Recent Trends in Nonlinear PDEs - MATRIX Satellite Workshop

Non-concavity of the Robin ground state

Ben Andrews (Australian National University, Australia)

Abstract

I will discuss recent work joint with Julie Clutterbuck (Monash) and Daniel Hauer (Sydney) in which we show that the first eigenfunction for the Robin problem on a convex domain is in general not log-concave, and indeed may even have non-convex super level sets. Our proof proceeds by an analysis of the case of polyhedral domains, where we show that concavity of the first Robin eigenfunction is a very rare phenomenon.

Diffusion with non-local boundary conditions

Wolfgang Arendt (University of Ulm, Germany)

Abstract

Non-local boundary conditions have occur if we tell a particle reaching the boundary to go back in the interior of the domain with a certain probability. There are non-local Dirichlet and non-local Robin boundary conditions. In both cases the space of all continuous functions living on the closure of the domain is the natural Banach space to work with. And indeed, In both cases, an elliptic operator subject to such non-local boundary conditions generates a positive, holomorphic semigroup (even though the holomorphic estimate is a challenge). In the Dirichlet case the semigoup is not continuous at 0, though, i.e. it is not a \(C_0\) semigroup. Still, it governs the diffusion equation subject to the non-local boundary conditions and it is a Feller semigroup with nice properties, just as in the Robin case (where it is \(C_0\)). We are able to determine the asymptotic behavior of the semigroup in terms of natural and intuitive properties of the measure defining the boundary condition.

The talk is based on joint work with Stefan Kunkel and Markus Kunze; the Dirichlet case appeared in J.Functional Anal. in 2017, the Robin case will appear in J. Japanese Math. Soc.

Propagation Phenomena in Non local Monostable Equation: Acceleration or not?

Jérôme Coville (INRA Avignon, France)

Abstract

I will report on some result obtained with Matthieu Alfaro (U. Montpellier) on the dichotomy between two possible propagation behaviours observed in nonlocal monostable equation: a propagation at a constant speed or an acceleration phenomenon. I will describe the fine interplay between the tail of the jumping probability and the growth rate of population that leads to each of these type of propagation behaviour and give some preliminary quantification of the speed of the levels sets.

The Fisher-KPP equation over simple graphs: Varied persistence states in river networks

Yihong Du (University of New England, Australia)

Abstract

In this talk I'll report a recent work with my collaborators, on the growth and spread of a new species in a river network with two or three branches via the Fisher-KPP advection-diffusion equation, over some simple graphs with every edge a half infinite line. We obtain a rather complete description of the long-time dynamical behavior for every case under consideration, which can be loosely described by a trichotomy, including two different kinds of persistence states as parameters vary. The phenomenon of "persistence below carrying capacity'' revealed here appears new, which does not occur in related models of the existing literature where the river network is represented by graphs with finite-lengthed edges, or the river network is simplified to a single infinite line.

On the number of critical points of solutions of PDE's in the plain

Massimo Grossi (Sapienza Università di Roma, Italy)

Abstract

We study the number of critical points of positive solutions of nonlinear partial differential equations in the plain in contractible domain. In particular we consider the case of semi-stable solutions defined in domains whose curvature change sign once.

The Sphere Covering Inequality and its applications

Changfeng Gui (University of Texas San Antonio, United States)

Abstract

In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two distinct surfaces with Gaussian curvature less than \(1\), 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 open problems in these areas will be presented.

The talk is based on joint work with Amir Moradifam from UC Riverside.

Generalized trigonometric functions and the p-Laplace equation

Petr Gurka (University of Life Sciences in Prague, Czech Republic)

Abstract

Generalized sine functions are eigenfunctions of \(p\)-Laplace (or \((p,q)\)-Laplace) operator. We present basic properties of these functions and discuss some applications.

On the existence of stable unduloids of dimension eight

David Hartley (University of Wollongong, Australia)

Abstract

The stability of constant mean curvature hypersurfaces is an important topic in mathematics and influences problems such as the dynamics of the volume preserving mean curvature flow. We consider the case where the hypersurfaces have free boundary contained in parallel planes, this results in the periodic Delaunay hypersurfaces. They include spheres (which are always stable), cylinders (which are stable at large radii), and nodoids (which are unstable). The final case of unduloids is more complex. All unduloids of dimension two through seven are unstable, but there exists stable unduloids of dimension nine and greater. In this talk, we will consider the missing case of whether there exists stable unduloids of dimension eight. We will also work towards finding a criterion to determine if a specific unduloid is stable or unstable.

Radial and non-radial solutions for the Lane-Emden problem in the planar ball

Isabella Ianni (Università degli Studi della Campania "Luigi Vanvitelli", Italy)

Abstract

We consider the Lane-Emden problem: \[ \begin{split} −\Delta u = |u|^{p−1}u &\qquad \text{in \(B\)}\\ u = 0\hspace{22pt} &\qquad \text{on \(\partial B\)} \end{split} \] where \(B\) is the unit ball of \(\mathbb{R}^2\) centered at the origin and \(p \in (1,+\infty)\). We discuss some recent results concerning the description of the asymtptotic behavior of the radial solutions as \(p\) goes to \(+\infty\). We also show the existence of quasi-radial solutions, namely non-radial solutions whose nodal line does not touch \(\partial B\), for certain values of the exponent \(p\).

Higher Order Convergence in Homogenizations

Ki-Ahm Lee (Seoul National University, Korea)

Abstract

In this talk, we are going to higher order convergence in Homogenization process through higher order interior and boundary correctors in various Nonlinear equations. Similar issues arises in vanishing viscous Hamiltonian Equations and Lower order convergence in Parabolic problems with various space and time scales. We also discuss how this idea can be applied to the other problem where some parameters approaches to zero: for example diffusive limit in Kinetic theory, asymptotic limit in parabolic flows, and so on.

The Cheeger constant of a Jordan domain without necks

Robin Neumayer (IAS Princeton, United States)

Abstract

In 1970, Cheeger established lower bounds on the first eigenvalue of the Laplacian on compact Riemannian manifolds in terms of a certain isoperimetric problem. The analogous problem on domains of Euclidean space has generated much interest in recent years, due in part to its connections to capillarity theory, image processing, and landslide modeling. In this talk, based on joint work with Leonardi and Saracco, we give an explicit characterization of minimizers in this isoperimetric problem for a very general class of planar domains.

Quasi-Linear Schrödinger equations in bounded domains: bounded, unbounded and singular coefficients

Benedetta Pellacci (Università della Campania "Luigi Vanvitelli", Italy)

Abstract

We will present some existence results concerning Mountain Pass solutions of quasi-linear Schröodinger equations in bounded domains via variational methods. In order to do this we will study non-differentiable, and possibly unbounded functionals. Part of the presented results have been obtained in collaboration with Marco Squassina and Lucio Boccardo

The Lane-Emden equation on a planar domain

Angela Pistoia (Sapienza Università di Roma, Italy)

Abstract

I will review some old and new results concerning existence, multiplicity and asymptotic behaviour of solutions to the classical Lane-Emden equation on a planar domain when the exponent of the non-linearity is large.

On uniqueness of diffusion processes

Derek W. Robinson (Australian National University, Australia)

Abstract

We discuss uniqueness of solutions of parabolic diffusion equations on \(L_1(\Omega)\) and \(L_2(\Omega)\) where \(\Omega\) is a domain in \(\mathbb{R}^d\). Specifically we consider equations \[ \partial_t\varphi_t+H\varphi_t=0 \] where \(H=-\sum^d_{k,l=1}\partial_k\, c_{kl}\,\partial_l\) with the coefficients \(c_{kl}\) real symmetric \(L_\infty(\Omega)\)-functions and \(C(x)=(c_{kl}(x))>0\) for almost all \(x\in \Omega\). We further assume that \[ a^{-1}d_\Omega^{\;\delta}\,I\leq C\leq a\,d_\Omega^{\;\delta}\,I, \] with \(a, \delta>0\), for \(d_\Omega\leq 1\) where \(d_\Omega\) is the Euclidean distance to the boundary of \(\Omega\).

Uniqueness depends on geometric properties, e.g. the Hausdorff dimension \(s\) of the boundary \(\partial\Omega\), and the degeneracy of the coefficients on the boundary, i.e. the value of \(\delta\). If \(s\) is small then boundary effects should be relatively insignificant. Similarly if the degeneracy \(\delta\) is strong then the boundary tends to be inaccessible and of little consequence to the diffusion. These effects are both important for uniqueness of the process. In particular one has \(L_1(\Omega)\)-uniqueness if and only if \(\delta\geq 1+(s-(d-1))\). This result is valid with mild regularity restrictions on the boundary and applies to Lipschitz domains and a wide class of domains with fractal boundaries. We also explain why one should expect \(L_2(\Omega)\)-uniqueness under similar circumstances if and only if \(\delta\geq (3+(s-(d-1)))/2\).

Curiously the condition for \(L_1(\Omega)\)-uniqueness is the condition for the failure of the corresponding Hardy inequality. A similar relation is expected for the conjectured \(L_2(\Omega)\)-condition and the Rellich inequality.

A heat equation with exponential nonlinearity and with singular data in \(\mathbb{R}^2\)

Bernhard Ruf (Università di Milano, Italy)

Abstract

We consider a semilinear heat equation with exponential nonlinearities and singular data in \(\mathbb{R}^2\).

In \(\mathbb{R}^N\), \(N \ge 3\), critical growth related to singular initial data is polynomial and has been studied by several authors. Existence and non-existence results for singular initial data in suitable $L^p$-spaces were obtained by F. Weissler and H. Brezis - T. Cazenave; furthermore, non-uniqueness results for certain singular initial data were given by W.-M. Ni - P. Sacks and E. Terraneo.

In \(N = 2\) critical growth is given by nonlinearities of exponential type (cf. N. Trudinger - J. Moser). With prove that similar phenomena, namely existence, non-existence and non-uniqueness, occur for suitable exponential nonlinearities and singular initial data in certain Orlicz spaces.

This is joint work with N. Ioku (Ehime University) and E. Terraneo (University of Milan).

Extremal metrics in conformal classes on Riemannian manifolds and applications to isoperimetry and minimal surfaces

Yannick Sire (Johns Hopkins University, United States)

Abstract

Critical points of eigenvalues (with respect to the metric) of differential operators in fixed area surfaces are important objects to understand the geometry and topology of surfaces, after the program initiated by Yau and collaborators. I will describe several results related to existence and regularity of extremal metrics for any eigenvalue of the Laplace-Beltrami on smooth surfaces. Then I will move on to their characterizations as generating minimal immersions by eigenvectors into round spheres. I will describe also several applications to isoperimetric inequalities on the 2-sphere. I will finally describe recent results in higher dimensions and some open problems in the complex case.

Diffusive Hamilton-Jacobi equations and their singularities

Philippe Souplet (University Paris XIII, France)

Abstract

We consider the diffusive Hamilton-Jacobi equation \[u_t-\Delta u=|\nabla u|^p \] with homogeneous Dirichlet boundary conditions. This equation arises in stochastic control theory, as well as in KPZ type models for interface growth in ballistic deposition processes, and it constitutes a model case of parabolic equations with first order nonlinearity.

The solutions display a variety of interesting behaviors and we will discuss two classes of phenomena:

• Gradient blow-up (GBU): localization of singularities on the boundary, single-point GBU, Bernstein estimates, time rate of GBU, spatial GBU profiles.
• Continuation in the viscosity sense after GBU: solutions with and without loss of boundary conditions, recovery of boundary conditions, regularization.

Segregated configurations involving the square root of the Laplacian and their free boundaries

Susanna Terracini (Università di Torino, Italy)

Abstract

We study the local structure and the regularity of free boundaries of segregated critical configurations involving the square root of the laplacian. We develop an improvement of flatness theory and, as a consequence of this and Almgren's monotonicity formula, we obtain partial regularity (up to a small dimensional set) of the nodal set, thus extending the known results by Caffarelli and Lin, Tavares and Terracini for the standard diffusion to some anomalous case.

This is a joint work with Daniela De Silva.

• Norman Dancer

• Daniel Hauer