# Analysis seminar

Università degli Studi Trento

38123 Povo (TN)

Tel +39 04 61/281508-1625-1701-3898-1980.

dept.math [at] unitn.it

Per partecipare agli eventi (telematici), contattare lo Staff di Dipartimento.

**Martedì 1 dicembre ore 14:30**

**Simone Di Marino**(Università di Genova)

**Minimal charge configurations, total charge discrepancy and grand-canonical optimal transport**

Based on a work in progress with L. Nenna and M. Lewin. Motivated by mathematical physics, in particular by the semi-classical limit of the grand-canonical density functional theory, we introduce the grand-canonical optimal transport. We minimize the expectation of the interaction energy between electrons, among probabilities in the configuration space which fix the average density of electrons. The key new feature of this model is that the number of electrons is not fixed. In discussing the features of this problem, we will describe the new phenomenon that is appearing, regarding the estimation of the number of electrons for the optimal configuration; we will see that that the estimate of the discrepancy of this number with respect to the average number of electrons is related to the problem of total charge discrepancy in minimal charge configurations.

**Martedì 15 dicembre ore 15:00**

**Francesco Rossi**(Università di Padova)

**From control of deterministic systems to control of transport partial differential equations**

Take a system of N interacting particles and control them with an external control. What is the corresponding mean-field problem, i.e. the problem of controlling the dynamics when N tends to infinity? It turns out that the answer only makes sense when the control is feedback and sufficiently regular. I will then answer two different questions: - what are the hypothesis on the original problem ensuring the required control regularity? - how can we translate our knowledge of control of finite dimensional deterministic systems into control of the mean-field PDE? These results are obtained with B. Bonnet, M. Duprez, M. Morancey.

**Martedì 12 gennaio ore 14:30**

**Paolo Baroni**(Università di Parma)

**On the relation between Morrey-type spaces of measures and regularity of solutions to related measure data problems**

In the talk I will show implementations, in different settings and various directions, of a general principle connecting regularity properties of measures (in terms of their decay) and good properties for solutions to related measure data problems; in particular, I will focus on quantitative integrability gain for the gradient.

Elliptic and parabolic equations, all modeled after the p-Laplacian, will be considered, as well as generalized Morrey conditions, where the total variation of balls decays in terms of a generic function of their radius.

**Martedì 19 gennaio ore 14:30**

**Nicholas Edelen**(Notre Dame)

**Regularity of minimal surfaces near quadratic cones**

Hardt-Simon proved that every area-minimizing hypercone C having only an isolated singularity fits into a foliation of the Euclidean space by smooth, area-minimizing hypersurfaces asymptotic to C. We prove that if a minimal hypersurface M in the unit ball lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), then, in the ball of radius 1/2, M is a C^{1,\alpha} perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of M, but of any nearby minimal surface. Our result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces in the Euclidean space asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation. This is joint work with Luca Spolaor.

**Martedì 26 gennaio ore 16:00**

**Francesco Maggi**(University of Texas at Austin)

**Soap films and capillarity theory**

Soap films are modeled, rather than as surfaces with zero mean curvature, as regions with small volume satisfying a spanning condition of homotopic type. We discuss qualitative properties of such soap films, and their convergence towards minimal surfaces, when the volume goes to zero. This talk is based on a series of joint works with Antonello Scardicchio (ICTP), Darren King (UT Austin) and Salvatore Stuvard (UT Austin).

**Martedì 2 febbraio ore 14:30**

**Mattia Fogagnolo**(Centro De Giorgi)

**The isoperimetric problem on manifolds with nonnegative Ricci curvature: geometric inequalities and asymptotic Gromov-Hausdorff analysis**

In this talk we discuss two basic tools in order to deal with the isoperimetric problem on manifolds with nonnegative Ricci curvature, namely the sharp isoperimetric inequality and the asymptotic behaviour of the minimizing sequences on these manifolds. The first implies that the sharp isoperimetric constant is given by the Asymptotic Volume Ratio of the manifold and that it is saturated only on balls of the Euclidean space. For what it concerns the latter, we show that the runaway volume is recovered in nonsmooth spaces arising as pointed Gromov-Hausdorff limits.

Examples of Riemannian manifolds enjoying existence and nonexistence of isoperimetric sets will be provided too.

The results are based on joint works with G.Antonelli, L. Mazzieri and M. Pozzetta.

**Martedì 9 febbraio ore 14:30**

**David Bate**(Warwick)

**Characterising rectifiable metric spaces using tangent measures**

A classical result of Marstrand and Mattila states that a set S in the Euclidean space (satisfying mild dimension assumptions) is n-rectifiable if and only if, for H^n-a.e. point x of S, all tangent spaces of H^n|_S at x are n-dimensional subspaces. Here a "tangent space" is defined using Preiss's tangent measures. This talk will present a generalisation of this result that replaces the ambient Euclidean space with an arbitrary metric space.

**Martedì 23 febbraio ore 14:30**

**Robin Neumayer**(Northwestern University)

**Quantitative stability for minimizing Yamabe metrics **

The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric. We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show—in a quantitative sense—that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.

**Martedì 2 marzo ore 14:30**

**Gareth Speight**(University of Cincinnati)

**Whitney Extension and Lusin Approximation in Carnot Group**

The classical Lusin theorem states that any measurable function can be approximated by a continuous function except on a set of small measure. Analogous results for higher smoothness give conditions under which a function can be approximated by a C^m function up to a set of small measure. Proving these results depends on applying a suitable Whitney extension theorem. After recalling the classical results in Euclidean spaces, we discuss recent work extending some of these results to Carnot groups. Based on joint work with Andrea Pinamonti and Marco Capolli.

**Martedì 9 marzo ore 14:30**

**Davide Vittone**(Università di Padova)

**Differentiability of intrinsic Lipschitz graphs in Carnot groups **

Submanifolds with intrinsic Lipschitz regularity in sub-Riemannian Carnot groups can be introduced using the theory of intrinsic Lipschitz graphs started by B. Franchi, R. Serapioni and F. Serra Cassano almost 15 years ago. One of the main related questions concerns a Rademacher-type theorem (i.e., almost everywhere existence of a tangent plane) for such graphs: in this talk I will discuss a recent positive solution to the problem in Heisenberg groups. The proof uses the language of currents in Heisenberg groups (in particular, a version of the celebrated Constancy Theorem) and a number of complementary results such as extension and smooth approximation theorems for intrinsic Lipschitz graphs. I will also show a recent example (joint with A. Julia and S. Nicolussi Golo) of an intrinsic Lipschitz graph in a Carnot group that is nowhere intrinsically differentiable. The talk will be kept at an introductory level.

**Martedì 16 marzo ore 16:00**

**Robert Young**(NYU Courant)

**Metric differentiation and embeddings of the Heisenberg group**

Pansu and Semmes used a version of Rademacher's differentiation theorem to show that there is no bilipschitz embedding from the Heisenberg groups into Euclidean space. More generally, the non-commutativity of the Heisenberg group makes it impossible to embed into any $L_p$ space for $p\in (1,\infty)$. Recently, with Assaf Naor, we proved sharp quantitative bounds on embeddings of the Heisenberg groups into $L_1$ and constructed a metric space based on the Heisenberg group which embeds into $L_1$ and $L_4$ but not in $L_2$; our construction is based on constructing a surface in $\mathbb{H}$ which is as bumpy as possible. In this talk, we will describe what are the best ways to embed the Heisenberg group into Banach spaces, why good embeddings of the Heisenberg group must be "bumpy" at many scales, and how to study embeddings into $L_1$ by studying surfaces in $\mathbb{H}$.

**Martedì 23 marzo ore 14:30**

**Andrea Malchiodi**(SNS Pisa)

**Prescribing Morse scalar curvatures in high dimension**

We consider the classical question of prescribing the scalar curvature

of a manifold via conformal deformations of the metric, dating back to works by Kazdan and Warner. This problem is mainly understood in low dimensions, where blow-ups of solutions are proven to be "isolated simple".We find natural conditions to guarantee this also in arbitrary dimensions, when the prescribed curvatures are Morse functions. As a consequence, we improve some pinching conditions in the literature and derive existence and non-existence results of new type. This is joint work with M. Mayer.

**Martedì 30 marzo ore 14:30**

**Andrea Mondino**(Oxford)

**Optimal transport and quantitative geometric inequalities**

The goal of the talk is to discuss a proof of the Levy-Gromov inequality for metric measure spaces (joint with Cavalletti), a quantitative version of the Levy- Gromov isoperimetric inequality (joint with Cavalletti and Maggi) as well as other geometric/functional inequalities (joint with Cavalletti and Semola). Given a closed Riemannian manifold with strictly positive Ricci tensor, one estimates the measure of the symmetric difference of a set with a metric ball with the deficit in the Levy- Gromov inequality. The results are obtained via a quantitative analysis based on the localisation method via L1-optimal transport.

**Martedì 13 aprile ore 14:30**

**Giorgio Saracco**(SISSA)

**The isoperimetric problem with a double density **

It is well-known that for any given volume, the sets that enclose said volume with the least perimeter are balls. What happens when one in place of the standard Euclidean volume and perimeter considers weighted counterparts? Given densities f: R^N \to R^+ and h:R^N \times S^{N-1} \to R^+ to weigh, resp., the volume and the perimeter, we shall discuss under which hypotheses isoperimetric sets exist for all volumes. Furthermore, we shall introduce the epsilon-epsilon^beta property, which readily allows to prove boundedness. If time allows, some regularity results shall be discussed. Based on joint works with A. Pratelli (Università di Pisa).

**Martedì 20 aprile ore 14:30**

**Sara Daneri**(GSSI)

**Symmetry breaking and pattern formation for local/nonlocal interaction functionals**

In this talk I will review some recent results obtained in collaboration with E. Runa and A. Kerschbaum on the one-dimensionality of the minimizers of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically a perimeter term or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used to model pattern formation, either in material science or in biology. One of the main difficulties in proving the emergence of such regular structures, together with nonlocality, is due to the fact that the functionals retain more symmetries (in this case symmetry with respect to permutation of coordinates) than the minimizers. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are (in general dimension) one-dimensional and periodic. In the discrete setting such results had been previously obtained for a smaller set of functionals with a different approach by Giuliani and Seiringer.

**Martedì 27 aprile ore 14:30**

**Stefano Borghini**(Milano Bicocca)

**Torsion problem for ring-shaped domains**

The torsion problem consists in the study of pairs (Ω, u), where Ω ⊂ R^{2 }is a bounded domain and u : Ω → R is a function with constant nonzero laplacian and such that u = 0 on the boundary ∂Ω.

A celebrated result due to Serrin states that, if one assumes the additional hypothesis that the normal derivative of u is constant on ∂Ω, then Ω must be a ball and u is rotationally symmetric.

We discuss the characterization of rotationally symmetric solutions to the torsion problem on a ring-shaped domain. In contrast with Serrin’s result, we show that having locally constant Neumann boundary data is not sufficient for this purpose. Nevertheless, we prove that rotational symmetry can be forced by means of an additional assumption on the number of maximum points.

**Martedì 4 maggio ore 14:30**

**Maria Colombo**(EPFL Losanna)

**Partial regularity for the supercritical surface quasigeostrophic equation**

The surface quasigeostrophic equation (SGQ) is a 2d physical model equation which emerges in meteorology and shares many of the essential difficulties of 3d fluid dynamics. In the supercritical regime for instance, where dissipation is modelled by a fractional Laplacian of order less than 1/2, it is not known whether or not smooth solutions blow-up in finite time. The goal of the talk is to show that every $L^2$ initial datum admits an a.e. smooth solution of the dissipative surface quasigeostrophic equation (SGQ); more precisely, we prove that those solutions are smooth outside a compact set (away from t=0) of quantifiable Hausdorff dimension. We draw analogies between SQG and other PDEs in fluid dynamics in several aspects, including the partial regularity results, and underline some extra structure that SQG enjoys. This is a joint work with Silja Haffter (EPFL).

**Martedì 11 maggio ore 14:30**

**Salvatore Stuvard**(University of Texas at Austin)

**Mean curvature flow with prescribed boundary: a dynamical approach to Plateau’s problem **

The Brakke flow is a measure-theoretic generalization of the mean curvature flow which describes the evolution by mean curvature of surfaces with singularities. In the first part of the talk, I am going to discuss global existence and large time asymptotics of solutions to the Brakke flow with fixed boundary when the initial datum is given by any arbitrary rectifiable closed subset of a convex domain which disconnects the domain into finitely many "grains". Such flow represents the motion of material interfaces constrained at the boundary of the domain, and evolving towards a configuration of mechanical equilibrium according to the gradient of their potential energy due to surface tension. In the second part, I will focus on the case when the initial datum is already in equilibrium (a generalized minimal surface): I will prove that, in presence of certain singularity types in the initial datum, there always exists a non-constant solution to the Brakke flow. This suggests that the class of dynamically stable minimal surfaces, that is minimal surfaces which do not move by Brakke flow, may be worthy of further study within the investigation on the regularity properties of minimal surfaces. Based on joint works with Yoshihiro Tonegawa (Tokyo Institute of Technology).

### Martedì 18 maggio ore 14:30

**Paolo Bonicatto**(Warwick)

**Decomposition of integral metric currents**

Currents are nowadays a widely used tool in geometric measure theory and calculus of variations, as they allow to give a weak formulation of a variety of geometric problems. The theory of normal and integral currents (initiated mostly by Federer and Fleming in the '60s) was developed in the context of Euclidean spaces. In 2000, Ambrosio and Kirchheim introduced metric currents, defined on complete metric spaces. The talk will be devoted to integral metric currents: we show that integral currents can be decomposed as a sum of indecomposable components and, in the special case of one-dimensional integral currents, we also characterise the indecomposable ones as those associated with injective Lipschitz curves or injective Lipschitz loops. This generalises to the metric setting a previous result by Federer. Joint work with Giacomo Del Nin (Warwick) and Enrico Pasqualetto (Scuola Normale Superiore).

### Martedì 25 maggio ore 14:30

**Chao Xia**(Xiamen University)

**Anisotropic Minkowski inequality for non-convex domains**

In this talk, we discuss the anisotropic Minkowski inequality, which is an isoperimetric type inequality between anisotropic mean curvature integral and anisotropic area, for star-shaped F-mean convex domains or outward F-minimizing domains. Our method is based on the inverse anisotropic mean curvature flow and the anisotropic capacity, respectively. Part of the work is joint with Dr. Jiabin Yin.

### Martedì 1 giugno ore 14:30

**Daniele Semola**(Oxford)

**Boundary regularity and stability under lower Ricci curvature bounds**

The theory of non smooth spaces with lower Ricci Curvature bounds has undergone huge developments in the last thirty years. On the one hand the impetus came from Gromov’s precompactness theorem and from the Cheeger-Colding theory of Ricci limit spaces. On the other hand “synthetic” theories of lower Ricci bounds have been developed, based on semigroup tools (the Bakry-Émery theory) and on Optimal Transport (the Lott-Sturm-Villani theory). The Cheeger-Colding theory did not consider manifolds with boundary, while in the synthetic framework even understanding what is a good definition of boundary is a challenge. The aim of this talk is to present some recent results obtained in collaboration with E. Bruè (IAS, Princeton) and A. Naber (Northwestern University) about regularity and stability for boundaries of spaces with lower Ricci Curvature bounds.

### Martedì 8 giugno ore 14:30

**Ivan Violo**(SISSA)

**Monotonicity formula for harmonic functions in RCD(0,N) spaces**

A classical result on Riemannian manifolds satisfying a lower bound on the Ricci curvature is the monotonicity of the Bishop-Gromov volume ratio. Colding and Minicozzi ('12-'14) realized that for non-negative Ricci curvature there exist

analogous monotone quantities involving the Green function. Recently this has been generalized by Agostiniani, Fogagnolo and Mazzieri ('18) from the Green function to the case of an electrostatic potential and has proven

to be fruitful in proving geometric inequalities. We will see that the same monotonicity formulas can be proven also in the setting of synthetic lower Ricci curvature bounds. This allows to prove some almost-rigidity results which are new also in the smooth case. This is a joint work with professor Nicola Gigli.