) bubbling points of the domain for any given \(k\). The delicate role of dimension is described. This is joint work with C. Cortázar and M. Musso.

Jean Dolbeault, Université Paris Dauphine
Sharp functional inequalities and nonlinear diffusions — Slides

Nonlinear diffusion flows are very interesting tools for the study of sharp functional inequalities, the stability of the optimal functions and for improvements of the inequalities. The lecture will be devoted to a series of recent results which provide a unified picture of rigidity methods in nonlinear elliptic equations. They have been obtained in a series of papers in collaboration with Maria J. Esteban, Michal Kowalczyk, and Michael Loss. The method is based on entropy methods and related with the Bakry-Emery method and various qualitative techniques for functional inequalities in probability theory, information theory and nonlinear analysis. As an illustration, a computation based on a well chosen nonlinear diffusion flow solves a longstanding conjecture on symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Results on spectral estimates in various settings can also be deduced.

Alberto Farina, Université de Picardie
On the classification of nonnegative solutions to nonlinear equations in half-spaces — Slides

We shall discuss the classification of nonnegative solutions to some semilinear elliptic problems on Euclidean half-spaces. Extensions and open questions will be also considered.

Filippo Gazzola, Politecnico di Milano
Torsional instability in suspension bridges: the Tacoma Narrows Bridge case — Slides

Most people have seen the videos of the spectacular collapse of the Tacoma Narrows Bridge (TNB). The torsional oscillations were considered the main cause of the collapse [AM,SC]. But the appearance of torsional oscillations is not an isolated event occurred only at the TNB, several suspension bridges manifested aerodynamic instability and uncontrolled oscillations leading to collapses.

Most attempts of explanations of the TNB collapse are based on aeroelastic effects such as the frequency of the vortex shedding, parametric resonance, and flutter theory. All these attempts received criticisms because the quantitative parameters do not fit the explanations and the experiments in wind tunnels do not confirm the underlying theory. In [AG] we gave an explanation in terms of a structural instability by modeling the bridge as a system of coupled oscillators.

In this talk we improve the results in [AG] from several points of view. We take into account the behavior of all the structural components of a suspension bridge: the deck, the sustaining cables, and the connecting hangers. We compute all the energies involved and we derive the differential equations from variational principles: the dynamics of a suspension bridge is described by a system of nonlinear nonlocal “hyperbolic'' equations. Then we introduce into the model the parameters of the TNB and we obtain quantitative responses; our numerical results confirm the behavior seen on the day of the collapse. Finally, we insert the obtained results into a suitable theoretical framework, both to justify the results and to explain the underlying phenomenon. The contents of the present talk are taken from two joint papers with Gianni Arioli (Milan).

References

[AM] O.H. Ammann, T. von Kármán, G.B. Woodruff, The failure of the Tacoma Narrows Bridge, Federal Works Agency, Washington D.C. (1941)

[AG] G. Arioli, F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge collapse, Appl. Math. Modelling 39, 901–912 (2015)

[SC] R. Scott, In the wake of Tacoma. Suspension bridges and the quest for aerodynamic stability, ASCE Press (2001)

Massimo Grossi, Università di Roma Sapienza
Multi-layer radial solutions for a supercritical neumann problem — Slides

We consider the problem \[\begin{cases} -\Delta u+u=u^p & \text{ in }B_1 \\ u > 0, & \text{ in }B_1 \\ \partial_\nu u=0 & \text{ on } \partial B_1, \end{cases} \] where \(B_1\) is the unit ball and we show the existence of radial solutions with \(k\) nodal zones for \(p\) large enough.

Hans Christoph Grunau, Universität Magdeburg
Minimising a relaxed Willmore functional for graphs subject to Dirichlet boundary conditions — Slides

The talk is based on joint work with Klaus Deckelnick (Magdeburg) and Matthias Röger (Dortmund) [DGR].


For a bounded smooth domain \(\Omega\subset \mathbb{R}^2\) and a smooth boundary datum \(\varphi:\overline{\Omega}\to \mathbb{R}\), we consider the minimisation of the Willmore functional \[ W(u) := \frac{1}{4} \int_{\Omega} H^2 \; \sqrt{1+ | \nabla u |^2} \, dx \] for graphs \(u: \overline{\Omega}\to \mathbb{R}\) with mean curvature \(H := \div\Bigl(\frac{\nabla u }{\sqrt{1+ |\nabla u|^2}}\Bigr)\) subject to Dirichlet boundary conditions, i.e. in the class \begin{equation*} \mathcal{M} := \bigl\{ u\in H^2(\Omega): (u-\varphi) \in H^2_0(\Omega)\bigr\} . \end{equation*} Making use of a celebrated result by L. Simon [Si, Lemma 1.2] we first show that in this class, bounds for the Willmore energy imply area and diameter bounds. Examples show that stronger bounds in terms of the Willmore energy are not available. This means that \(L^\infty\cap BV(\Omega)\) is the natural solution class where, however, the original Willmore functional is not defined. So, we need to consider its \(L^1\)-lower semicontinuous relaxation. Our main result states that this relaxation coincides on \(\mathcal{M}\) with the original Willmore functional so that the relaxed functional is indeed its largest possible \(L^1\)-lower semicontinuous extension to \(BV(\Omega)\). Moreover, finiteness of the relaxed energy encodes attainment of the Dirichlet boundary conditions in a suitable sense. Finally, we obtain the existence of a minimiser in \(L^\infty\cap BV(\Omega)\) for the relaxed/extended energy.

The major benefit of our non-parametric approach is the validity of a-priori diameter and area bounds, which are not available in the general setting of R. Schätzle's work [Sc]. On the other hand we need to leave open most of the regularity issues.

References

[DGR] Klaus Deckelnick, Hans-Christoph Grunau, Matthias R\”oger, Minimising a relaxed Willmore functional for graphs subject to boundary conditions, Preprint 2015, arxiv:1503.01275.

[Sc] Reiner Schätzle, The Willmore boundary problem, Calc. Var. Partial Differential Equations, 37, 275–302, 2010.

[Si] Leon Simon, Existence of surfaces minimizing the {W}illmore functional, Comm. Anal. Geom. 1, 281–326, 1993.

Francois Hamel, Université d'Aix-Marseille & Institut Universitaire de France
Transition fronts for the Fisher-KPP equation — Slides

The standard notions of reaction-diffusion fronts can be viewed as examples of generalized transition fronts. These notions involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. The existence of transition fronts has been proved in various contexts where the standard notions of fronts make no longer sense. Even for homogeneous equations, fronts with varying speeds are known to exist. In this talk, I will report on some recent existence results and qualitative properties of transition fronts for monostable homogeneous and heterogeneous one-dimensional equations. I will also discuss their asymptotic past and future speeds. The talk is based on some joint works with Luca Rossi [hr1,hr2].

References

[hr1] F. Hamel, L. Rossi, Transition fronts for the Fisher-KPP equation, Trans. Amer. Math. Soc., forthcoming.

[hr2] F. Hamel, L. Rossi, Admissible speeds of transition fronts for time-dependent KPP equations, SIAM J. Math. Anal., forthcoming.

Michael Loss, Georgia Institute of Technology
The phase diagram of the Caffarelli-Kohn-Nirenberg inequalities — Slides

The Caffarelli-Kohn-Nirenberg inequalities form a two parameter family of inequalities that interpolate between Sobolev's inequality and Hardy's inequality. The functional whose minimization yields the sharp constant is invariant under rotations. It has been known for some time that there is a region in parameter space where the optimizers that yield the sharp constant are not radial. In this talk I indicate a proof that in the remaining parameter region the optimizers are in fact radial. The proof will proceed via a well chosen flow that decrases the functional unless the function is a radial optimizer. This is joint work with Jean Dolbeault and Maria Esteban.

Angela Pistoia, Università di Roma Sapienza
Blowing-up solutions for Yamabe-type problems — Slides

The Yamabe equation is one of the most natural and well-studied second-order semilinear elliptic equations arising in geometric variational problems. The issue of the compactness of the set of solutions of the geometric Yamabe equation has been recently studied and it is strictly related to the existence of solutions blowing-up at one or more points in the manifold. In this lecture, I will review these results and present more recent works on the Yamabe problem, where solutions blowing-up at multiple (clustering and towering) points have been found.

Pavol Quittner, Comenius University
Liouville theorems for superlinear parabolic problems — Slides

Liouville theorems for scaling-invariant nonlinear parabolic problems (guaranteeing nonexistence of positive bounded entire solutions) imply universal estimates for solutions in general domains. We prove such Liouville theorems for two classes of problems. First, we consider several parabolic equations and systems with gradient structure and show that each positive bounded entire solution has to be time-independent, see [1]. Second, we consider a class of two-component parabolic systems without gradient structure and show that the components of any positive bounded entire solution have to be proportional, see [2].

It is known that the universal estimates guaranteed by our Liou­vil­le theorems yield optimal blow-up rate estimates and also imply boundedness of threshold solutions lying on the borderline between global existence and blow-up. We use these universal estimates to prove the existence of positive periodic solutions of strongly cooperative parabolic Lotka-Volterra systems with equal diffusion coefficients.

References

[1] Quittner P., “Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure,'' Math. Ann., DOI 10.1007/s00208-015-1219-7.

[2] Quittner P., “Liouville theorems, universal estimates and periodic solutions for cooperative parabolic Lotka-Volterra systems,'' Preprint arXiv:1504.07031.

Philippe Souplet, Université Paris 13
Space profile of single-point gradient blow-up on the boundary for the diffusive Hamilton-Jacobi equation — Slides

We study the asymptotic behavior of gradient blow-up solutions for the diffusive Hamilton-Jacobi equation \(u_t-\Delta u=|\nabla u|^p\) (\(p>2\)) in planar domains, with Dirichlet boundary conditions. We will present results on the concentration of singularities of the gradient at a single point of the boundary in finite time. In particular we obtain the precise final space profile near the singularity. Interestingly, unlike in other related problems, the profile turns out to be strongly non-isotropic, being more singular in the tangential direction than in the normal direction. Joint work with Alessio Porretta.

Susanna Terracini, Università di Torino
Existence and regularity of solutions to optimal partition problems involving Laplacian eigenvalues — Slides

Let \(\Omega\subset \mathbb{R}^N\) be an open bounded domain and \(m\in \mathbb{N}\). Given \(k_1,\ldots,\linebreak[2] k_m\in \mathbb{N}\), we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, including the following \[ \inf\biggl\{ \Phi(\omega_1,\ldots,\omega_m) := \sum_{i=1}^m \lambda_{k_i}(\omega_i) :\ (\omega_1,\ldots, \omega_m)\in \mathcal{P}_m(\Omega) \biggr\}, \] where \(\lambda_{k_i}(\omega_i)\) denotes the \(k_i\)-th eigenvalue of \(\bigl(-\Delta,H^1_0(\omega_i) \bigr)\) counting multiplicities, and \(\mathcal{P}_m(\Omega)\) is the set of all open partitions of \(\Omega\), namely \[ \mathcal{P}_m(\Omega)=\bigl\{(\omega_1,\ldots,\omega_m) :\ \omega_i\subset \Omega \text{ open},\ \forall i\neq j, \omega_i\cap \omega_j=\emptyset \bigr\}. \] We prove the existence of an open optimal partition \((\omega_1,\ldots, \omega_m)\), proving as well its regularity in the sense that the free boundary \(\bigcup_{i=1}^m \partial \omega_i\cap \Omega\) is, up to a residual set, locally a \(C^{1,\alpha}\) hypersurface.

In order to prove this result, we first treat some general optimal partition problems involving all eigenvalues up to a certain order. This class of problems includes the one with cost function \(\Phi_p(\omega_1,\ldots, \omega_m) := \sum_{i=1}^m \Bigl(\sum_{j=1}^{k_i} \bigl(\lambda_j(\omega_i)\bigr)^{p} \Bigr)^{1/p}\), whose solutions approach the solutions of the original problem as \(p\to +\infty\). The study of this new class of problems is done via a singular perturbation approach with a class of Schr\”odinger-type systems which models competition between different groups of possibly sign-changing components. An optimal partition appears in relation with the nodal set of the limiting components, as the competition parameter becomes large. This is a joint work with Miguel Ramos and Hugo Tavares.

Jean Van Schaftingen, Université catholique de Louvain
Nonlinear Schrödinger equation under a magnetic field kept in the foreground — Slides

I shall present results on standing waves for the nonlinear Schrö­din­ger equation in the strong magnetic field régime. In contrast with the well-studied mean magnetic field régime in which the behaviour of solutions in the semiclassical limit is governed by the electric field alone, the concentration of solutions depends on both the electric and magnetic fields and allows to recover a semiclassical formula for the Lorentz electromagnetic field. I shall explain how we have overcome the lack of knowledge of the solutions of the limiting problem.

I shall also describe the symmetry and decay properties of the solutions of the limiting problem when the magnetic field is not too large. We have obtained these properties by an asymptotic analysis that we could perform along a family of functional spaces that depend on the magnetic field.

References

[BNVS] D. Bonheure, M. Nys and J. Van Schaftingen, Properties of groundstates of nonlinear Schr\”odinger equations under a constant magnetic field, in preparation.

[DCV] J. Di Cosmo and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field, J. Differential Equations 259 (2015), no. 2, 596–627.

Juncheng Wei, University of British Columbia & Chinese University of Hong Kong
On Type II Singularity Formation of Harmonic Map Flow — Slides

We consider the following harmonic map \[ u_t=\Delta u + |\nabla u|^2 u \] \[ u: \Omega \to S^2 \] where \(\Omega\) is a general two dimensional domain. We show that the Type II blow up rate \[ \frac{T-t}{\log^2 (T-t)}\] is generic and stable. We also discuss the multiple bubbling and reverse bubbling. We develop gluing techniques for parabolic equations. (Joint work with Manuel del Pino and Juan Davila).

Session speakers

Schrödinger equations

chaired by Silvia Cingolani, Louis Jeanjean, Marco Squassina
Riccardo Adami, Politecnico di Torino
Ground states for the NLS on quantum graphs: the \(L^2\)-critical case — Slides

The search for ground states (namely, minimizers of the energy with fixed \(L^2\)-norm) for the quintic NLS on the line has a particular feature: solutions exist only for a particular value of the \(L^2\)-norm. We consider the same problem on some quantum graphs and show a much richer structure of the set of ground states. Such a structure strongly depends on the topology of the graph. This is a joint work with E. Serra and P. Tilli.

Eduardo Colorado, Universidad Carlos III de Madrid
Bound and Ground states to systems of coupled NLS-KdV equations — Slides

We will show some results on existence and multiplicity of solutions for the system of coupled NLS-KdV equations, \begin{equation*} (S) \begin{cases} \I f_t + f_{xx} -\beta fg+ |f|^2f = 0, & x\in \mathbb{R}, t>0\\[0.5ex] g_t +g_{xxx} +gg_x -\frac 12\beta \bigl(|f|^2\bigr)_x = 0, & x\in \mathbb{R}, t>0, \end{cases} \end{equation*} where \(\I\) denotes the complex unit, \(f(x,t)\in \mathbb{C}\), \(g(x,t)\in \mathbb R\), \(\beta\in \mathbb{R}\) and \(|f|,\, |g|\to 0\) as \(|x|\to \infty\). This problem appears in phenomena of interactions between short and long dispersive waves, arising in fluid mechanics, such as the interactions of capillary-gravity water waves. Indeed, \(f\) represents the short-wave, while \(g\) stands for the long-wave.

The main results that we will show deals with existence of positive bound and ground states for the corresponding stationary system when one looks for solitary-traveling wave solutions of \((S)\).

It is remarkable that the methods, we use here, have been previously used and developed in [AA], and firstly applied to this kind of problems in [C1,C2]. Also the methods previously used to study system \((S)\) could be applied to the more studied systems of NLS equations.


The results presented in this talk are contained in papers [C1,C2].

References

[AA] A. Ambrosetti, E. Colorado, Standing waves of some coupled nonlinear Schr\”odinger equations. J. Lond. Math. Soc. (2) 75 (2007), no. 1, 67-82.

[C1] E. Colorado, Existence of Bound and Ground States for a System of Coupled Nonlinear Schr\”odinger-KdV Equations. C. R. Acad. Sci. Paris S\'er. I Math. 353 (2015), no. 6, 511-516.

[C2] E. Colorado, On the existence of bound and ground states for a system of coupled nonlinear Schr\”odinger-KdV equations. Preprint ArXiv:1411.7283v3.

Pietro d'Avenia, Politecnico di Bari
Born-Infeld equations in the electrostatic case — Slides

We consider the problem \begin{equation}\label{eq:BI-abs} \tag{\(\mathcal{BI}\)} \begin{cases} -\operatorname{div}\left(\displaystyle\frac{\nabla \phi}{\sqrt{1-|\nabla \phi|^2}}\right)= \rho, & x\in\mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty}\phi(x)= 0. \end{cases} \end{equation} The equation in \eqref{eq:BI-abs} appears for instance in the Born-Infeld nonlinear electromagnetic theory: in the electrostatic case it corresponds to the Gauss law in the classical Maxwell theory and so \(\phi\) is the electric potential and \(\rho\) is an assigned extended charge density.

We discuss existence, uniqueness and regularity of the solution of \eqref{eq:BI-abs}. The results have been obtained in a joint work with Denis Bonheure and Alessio Pomponio.

Veronica Felli, University of Milano-Bicocca
Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles — Slides

In this talk, I will present some results obtained in collaboration with L. Abatangelo about the behavior of eigenvalues of a magnetic Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary conditions in a planar domain \(\Omega\), i.e. of an operator of the form \((\mathbf{i}\nabla +A_{a})^2\) with \(a=(a_1,a_2)\in \Omega\) and \begin{equation*} A_{a}(x_1,x_2) =\frac12 \biggl(\frac{-(x_2-a_2)}{(x_1-a_1)^2+(x_2-a_2)^2}, \frac{x_1-a_1}{(x_1-a_1)^2+(x_2-a_2)^2} \biggr). \end{equation*} An approach based on energy estimates obtained through Alm­gren-type monotonicity arguments for magnetic operators together with a sharp blow-up analysis is developed to provide sharp asymptotics for eigenvalues as the pole \(a\) is moving in the interior of the domain \(\Omega\), approaching a zero of an eigenfunction of the limiting problem along a nodal line.

Stefan Le Coz, University of Toulouse 3
On the blow-up speed for nonlinear Schrödinger equations

The formation of singularities in nonlinear Schrödinger equations is a fascinating yet still little understood phenomenon. So far, only two blow-up regimes have been observed for NLS equations: the pseudo-conformal regime and the log-log regime.

Our goal in this talk is to present a construction of a blowing up solution whose blow-up speed is neither the log-log speed nor the pseudo-conformal speed, but is of the type \(|t|^{-s}\) with \(s\) varying between \(1/2\) and \(1\).

This is based on a joint work with Yvan Martel and Pierre Raphael.

Mihai Maris, Université Paul Sabatier - Toulouse 3
Some remarks on the concentration-compactness principle — Slides

We present some recent improvements of the concentration-com­pact­ness principle and show that they give a new insight in some minimization problems arising in the study of solitary waves for nonlinear dispersive equations.

Jarosław Mederski, Nicolaus Copernicus University
Ground states of time-harmonic semilinear Maxwell equations — Slides

We investigate the existence and the nonexistence of solutions \(E:\mathbb{R}^3\to\mathbb{R}^3\) of the time-harmonic semilinear Maxwell equation \[\nabla\times(\nabla\times E) + V(x) E = \partial_E F(x,E) \hbox{ in }\mathbb{R}^3\] where \(V:\mathbb{R}^3\to\mathbb{R}\), \(V(x)\leq 0\) a.e. on \(\mathbb{R}^3\), \(\nabla\times\) denotes the curl operator in \(\mathbb{R}^3\) and \(F:\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}\) is a nonlinear function in \(E\). In particular we find a ground state solution provided that suitable growth conditions on \(F\) are imposed and \(L^{3/2}\)-norm of \(V\) is less than the best Sobolev constant. In applications \(F\) is responsible for the nonlinear polarization and \(V(x)=-\mu\omega^2\varepsilon(x)\) where \(\mu>0\) is the magnetic permeability, \(\omega\) is the frequency of the time-harmonic electric field \(\Re\{E(x) \e^{\I\omega t}\}\) and \(\varepsilon\) is the linear part of the permittivity in an inhomogeneous medium.

References

[BM] T. Bartsch, J. Mederski, Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain, Arch. Ration. Mech. Anal. 215 (1), (2015), 283–306.

[M] J. Mederski Ground states of time-harmonic semilinear Maxwell equations in \(\mathbb{R}^3\) with vanishing permittivity, arXiv:1406.4535.

Wolfgang Reichel, Karlsruhe Institute of Technology
Standing waves for nonlinear curl-curl wave equations — Slides

We consider a class of nonlinear wave equations arising in the theory of nonlinear electromagnetics. The equations may be quasilinear \[ \nabla\times \nabla \times U + \partial_t^2 \bigl(V(x) U + \Gamma(x) |U|^{p-1}U\bigr) = 0 \] or semilinear \[ \nabla\times \nabla \times U + V(x) \partial_t^2 U + \Gamma(x) |U|^{p-1}U = 0 \] for given coefficients \(V,\Gamma: \mathbb{R}^3\to \mathbb{R}\) and exponents \(p>1\). We prove the existence of some classes of standing waves. Depending on the model the solutions \(U(x,t)\) defined for \((x,t)\in \mathbb{R}^3\times \mathbb{R}\) may take either complex or real vector-values. In the case of monochromatic complex valued standing waves we use variational methods. In the case of real valued so-called breathers we use a specific, partly explicit construction based on a simple phase-plane argument.

Simone Secchi, Università di Milano Bicocca
Some recent results on pseudo-relativistic Hartree equations — Slides

In this talk I will present some results on nonlinear fractional field equations, and in particular on the pseudo-relativistic Hartree equation \[ \sqrt{-\varepsilon^2 \Delta+m^2} \; u + Vu = \left( I_\alpha * |u|^p \right) |u|^{p-2} u \quad \hbox{in \(\mathbb{R}^N\)}, \] where \(V \colon \mathbb{R}^N \to \mathbb{R}\) is an external scalar potential and \(I_\alpha(x) = |x|^{\alpha -N}\) up to some multiplicative constant. This is a joint work with Silvia Cingolani.

Kazunaga Tanaka, Department of Mathematics, Waseda University
Singular perturbation problems for nonlinear elliptic problems

In this talk we study singular perturbation problems for a system of nonlinear Schrödinger equations and for a nonlinear elliptic problem in perturbed cylindrical domains. We show the existence of a family of solutions which concentrate at a prescribed part of domain — especially we consider the situation where the prescribed part is corresponding to “local maxima''.

Higher order PDEs and systems

chaired by Ederson Moreira dos Santos, Benedetta Noris, Hugo Tavares
Elvise Berchio, Politecnico di Torino
Structural instability of nonlinear plates modelling suspension bridges — Slides

We analyze the structural instability arising in a nonlinear fourth order hyperbolic-like problem which describes the dynamics of suspension bridges. With a finite dimensional approximation, two different kinds of oscillating modes are found: longitudinal and torsional. We show the existence of an energy threshold where stability is lost and part of the energy is suddenly transferred from longitudinal to torsional modes giving rise to wide torsional oscillations. An explanation of the precise mechanism which governs the energy transfer is also provided. The results stated are contained in the joint papers [BFG] and [BGZ].

References

[BFG] E. Berchio, A. Ferrero, F. Gazzola, Structural instability of nonlinear plates modelling suspension bridges: mathematical answers to some long-standing questions, arXiv:1502.05851

[BGZ] E. Berchio, F. Gazzola, C. Zanini, Which residual mode captures the energy of the dominating mode in second order Hamiltonian systems?, arXiv:1410.2374

Norman Dancer, University of Sydney & Swansea University
Multiplicity of Solutions of Certain Systems — Slides

Suppose \(D\) is a bounded domain in 3 dimensional space.

We discuss how the shape of \(D\) can affect the multiplicity of solutions for large \(k\) of \begin{equation*} \begin{cases} -\Delta u= a u -k u v^2 & \text{in } D, \\ -\Delta v= d v - k u^2 v & \text{in } D, \\ u,v\geq0 & \text{in } D, \\ u=v=0 & \text{on } \partial D. \end{cases} \end{equation*}

Djairo de Figueiredo, Universidade Estadual de Campinas
On semipositone problems for the p-Laplacian — Slides

The existence of positive solutions to a semipositone p-Laplacian problem is proved combining variational methods and a priori estimates.

Boyan Sirakov, PUC - Rio de Janeiro
A priori bounds for elliptic PDEs with natural growth in the gradient — Slides

We present new results and methods for proving a priori bounds and existence/multiplicity result for the Dirichlet problem for a general class of elliptic operators in which the first and the second order terms have the same scaling with respect to dilations.

Nicola Soave, Justus-Liebig-Universität Gießen
Geometric aspects and asymptotic analysis in phase separation of coupled elliptic equations — Slides

We consider a family of positive solutions to the system of \(k\) components \[ -\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^2 \qquad \text{in \(\Omega\)}, \] where \(\Omega \subset \mathbb R^N\) with \(N \ge 2\). It is known that uniform bounds in \(L^\infty\) of \(\{\mathbf{u}_{\beta}\}\) imply convergence of the densities to a segregated configuration, as the competition parameter \(\beta\) diverges to \(+\infty\). In this paper we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of \(\mathbf{u}_\beta\) in terms of entire solutions to the limit system \[ \Delta U_i = U_i \sum_{j\neq i} a_{ij} U_j^2. \] Moreover, we develop a uniform-in-\(\beta\) regularity theory for the interfaces. This is a joint work with Alessandro Zilio (EHESS, Paris).

Parabolic PDEs

chaired by Juraj Foldes, Alberto Saldaña
Marek Fila, Comenius University
Positive solutions of an elliptic equation with a dynamical boundary condition — Slides

We consider a semilinear elliptic equation in the half-space. On the boundary, a linear dynamical boundary condition is imposed. We present sharp results on existence and non-existence of positive solutions. We also study the large-time behavior of small solutions. This is a joint work with Kazuhiro Ishige and Tatsuki Kawakami.

References

[FIK1] Fila M., Ishige K., Kawakami T., “Large-time behavior of solutions of a semilinear elliptic equation with a dynamical boundary condition,'' Adv. Differential Equations 18, 69–100 (2013).

[FIK2] Fila M., Ishige K., Kawakami T., “Large-time behavior of small solutions of a two-dimensional semilinear elliptic equation with a dynamical boundary condition,'' Asymptotic Analysis 85, 107–123 (2013).

[FIK3] Fila M., Ishige K., Kawakami T., “Existence of positive solutions of a semilinear elliptic equation with a dynamical boundary condition,'' Calc. Var. Partial Differential Equations, DOI: 10.1007/s00526-015-0856-8.

[FIK4] Fila M., Ishige K., Kawakami T., “Minimal solutions of a semilinear elliptic equation with a dynamical boundary condition,'' preprint.

Isabella Ianni, Seconda Università degli Studi di Napoli
Existence of sign-changing solutions to the Lane-Emden problem via parabolic approach — Slides

We consider the semilinear elliptic Lane-Emden problem \begin{equation} \label{problemAbstract} \begin{cases} -\Delta u= |u|^{p-1}u&\text{in }\Omega,\\ u=0 &\text{on }\partial \Omega, \end{cases} \tag{\(\mathcal E_p\)} \end{equation} where \(p>1\) and \(\Omega\) is a smooth bounded simply connected domain of \(\mathbb R^2\). By imposing some symmetry on the domain we show the existence of sign-changing solutions \(u_p\) having two nodal regions and whose nodal line doesn't touch the boundary. The proof relies on a dynamical approach which combines the use of the parabolic flow associated to the problem together with some geometrical arguments.

We also discuss the blow-up in finite time of the solutions of the parabolic problem associated to \eqref{problemAbstract} having initial condition \(u_0=\lambda u_p\) with \(\lambda\in \mathbb R\).

The results presented are obtained in collaboration with F. De Mar­chis and F. Pacella (Università Sapienza, Roma).

Christian Stinner, Technische Universität Kaiserslautern
Finite time versus infinite time blowup for a fully parabolic Keller-Segel system — Slides

Several variants of the Keller-Segel model are used in mathematical biology to describe the evolution of cell populations due to both diffusion and chemotactic movement. In particular, the emergence of cell aggregation is related to blowup of the solution. Critical nonlinearities with respect to the occurrence of blowup had been identified for a quasilinear parabolic-parabolic Keller-Segel system, but it was not known whether the solution blows up in finite or infinite time. We show that indeed both blowup types appear and that the growth of the chemotactic sensitivity function is essential to distinguish between them. We provide conditions for the existence of each blowup type and discuss their optimality. An important ingredient of our proof is a detailed analysis of the Liapunov functional. This is a joint work with T. Cieślak (Warsaw).

Tuoc Van Phan, University of Tennessee
Gradient estimates and global existence of smooth solutions to a cross-diffusion system — Slides

We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global \(W^{1,p}\)-estimates of Calderón-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.


The talk is based on the joint work with L. Hoang (Texas Tech) and T. Nguyen (U. of Akron).

Michael Winkler, Universität Paderborn
How far can chemotactic cross-diffusion enforce exceeding carrying capacities? — Slides

We consider variants of the Keller-Segel system of chemotaxis which contain logistic-type source terms and thereby account for proliferation and death of cells. We briefly review results and open problems with regard to the fundamental question whether solutions \vadjust{\pagebreak[2]}exist globally in time or blow up. The primary focus will then be on the prototypical parabolic-elliptic system \begin{equation*} \begin{cases} u_t=\varepsilon u_{xx} - (uv_x)_x + ru - \mu u^2, \\[1mm] 0= v_{xx}-v+u, \end{cases} \end{equation*} in bounded real intervals. The corresponding Neumann initial-boundary value problem, though known to possess global bounded solutions for any reasonably smooth initial data, is shown to have the property that the so-called carrying capacity \({r}/{\mu}\) can be exceeded dynamically to an arbitrary extent during evolution in an appropriate sense, provided that \(\mu<1\) and that \(\varepsilon>0\) is sufficiently small. This is in stark contrast to the case of the corresponding Fisher-type equation obtained upon dropping the term \(-(uv_x)_x\), and hence reflects a drastic peculiarity of destabilizing action due to chemotactic cross-diffusion, observable even in the simple spatially one-dimensional setting. Numerical simulations underline the challenge in the analytical derivation of this result by indicating that the phenomenon in question occurs at intermediate time scales only, and disappears in the large time asymptotics.

Communications

The contributed talks will be scheduled on Monday September 7.