Menu (preliminary)

Enno Lenzmann - On uniqueness for the prescribed Q​​-curvature problem in one dimension

We consider the nonlocal Liouville equation √-Δ w = K ew​​ on the real line, where K​​ is a given function that can be regarded as a prescribed Q​​-curvature. For positive and constant K = const. > 0, the solutions are explicitly known and unique up to symmetries. In this talk, we shall present a uniqueness result for a broad class of positive and non-constant K’s by exploiting a surprising connection to completely integrable PDEs of Calogero-Moser type. This talk is based on joint work with Maria Ahrend (Basel) and Patrick Gérard (Orsay).

Tuomo Kuusi - Unique continuation for periodic elliptic equations

I will discuss recent works on quantitative unique continuation for solutions of periodic elliptic equations on large scales. The results range from large-scale analyticity to nearly optimal doubling inequalities and three-sphere type theorems. Based on joint works with S. Armstrong and C. Smart.

Sławomir Kolasiński - The Atomic Condition in varifold theory

A smooth positive function F​​ on the Grassmannian of k​​-planes in Rn​​ gives rise to a functional ΦF​​ on k​​-dimensional rectifiable surfaces in Rn​​ given by

ΦF(S)​​ = ∫S F(Tan(S,x))  d Hk(x)​  whenever S ⊆ ℝn​​ is (Hk,k)​​ rectifiable.

Such energies arise naturally in anisotropic spaces, e.g., the k​​-dimensional Hausdorff measure Hkϕ​​ constructed over a non-Euclidean normed space (ℝn,ϕ)​​ coincides with ΦF​​ on rectifiable sets for certain choice of F​​.

If the integrand F​​ is elliptic, one expects critical points of ΦF​​ to be regular. However, it is not clear what ``ellipticity'' should mean in this context. In 1968 Almgren introduced his notion of ellipticity tailored for proving existence and regularity of minimisers. Alas, the condition is rather hard to check for any particular F​​ and there are virtually no non-trivial examples of integrands satisfying it.

The Atomic Condition (AC) has been introduced by De Philippis, De Rosa, and Ghiraldin, in a paper published in 2018, for proving rectifiability of critical points (considered in the class of all k​​-varifolds with lower density bound). This condition is more geometric and there are reasons to believe it is exactly the right requirement.

In my talk I will survey results considering ellipticity with special focus

on AC. I will also present my own perspective on the geometric nature of this condition and, possibly, present a construction of AC integrands.

Valentino Magnani - Surface measure on, and the local geometry of, sub-Riemannian manifolds

In equiregular sub-Riemannian manifolds, we relate the perimeter measure of a smooth bounded open set with the spherical measure of its boundary. Our arguments rely on uniform asymptotic estimates of the diamter of sub-Riemannian balls, which require a careful study of the local geometry of the ambient space. The density of the perimeter measure is a geometric invariant, which can be explicitly related to different objects, like the nilpotent approximation, the tangent Riemannian metric and the shape of the tangent sub-Riemannian unit ball. The area formula for the perimeter measure is finally achieved by showing that this invariant exactly equals the Federer density. These results are a joint work with Sebastiano Don (University of Brescia).

Daniel Cameron Campbell - Injectivity in second-gradient Nonlinear Elasticity

We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that Ω⊂ℝn​​ is a domain, f∈ W2,q(Ω,ℝn)​​ satisfies |Jf|-a∈ L1​​ and that f​​ equals a given homeomorphism on ∂Ω​​. Under suitable conditions on q​​ and a​​ we show that f​​ must be a homeomorphism. As a main new tool we find an optimal condition for a​​and q​​ that imply that ℋn-1​​({Jf=0})=0​ and hence Jf​​ cannot change sign. We further specify in dependence of q​​ and a​​ the maximal Hausdorff dimension d​​ of the critical set {Jf=0}​​. The sharpness of our conditions for d​​ is demonstrated by constructing respective counterexamples.

Azahara de la Torre - Symmetry and symmetry-breaking for the fractional Caffarelli-Kohn-Nirenberg inequality

In this talk we will consider a fractional version of the Caffarelli-Kohn-Nirenberg inequality which represents an interpolation between the fractional Sobolev inequality and the (usual or weighted) fractional Hardy inequality. Using some tools developed in conformal geometry, we will focus on three different goals. First, we review the existence and nonexistence of extremal solutions. Next, we prove some new results on the symmetry and symmetry-breaking region for the minimizers, where we will observe that the non-local version presents a contrasted behaviour from its local counterpart. Finally, we will show non-degeneracy of critical points and uniqueness of minimizers in the radial symmetry class. 

This is a joint work with W. Ao and MdM. González.

Julian Scheuer - Stability for the constant mean curvature problem in warped product spaces

Closed, embedded constant mean curvature (CMC) hypersurfaces in the Euclidean, spherical and hyperbolic spaces have been classified as round spheres in a nowadays classical and famous paper by Alexandrov from the 1960s. In a class of warped product spaces, such as the de-Sitter-Schwarzschild spaces, such a classification is due to Simon Brendle and about 10 years old. The stability question for these results ask, whether closed and embedded "almost" CMC hypersurfaces must be Hausdorff-close to geodesic spheres. While in the Euclidean, spherical and hyperbolic spaces this problem has been studied with very general results by Magnanini-Poggesi, Ciraolo-Vezzoni and several others, in this talk we want to present the proof of the corresponding counterpart in a class of warped product spaces with non-constant sectional curvature. This is joint work with Chao Xia (Xiamen University). 

Zofia Grochulska - Approximately differentiable homeomorphisms with prescribed derivative

In recent years, there has been a lot of interest in properties of non-classically differentiable homeomorphisms, i.e., having a Sobolev or BV derivative. We will focus on the notion of approximate differentiability, which is a~weaker property and gives rise to surprising examples. Goldstein and Hajłasz showed in 2017 that there exists an approximately differentiable a.e. homeomorphism of the unit cube Q​​, which coincides with identity on the boundary but whose approximate derivative equals the reflection matrix a.e.

In this talk, we will see that this construction can be significantly generalized and that under some mild assumptions on a map T: Q → GL(n)​​ we can find an approximately differentiable a.e. homeomorphism Φ​​ of the unit cube such that Φ(x) = x​​ for x ∈ ∂Q​​ and DΦ = T​​ a.e. I will also present the connections between this problem and the Oxtoby--Ulam theorem about homeomorphic measures and the Dacorogna--Moser theory of mappings with prescribed Jacobian. This is joint work with Paweł Goldstein and Piotr Hajłasz.

Tobias Lamm - Estimates for sequences of harmonic maps

We study the limiting behavior of the index and the nullity of sequences of harmonic maps from a two-dimensional Riemann surface into a general target manifold. We show upper and lower bounds for the index of the sequence in terms of the index of the so called bubble limit. This is a joint work with Jonas Hirsch (Leipzig).

Jonas Hirsch - Interior regularity for two-dimensional stationary Q-valued maps

We consider 2-dimensional Q-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy. In this talk I would like to present some idea’s that can be used to show that these maps are actually Hölder continuous and that the dimension of their singular set is at most one.

I would like to highlight how one can adapt ideas of M. Grüter to “reverse” the Douglas and Rado approach. We will use properties of minimal surfaces to show continuity of stationary “harmonic” functions. This idea helps us to localise in the target space AQ​​ and overcome the problem of the absence of an “honest” PDE.

Joint work with L. Spolaor.

Elena Maeder-Baumdicker - Alexandrov immersed extrinsic flows

Originated in Alexandrov's work on surfaces of constant mean curvature, an Alexandrov immersion bounds a domain that can overlap itsself. We explain that this property is of interest also in the world of extrinsic geometric flows. While evolving by the area preserving curve shortening flow, for example, a closed Alexandrov immersed curve keeps this property as long as the solution exists. We will give an overview of recent results about Alexandrov immersed flows, in particular for the Volume preserving Mean curvature flow. This talk is based on joint work with Ben Lambert.

Lorenzo Brasco - Uniqueness and non-uniqueness for the Lane-Emden equation

We discuss the uniqueness of positive solutions for the Lane-Emden equation for the p-​​Laplacian, i.e.

-Δp u=α uq-1.​​

We consider this in an open set Ω​​, coupled with homogeneous Dirichlet boundary conditions.

We present some results both for the sub-homogeneous case (i.e., 1<q<p​​) and the super-homogeneous one (i.e., q>p​​).

Some of the results presented have been obtained in collaboration with Erik Lindgren (KTH), Francesca Prinari (Pisa) and Anna Chiara Zagati (Parma).

Ryan Alvarado - Function spaces and the geometry of sets

The primary focus of this talk is on how the geometric makeup of a given ambient can directly influence the amount of analysis that the underlying space can support. As an illustration of the interplay between these two branches of mathematics, we will survey some recently obtained results pertaining to the properties of extension and embedding domains for the scale of Besov and Triebel-Lizorkin spaces (Nsp,q​​and Msp,q​​ spaces) in the general setting of quasi-metric spaces. In particular, we will provide examples of several environments, including fractal sets, which highlight how the range of the smoothness parameter, s​​, for which these embedding and extension results hold is intimately linked to the geometry of underlying quasi-metric space. This talk is based on joint work with Dachun Yang and Wen Yuan.

Eleonora Cinti - Flatness results for stable solutions to some nonlocal problems

We study stable critical points to two, closely related, nonlocal functionals: the fractional perimeter and the fractional Allen-Cahn energy. In particular, we establish optimal energy estimates, whose local analogue is still unknown. These estimates, together with other ingredients (such as density estimates and a monotonicity formula) allow to reduce the classification of stable critical points for both functionals to the classification of stable hypercones. These results have been obtained in collaboration with X. Cabré and J. Serra.

Ben Sharp - A gap theorem for H-surfaces in ℝ3​

The H-energy is a natural variant of the Dirichlet energy along maps from a Riemann surface into ℝ3​​, whose critical points (H-surfaces) include conformally parameterised constant mean curvature surfaces. We will show that the minimum H-energy level of a fixed non-spherical Riemann surface must be strictly larger than that of the sphere - in particular showing that the optimal constant in Wente’s famous L2​​-estimate cannot be approached by critical values on non-spherical surfaces. This is a joint work with Andrea Malchiodi (SNS Pisa) and Melanie Rupflin (Oxford).

Vincent Millot - Torus and split solutions of the Landau-de Gennes model for nematic liquid crystals

In this talk, I will present the Q-tensor model of Landau-de Gennes for nematic liquid crystals in the so called Lyutsyukov regime dealing with maps with values in the 4-dimensional sphere. This model describes stable configurations of a liquid crystal as minimizers of a Ginzburg-Landau type energy in which the potential well is the real projective plane, seen as a submanifold of S4.  In the case where the 3D domain is the unit ball and the Dirichlet boundary data is radially symmetric (equivariantly), one may expect that a minimizer inherits such symmetry. Classical simulations show that this is not the case and a certain toroidal structure appears. If (equivariant) axial symmetry is imposed to reduce the complexity of the problem, another type of « singular » solutions appears, the split solutions. By means of regularity results on this model, I will discuss the existence / geometry of torus and split solutions and explain the strong dependence of the type of solutions with respect to the boundary condition and the shape of the domain. This talk is based on recent works in collaboration with Federico Dipasquale and Adriano Pisante. 

Simon Bortz - A quantitative free boundary problem for caloric measure

Through a series of works (in the years ~1988-2005) John Lewis and his collaborators (Hofmann, Murray, Nyström and Silver) studied the caloric measure different time-varying domains. In particular, Lewis and Murray showed that the region above a `regular’ parabolic Lipschitz graph has Caloric measure (locally) in the Muckenhoupt A∞​​ class. The additional regularity in `regular’ graphs is a non-local condition and the local A∞​​ condition is equivalent to the solvability of the Dirichlet problem with Lp​​ estimates for some p > 1.

Together with Hofmann, Nystorm and Martell, we have shown that this extra regularity is, in fact, necessary (assuming the boundary is a parabolic Lipschitz graph). In particular, the A∞​​ condition implies that the graphs are `regular’. In rougher settings, one quantitative substitute for `regular’ Lipschitz graphs is parabolic uniform rectifiability, a notion introduced by Hofmann, Lewis and Nyström. For instance, it is know that a parabolic Lipschitz graph is ‘regular’ if and only if it is parabolic uniformly rectifiable. We have also made progress towards showing the (weak-)A∞​​ condition implies parabolic uniform rectifiability in vast generality. I will discuss the techniques and the difficulties in these problems.

Cristiana de Filippis - Schauder estimates for any taste

So-called Schauder estimates are a standard tool in the analysis of linear elliptic and parabolic PDE. They have been originally obtained by Hopf (1929, interior case), and by Schauder and Caccioppoli (1934, global estimates). The nonlinear case is a more recent achievement from the ’80s (Giaquinta & Giusti, Ivert, Lieberman, Manfredi). All these classical results hold in the uniformly elliptic framework. I will present the solution to the longstanding problem, open since the ‘70s, of proving estimates of such kind in the nonuniformly elliptic setting. I will also cover the case of nondifferentiable functionals and provide a complete regularity theory for a new double phase model. From joint work with Giuseppe Mingione (University of Parma).