Analysis Seminars

Adding numbers and shuffling cards

Geodesic Levy Flight and The Foraging Hypothesis

Phase transitions for everybody 2/2

Phase transitions for everybody 1/2

Continuity of derivatives of a convex solution to a total variation equation perturbed by p-Laplacian

Viscosity & reversibility in fluid flows

B^1_-: Analysis and PDE from the South: The Malmheden Theorem and the geometry of harmonic functions 3/3

B^1_-: Analysis and PDE from the South: The Malmheden Theorem and the geometry of harmonic functions 2/3

B^1_-: Analysis and PDE from the South: The Malmheden Theorem and the geometry of harmonic functions 1/3

One-phase problems for inhomogeneous equations governed by the p(x)-Laplace operator

Vortex filaments for Euler equations

Long-range phase coexistence models

Integrability of normal distributions: A Stefan-Sussmann theorem and neat foliations by manifolds with boundary

The stickiness property for nonlocal minimal surfaces

Nonlocal minimal surfaces are generically sticky

Systolic questions in metric and symplectic geometry 

(Non)local logistic equations with Neumann conditions

The evolution of atomic dislocations in crystals

Symbolic dynamics and atom dislocation in crystals

Recovering obstacles from their travelling times

MHD equilibria with nonconstant pressure in nondegenerate toroidal domains

Nonlocal minimal graphs are generically sticky

The evolution of atomic dislocations in crystals

Symbolic dynamics and atom dislocations in crystals

Linearization and rigidity, from circle diffeos to higher genus

The Lévy flight foraging hypothesis

First-passage time of confined non-markovian random walks

Partial regularity for the supercritical surface quasigeostrophic equation

Under what conditions are inverse square Lévy walk searches optimal?

A semilinear spectrum for the Dirichlet-Laplacian

A topological toolbox for Sobolev maps

The best constant in Sobolev's inequality

The equilibrium of floating elastic bodies

Inverse problems and rigidity questions in Billiard Dynamics

Analytical aspects of local and nonlocal minimal surfaces (Course at the University of Milan) 4/4

Topological and variational methods for the Moser-Trudinger equation

Analytical aspects of local and nonlocal minimal surfaces (Course at the University of Milan) 3/4

Analytical aspects of local and nonlocal minimal surfaces (Course at the University of Milan) 2/4

Asymptotic behavior of the volume preserving mean curvature flow

Analytical aspects of local and nonlocal minimal surfaces (Course at the University of Milan) 1/4

The Beat of Math

Symmetry properties for the Euler equations and semilinear elliptic equations

Fractional fully nonlinear degenerate elliptic operators: surprising phenomena and Liouville results

A mathematical model for civil wars: a new Lotka-Volterra competitive system

Should I stay or should I go? Zero-size jumps in random walks for Lévy flights

A survey on recent results of isoperimetric inequalities for eigenvalues

Boundary behavior of nonlocal minimal surfaces

Nonlocal Minimal Surfaces: interior regularity, boundary behavior and stickiness phenomena

[Part 2]

Nonlocal Minimal Surfaces: interior regularity, boundary behavior and stickiness phenomena

[Part 1]

A broad look at elliptic partial differential equations - Lecture 2

We will discuss some classical (and less classical) topics related to elliptic partial differential equations, related also to symmetry, classification and regularity theory. Some applications arising in geometry and material sciences will also be presented.

A broad look at elliptic partial differential equations - Lecture 2

We will discuss some classical (and less classical) topics related to elliptic partial differential equations, related also to symmetry, classification and regularity theory. Some applications arising in geometry and material sciences will also be presented.

A broad look at elliptic partial differential equations - Lecture 1

We will discuss some classical (and less classical) topics related to elliptic partial differential equations, related also to symmetry, classification and regularity theory. Some applications arising in geometry and material sciences will also be presented.

Boundary behavior of nonlocal minimal surfaces

We present some results, and some pictures, related to some special phenomena exhibited by nonlocal minimal surfaces, with particular emphasis on the boundary stickiness properties.

Almost periodic solutions of NLS

In general, the phase space of a completely integrable Hamiltonian System (HS) of n degrees of freedom is foliated by invariant n-dimensional tori on which the motion is quasi-periodic. Kolmogorov-Arnold-Moser (KAM) theory deals with persistence, under (hamiltonian) perturbation, of (the majority of) such tori. Starting from the late eighties KAM Theory was applied to infinite dimensional HS, e.g. to PDEs with Hamiltonian structure. While the Theory for finite dimensional tori is well established, very few results (essentially due to Bourgain) are know for infinite dimensional tori supporting almost-periodic solutions. I will discuss some recents developments on the subject, obtained in collaboration with J. Massetti and M. Procesi.

Symmetry results for Plateau's surfaces

By employing the method of moving planes in a novel way, we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films.

Analyticity of solutions to fractional partial differential equations

We will discuss an old topic in the field of partial differential equations in a new context: The question of analyticity of solutions to elliptic equations.

While first results for classical elliptic partial differential equations were already obtained by Bernstein in 1904, in the context of fractional and non-local equations only partial results or results for very special cases like the Hartree-Fock equations and the Boltzmann equation are known up to now.

After presenting some known results, we will discuss our recent findings for so-called knot energies and general semi-linear integro-differential equations. The main ingredients in the proof of these results are Cauchy's method of majorants and a new estimate for the long range interactions of these equations

Double Bubbles and Densities

The round sphere provides the least-perimeter way to enclose prescribed volume in ℝm. The n-bubble problem seeks the least-perimeter way to enclose and separate n prescribed volumes in ℝm. The solution is also known only for n = 2 in ℝm (the standard double bubble) and n = 3 in ℝ2 (the standard triple bubble). If you give ℝm Gaussian density, the solution was recently proved by Milman and Neeman for n ≤ m. There is further news for other densities.

In 2000 Hales proved that regular hexagons provide a least-perimeter way to partition the plane into unit areas. Undergraduates recently obtained a partial extension to closed hyperbolic manifolds. The 3D Euclidean case remains open. The best tetrahedral tile was proved recently. (Despite what Aristotle said, the regular tetrahedron does not tile.)

We'll describe many such results and open questions. Students welcome.

The Bernstein technique for integrodifferential equations

We present a version of the classical Bernstein technique for integrodifferential operators. We provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully nonlinear equations of order smaller than two, for which we prove uniform estimates as their order approaches two. Our method is robust enough to be applied to some Pucci-type extremal equations and to obstacle problems for fractional operators, although several of the results are new even in the linear case. We also raise some open questions. The result discussed come from a joint work with Xavier Cabré and Serena Dipierro.

The Bernstein technique for integrodifferential equations

We present a version of the classical Bernstein technique for integrodifferential operators. We provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully nonlinear equations of order smaller than two, for which we prove uniform estimates as their order approaches two. Our method is robust enough to be applied to some Pucci-type extremal equations and to obstacle problems for fractional operators, although several of the results are new even in the linear case. We also raise some open questions. The result discussed come from a joint work with Xavier Cabré and Serena Dipierro.

The stickiness property of nonlocal minimal surfaces

Surfaces which minimize a nonlocal perimeter functional exhibit quite different behaviors than the ones minimizing the classical perimeter. Among these peculiar features, an interesting property, which is also in contrast with the pattern produced by the solutions of linear equations, is given by the capacity, and the strong tendency, of adhering at the boundary.

Some recent Liouville type results and their applications

We will discuss a number of recent Liouville type theorems and their applications, in the context of nonlinear elliptic and parabolic equations. 

The problems under consideration include diffusive Hamilton-Jacobi, Lane-Emden and Fujita equations.

Fractional Sobolev and BV spaces: a distributional approach

Differently from their integer versions, the fractional Sobolev spaces Wα,p(ℝn) do not seem to have a clear distributional nature. By exploiting suitable notions of fractional gradient and of fractional divergence already existing in the literature, in recent papers in collaboration with G. Stefani we introduce the new space BVα(ℝn) of functions with bounded fractional variation in ℝn of order α ∈ (0, 1) via a new distributional approach. Thanks to the continuous inclusion Wα,1(ℝn) ⊆ BVα(ℝn), our theory provides a natural extension of the known fractional framework. In addition, we define in a similar way the distributional fractional Sobolev spaces Sα,p(ℝn). In analogy with the classical BV theory, we define sets with (locally) finite fractional Caccioppoli α-perimeter and we partially extend De Giorgi's Blow-up Theorem to such sets, proving existence of blow-ups on points of the naturally defined fractional reduced boundary. In addition, we investigate the asymptotic behaviour of these fractional differential operators as α converges to 1- and 0+. In particular, we prove that the fractional α-variation weakly and Γ-converges to the standard De Giorgi’s variation as α →  1-, in perfect analogy with the well-known Γ-convergence result by Ambrosio-De Philippis-Martinazzi.

Asymptotic stability for the gradient flow of some non local energies

I will start by discussing some recent results on the asymptotic stability of the H-1-gradient flow of the perimeter (surface diffusion). Then I will also consider the  H-1-gradient flow of some energy functionals given by the area of an interface plus a non local volume term. 

Regularity of the two-phase free boundaries

In this talk, we will present some new results on the regularity of the free boundary of local minimizers u of the two-phase free boundary functional. Precisely, we will show that:

• in dimension d= 2, the free boundaries are C1,α-regular curves (this result was proved in Spolaor-V. [CPAM, 2019]);

• in dimension d >2, the free boundaries are C1,α-regular, up to a (possibly empty) one-phase singular set of lower dimension (De Philippis-Spolaor-V.);

In particular, these results complete (in any dimension) the analysis of the two-phasefree boundaries started by Alt, Caffarelli and Friedman in 1984.

Finally, we will discuss the applications of this result to a shape optimization problem for the second eigenvalue of the Dirichlet Laplacian.

On the relations between principal eigenvalue and torsional rigidity

The relations between principal eigenvalue of the Laplace operator and torsional rigidity are studied in the class of general domains, convex domains, and domains with a small thickness. This is of help to provide some bounds for the Blasche-Santaló diagram of the two quantities. The results have been obtained in a joint work with Michiel van den Berg (Bristol) and Aldo Pratelli (Pisa).

The singular set in the Stefan problem

The Stefan problem, dating back to the XIXth century, is probably the most classical and well-known free boundary problem. The regularity of free boundaries in such problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is C∞  in space and time, outside a certain set of singular points.

The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs and Geometric Analysis. In particular, a major question in such context is to establish estimates for the size and structure of the singular set. The goal of this talk is to present some new results in this direction for the Stefan problem. This is a joint work with A. Figalli and J. Serra.

Flatness results for nonlocal minimal cones

In this  talk we will present some recent results in the classification of nonlocal minimal cones (that is, cones which minimize the nonlocal, or fractional, perimeter)  in low dimensions. More precisely, we will prove flatness of s-minimal cones in dimension n=3, when the fractional parameter s is sufficiently close to 1.

This result, obtained in collaboration with X. Cabré and J. Serra, relies on several ingredients, such as some perimeter estimates for nonlocal minimal surfaces (obtained in collaboration with J. Serra and E. Valdinoci) and a fractional Hardy inequality.

Nonlocal Models in Replicator Dynamics 

In this talk we present a degenerate non-local parabolic model arising in Evolutionary  Game Theory. In particular the constructed model  is associated with replicator dynamics and actually reflects the fact that the game players tend to update their strategy  comparing it with that of the averaged gamers' population. The qualitative behaviour of the underlying model, including the investigation of possible  blow-up phenomena and the convergence towards the steady-states (Nash Equilibria), by using a PDE approach is delivered. We close our presentation by the construction of a second related model involving the Fractional Laplacian operator and which describes the replicator dynamics approach when the average of the game players choose a more risky strategy.

The Maths Behind Pandemics

Maths is playing a pivotal role in the understanding and fight against epidemics. We will understand together how differential equations can be useful to predict the evolution of infectious diseases and to help governments in taking farsighted decisions.

What Can Networks Tell Us about Disease Transmission – in Cities and in the Outback?

Social networks are familiar to all of us: from our networks of friends and co-workers to Facebook and Instagram. Networks also provide a mechanism for modelling the transmission of information – or disease. In this talk I will describe how network science is being used to model the risk of transmission of coronavirus and the effect of various control strategies in cities like Perth. What is the risk posed by gathering 30,000 spectators in a football ground, and what is the benefit of working from home or installing COVIDSafe? In remote areas the challenges of infectious diseases are acute and very complex. We study the network of movement of people between these remote communities and find that they act like a large virtual city, highly vulnerable to infection.

The stochastic Cahn–Hilliard/Allen–Cahn equation from phase transitions

This SPDE is a combination of Cahn–Hilliard and Allen–Cahn type operators. It involves a small positive parameter ε which stands as a measure of the width of the transition layers that are generated during the phase separation of a binary alloy. Thermal fluctuations or impurities of the alloy motivate a noise additive or multiplicative to the equation. The stochastic Cahn-Hilliard equation is always a special case. We discuss first the physical model, and various definitions of applicable noise therein. Then, for the case of non smooth in space and time noise with unbounded diffusion, we present the mains steps for investigating existence and regularity of stochastic solutions in dimensions d=1,2,3, and by Malliavin calculus existence of density for d=1.

Variational Interpolants: a very simple special case

Given a sequence of points in some ambient space, and possibly some velocity information, the task is to find a suitable curve through the points at given times. The curve is a variational interpolant when it satisfies an optimality condition. Variational interpolants are significant in classical approximation theory, where they are much used and well understood, especially when the ambient space is Euclidean. They are only partially understood when the ambient space is non-Euclidean, for example when the curve describes the trajectory of a rigid body in 3-space. Our talk stays well away from such examples, focusing instead on simpler cases.  

By way of introduction, we consider the task of interpolating between two given affine maps from R to R. This can be seen as absolutely trivial, but from our perspective it amounts to elementary hyperbolic geometry, which is to say nontrivial and well-known. This sets the tone for the rest of the talk, which concerns Hermite interpolation between two given affine maps from R to itself. When the slopes of the maps have the same sign, our task reduces to a boundary value problem for a very special 8-dimensional system of ODEs (a little bit of differential geometry is needed to explain why). We describe some theoretical results about solutions, some work in progress (asymptotics), and some magic (generalized de Casteljau constructions).

Fractional Laplacians on ellipses 

We will illustrate some recent findings concerning explicit evaluations of (higher-order) fractional Laplacians on elliptic domains. This will enable us to give the explicit expression of the torsion function and to construct elementary counterexamples to maximum principles. All the results have been obtained in collaboration with S. Jarohs (Frankfurt a.M.) and A. Saldana (Mexico City). 

The Bernstein problem for parametric elliptic functionals

The Bernstein problem asks whether entire minimal graphs in ℝn+1 are necessarily hyperplanes. This problem was completely solved by the late 1960s in combined works of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti. We will discuss the analogue of this problem for more general elliptic functionals, and some recent progress in the case n = 6.

Nonlocal logistic equations with Neumann conditions

We consider a problem of population dynamics modelled on a logistic equation with both classical and nonlocal diffusion, possibly in combination with a pollination term. The environment considered is a niche with zero-flux, according to a new type of Neumann condition. We discuss the situations that are more favourable for the survival of the species, in terms of the first positive eigenvalue. The eigenvalue analysis for the one-dimensional case is structurally different than the higher dimensional setting, and it sensibly depends on the nonlocal character of the dispersal. We also analyze the role played by the optimization strategy in the distribution of the resources, also showing concrete examples that are unfavourable for survival, in spite of the large resources that are available in the environment.