# Incontri di Analisi MaTÈmatica

Giovedì 19 Dicembre, Aula 1B1

14:30 Gianmaria Verzini (Politecnico di Milano)

Normalized solutions to semilinear elliptic equations and systems

Abstract: We study the existence of solutions having prescribed L^2 norm to some semilinear elliptic problems in bounded domains. These kind of solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. When the nonlinearity is critical or supercritical with respect to the Gagliardo-Nirenberg inequality, though not Sobolev subcritical, we show that solutions having Morse index bounded from above can exist only when the mass is sufficiently small. On the other hand, we provide sufficient conditions for the existence of such solutions, also in the Sobolev critical case. Based on joint works with Benedetta Noris, Dario Pierotti and Hugo Tavares

15:45 Marco Cirant (Università di Parma)

Gradient regularity of viscous Hamilton-Jacobi equations with rough data

Abstract: The talk will be devoted to the study of Lipschitz regularity of solutions to viscous Hamilton-Jacobi equations, with data in Lebesgue spaces. I will first present some results obtained in collaboration with A. Goffi (GSSI-Padova). These are based on adjoint methods, namely on the analysis of regularity to dual equations of Fokker-Planck type. Finally, I will discuss further perspectives regarding the so-called problem of "maximal regularity".

Giovedì 7 Novembre 2019

Giuseppe Viglialoro (Università di Cagliari)

Basic concepts and analyses on Keller-Segel-type systems modeling chemotaxis

Chemotaxis is the motor reaction of an organism (cells, bacteria, and other single-cell or multicellular organisms) distributed in a certain environment in response to a chemical stimulus also present therein. In this seminar we present the landmarking chemotaxis model, proposed by Keller and Segel in 1970, and described by means of two coupled partial differential equations. Once some of its generalizations are briefly discussed, for a simplified version we give mathematical details indicating how the behavior of solutions to the corresponding system is intimately connected to the data of the problem.

Helicoidal vortex filaments for the three dimensional Ginzburg-Landau equation

In this talk we will construct new entire solutions of the Ginzburg-Landau equation in dimension 3. These solutions are screw-symmetric and have two vortex filaments clustering near the e3-axis

Mercoledì 29 Maggio 2019

Begoña Barrios Barrera (Universidad de La Laguna)

The sharp exponent in the study of the nonlocal Hénon equation in RN.

We will consider the nonlocal Hénon equation (−Δ)su=|x|αup,RN, where (−Δ)s is the fractional Laplacian operator with 0<s<1, −2s<α, p>1 and N>2s. We prove a Liouville result for positive solutions in the optimal range of the nonlinearity, that is, when 1<p<p∗α,s:=N+2α+2sN−2s. Moreover, we prove that a kind of bubble solution, that is, a fast decay positive radially symmetric solutions, exists when p=p∗α,s.

Mercoledì 3 Aprile 2019

Azahara DelaTorre (Albert-Ludwigs-Universität Freiburg)

Concentration phenomena for the fractional Q-curvature equation in dimension 3 and fractional Poisson formulas

We study the compactness properties of metrics of prescribed fractional Q-curvature of order 3 in R3. We will use an approach inspired from conformal geometry, seeing a metric on a subset of R3 as the restriction of a metric on R4+ with vanishing fourth-order Q-curvature. In particular, we will show that a sequence of such metrics with uniformly bounded fractional Q-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function Φ in R4+, in analogy with a 4-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest. This is a work done in collaboration with María del Mar González, Ali Hyder and Luca Martinazzi.

Luca Battaglia (Università di Roma Tre)

A double mean field approach for a curvature prescription problem

I will consider a double mean field-type Liouville PDE on a compact surface with boundary, with a nonlinear Neumann condition. This equation is related to the problem of prescribing both the Gaussian curvature and the geodesic curvature on the boundary. I will discuss blow-up analysis, a sharp Moser-Trudinger inequality for the energy functional, existence of minmax solution when the energy functional is not coercive. The talk is based on a work in progress with Rafael Lopez-Soriano (Universitat de Valencia).

27 Febbraio 2019

Giusi Vaira (Università degli studi della Campania “Luigi Vanvitelli”)

"On an elliptic equation with critical growth and Hardy potential"

I will discuss classification results for the critical p-Laplace equation in the

whole space. In particular I shall present some new results in collaboration with F. Oliva and B. Sciunzi regarding the doubly critical equation involving the Hardy potential.

Michal Kowalczyk (CMM, Chile)

"New examples of multiple end solutions in the Cahn-Hilliard theory of phase transitions"

In very general terms multiple end solutions of a semilinear elliptic PDE in $\mathbb R^N$ are entire solutions which along some specific directions such as hyperplanes approach asymptotically lower dimensional solutions of the same equation. For instance for the Allen-Cahn equation $\Delta u=-u+u^3$ in $\mathbb R^2$ there exists an odd symmetric solution, known as the saddle solution such that $\lim_{y\to \pm \infty} u(x,y)=\pm H(x)$, where $H$ is the unique, odd, increasing solution of the ODE $u''=-u+u^3$ (Fife, Dang, Peletier '92). Many other examples are known for this and other equations. In this talk I will discuss two new examples: one for the Cahn-Hilliard equation in $\mathbb R^3$ and the other for the generalized second Painlev\'e equation on $\mathbb R^2$.

Mercoledì 30 Gennaio 2019, Aula Seminari

Seunghyeok Kim (Hanyang University)

Elliptic systems with nearly critical exponents on general domains

In 2008, Guerra studied the asymptotic behavior of positive minimal energy solutions of nearly critical Lane-Emden systems on smooth bounded convex domains, as the exponents of the nonlinear terms approach the critical Sobolev hyperbola. In his work, in addition to the convexity assumptions on domains, there was a restriction on the range of the exponents. In this talk, we remove all these technical assumptions, thereby completing the analysis under the whole subcritical regime. This is a joint work with Professor Woocheol Choi (Incheon National University).

Michał Łasica (University of Warsaw)

Existence of 1-harmonic map flow

Given a complete Riemannian manifold, we consider the problem of constructing an L2-steepest descent flow of the total variation energy of maps taking values in the manifold. In the case that the domain is a convex, bounded subset of a Euclidean space or a compact, orientable Riemannian manifold, we are able to prove local existence of a suitably defined flow on Lipschitz maps, which under some conditions can be continued indefinitely. This flow is unique. If the domain is an interval, we can define and construct a globally existing flow on maps (parametrized curves) of bounded variation. If the target manifold has non-positive sectional curvatures, this flow coincides with the one obtained by the general construction of Ambrosio, Gigli, Savare and is unique.

Mercoledì 30 Gennaio 2019, Aula Seminari

Seunghyeok Kim (Hanyang University)

Elliptic systems with nearly critical exponents on general domains

In 2008, Guerra studied the asymptotic behavior of positive minimal energy solutions of nearly critical Lane-Emden systems on smooth bounded convex domains, as the exponents of the nonlinear terms approach the critical Sobolev hyperbola. In his work, in addition to the convexity assumptions on domains, there was a restriction on the range of the exponents. In this talk, we remove all these technical assumptions, thereby completing the analysis under the whole subcritical regime. This is a joint work with Professor Woocheol Choi (Incheon National University).

Michał Łasica (University of Warsaw)

Existence of 1-harmonic map flow

Given a complete Riemannian manifold, we consider the problem of constructing an L2-steepest descent flow of the total variation energy of maps taking values in the manifold. In the case that the domain is a convex, bounded subset of a Euclidean space or a compact, orientable Riemannian manifold, we are able to prove local existence of a suitably defined flow on Lipschitz maps, which under some conditions can be continued indefinitely. This flow is unique. If the domain is an interval, we can define and construct a globally existing flow on maps (parametrized curves) of bounded variation. If the target manifold has non-positive sectional curvatures, this flow coincides with the one obtained by the general construction of Ambrosio, Gigli, Savare and is unique.

## Mercoledì 12 Dicembre 2018:

Dicembre 2018

Arrigo Cellina

Universita` degli Studi di Milano-Bicocca

On the regularity of solutions to some classes of variational problems

Susanna Terracini

Università di Torino

The nodal set of solutions to sublinear, discontinuous and singular equations

### Mercoledì 24 Ottobre 2018

Marco Degiovanni

Università Cattolica del Sacro Cuore

Multiple solutions of quasilinear equations with natural growth conditions and related problems

Manuel Gnann

Technical University of Munich

Analysis of moving contact line motion

### Mercoledì 20 Giugno 2018

François Murat

Laboratoire Jacques-Louis Lions, Sorbonne Université e CNRS

Problemi ellitici semilineari con un termine noto con singolarità in u=0

Luigi Orsina

Sapienza Università di Roma

Il principio del massimo per le equazioni di Schrödinger

### Mercoledì 16 Maggio 2018:

Sobolev inequality in arbitrary domains

Anatoli F. Tedeev

Some qualitative results for degenerate parabolic equations with inhomogeneous density

### Mercoledì 21 Marzo 2018:

Andrea Moiola

Università di Pavia

Scattering by fractal screens: functional analysis and computation

Daniele Cassani

Università degli Studi dell'Insubria

Choquard type equations with Hardy-Littlewood-Sobolev upper- and lower-critical growth

PosterFebbraio2.pdf

### Mercoledì 28 Febbraio 2018: Poster.pdf

Marco Rigoli

Università di Milano

Mean curvature flow solitons

Paolo Mastrolia

Università di Milano

From Ricci solitons to f-structures: “potential” generalization of some canonical Riemannian metrics

PosterFebbraio.pdf

### Mercoledì 7 Febbraio 2018 Poster.pdf

Marco Squassina

Università Cattolica di Brescia

Approximation results for magnetic Sobolev norms

Sergio Segura de León

Regularizing effects concerning elliptic problems having a superlinear gradient term

PosterGennaio.pdf

### Mercoledì 31 Gennaio 2018 Poster.pdf

María Medina

Benedetta Pellacci

Università degli studi della Campania “Luigi Vanvitelli”

Nonlinear Helmholtz equations: some existence results

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Organizzatori: Tommaso Leonori, Francesco Petitta, Angela Pistoia