RELAZIONE ATTIVITA’ ANNUALE DEI PERFEZIONANDI/DOTTORANDI – PRIMO ANNO

REPORT ON THE PHD ACTIVITY – FIRST YEAR

NOME E COGNOME

NAME AND SURNAME

Sergio Cruz Blázquez

DISCIPLINA/PHD COURSE Mathematics

CORSI FREQUENTATI CON SOSTENIMENTO DI ESAME FINALE

ATTENDED COURSES (WITH FINAL EXAM)

VOTAZIONE

RIPORTATA

MARK

NUMERO

DI ORE

HOURS

Geometric Flows – Andrea Malchiodi 29 40

CORSI FREQUENTATI SENZA SOSTENIMENTO DI ESAME FINALE

ATTENDED COURSES (ATTENDANCE ONLY)

NUMERO

DI ORE

HOURS

Riemannian Geometry – Luciano Mari 40

Elliptic Partial Differential Equations – Luigi Ambrosio 40

Initial Data Sets in General Relativiti – Alessandro Carlotto 20

ALTRE ATTIVITÀ FORMATIVE (SEMINARI, WORKSHOP, SCUOLE ESTIVE, ECC.) –

DESCRIZIONE

OTHER PHD ORIENTED ACTIVITIES (SEMINARS, WORKSHOPS, SUMMER SCHOOLS,

ETC) – DESCRIPTION

NUMERO DI

ORE

HOURS

Conference: Geometric Aspects of Partial Differential Equations (Università degli

studi di Firenze, 10/11 – 11/11) – Full description of the event available here:

https://sites.google.com/view/geometricaspectsofpdes/home?authuser=0

Aprox. 13

(The

program is

attached)

Simposio di Analisi Matematica in Onore di Ferruccio Colombini (Universitá di Pisa,

30/11 - 1/12) – Full description of the event available here:

https://www.dm.unipi.it/webnew/it/content/convegno-onore-di-ferruccio-

colombini

Aprox. 12

(The

program is

attached)

Conference: Variational Methods in Analysis, Geometry and Physics (Scuola

Normale Superiore, 12/2 – 16/2) – Full description of the event available here:

http://webtheory.sns.it/variational-methods/index.php

Aprox. 25

(The

program is

attached)

Conference with mini courses: INDAM - Nonlinear PDEs in Geometry and Physics

(Palazzone di Cortona, 11/6 – 15/6) – More information available in the website of

the event: https://sites.google.com/a/uniroma1.it/cortona_2018/home

Aprox. 23

(The

program is

attached)

Summer school: CIME - Geometric Analysis (Centro Internazionale Matematico

Estivo, 18/6 - 22/6) - The details of the event are available here:

http://web.math.unifi.it/users/cime/

Aprox. 28

(The

program is

attached)

Conference: Recent advances in Geometric Analysis (Scuola Normale Superiore,

4/6 - 8/6) - The complete information about the event is available here:

http://www.crm.sns.it/event/435/

Aprox. 21

(The

program is

attached)

Attendance to some seminars of the Calculus of Variations and Geometric

Measure Theory research group. The precise list is attached without the

certificates of attendance.

16

ATTIVITÀ DI RICERCA EVENTUALMENTE SVOLTA (MAX. 3.000 CARATTERI)

RESEARCH ACTIVITY (MAX. 3000 CHARACTERS)

I continued my work on the problem of prescribing the Gaussian and geodesic curvature on a disk

and its boundary under the supervision of David Ruiz. Using a completely new variational

framework and a deep study on the Moser-Trudinger type inequalities, we were able to find

minimizers under symmetry assumpions for non-negative functions. Compactness of solutions

holds and we extend the existence results to small perturbations of non-negative functions. The

results have already been published.

Together with Andrea Malchiodi, we decided to address two natural generalizations of this

problem. The first one is the problem of prescribing Q and T curvature on a fourth dimensional

manifold with boundary, starting from the case 𝑀 = ℝ+4

with constant curvatures, and the second

corresponds with prescribing scalar curvature on a compact manifold and mean curvature on its

boundary. Concerning the last one, we firstly work under the assumption that the border is a

minimal surface.

The study of the energy functionals require to learn several techniques from the Calculus of

Variations, such as the Mountain-Pass theorem, the Morse Theory and the Monotonicity Method,

which is useful when standard compactness conditions are not guaranteed.

EVENTUALI PUBBLICAZIONI

PUBLICATIONS (IF AVAILABLE)

S. Cruz-Blázquez, D. Ruiz, Prescribing Gaussian and geodesic curvatures on disks, Advanced Nonlinear Studies 18 (2018), 453-468. Published Online: 2018-07-20 | DOI: https://doi.org/10.1515/ans-2018-2021 | ArXiv version: https://arxiv.org/abs/1806.06292

DATA

DATE 9 / 10 / 2018

FIRMA

SIGNATURE 26/11/2018

SIMPOSIO di ANALISI MATEMATICA

in occasione dei 70 anni di

FERRUCCIO COLOMBINI

30 Novembre - 1 Dicembre 2017

Aula Magna, Dipartimento di Matematica, Pisa

• 30 Novembre, mattino

– 10.00 - 10.15 S. Spagnolo (Pisa)

Opening

– 10.15- 11.05 C. Zuily (Orsay)

The stationary phase estimate revisited

– 11.05 - 11.40. Coffee Break

– 11.40 - 12.30 L. Pernazza (Pavia)

A few attempts non-solving non-solvable equations

• 30 Novembre, pomeriggio

– 15.00 - 15.50 F. Fanelli (Lyon)

Asymptotic behaviour of non-homogenous fluids in fast rotation

– 16.00 - 16.50 T. Nishitani (Osaka)

On the Cauchy problem for D2t − b(t)Dxa(x)Dx

– 17.00 - 17.30 Coffee Break

– 17.30 - 18.20 J.-M. Bony (Ecole Polytechnique)

On the decomposition of smooth nonnegative functions into sumsof squares

1

• 1 Dicembre, mattino

– 09.00 - 09.50 V. Petkov (Bordeaux)

Cauchy problem for hyperbolic operators with triple characteristics

– 10.00 - 10.50 F. Ricci (SNS)

Continuity and smoothness of spectral multipliers in the functionalcalculus of sublaplacians on Lie groups

– 11.00 - 11.30 Coffee Break

– 11.30 - 12.20 E. Jannelli (Bari)

Dissipative higher order hyperbolic equations

• 1 Dicembre, pomeriggio

– 15.00 - 15.50 N. Orru (Cagliari)

Levi conditions for some third order hyperbolic equations

– 16.00 - 16.30 Coffee Break

– 16.30 - 17.20 N. Visciglia (Pisa)

Large data scattering for the generalized KdV equation

2

SCHEDULE

Geometric AnalysisJune 18 - June 22, 2018 - Cetraro

Time Monday Tuesday Wednesday Thursday Friday

9:00 - 10:30 AN 1 AN 2 AN 3 AF 3 AF 411:00 - 12:30 PT 1 PT 2 PT 3 PY 3 PY 4

Lunch Lunch Lunch Lunch Lunch16:00 - 17:30 AF 1 AF 2 Open AN 418:00 - 19:30 PY 1 PY 2 Open PT 4

Ailana Fraser: Minimal surfaces and extremal eigenvalue problems on surfaces

Andre Neves: Min-max Theory, volume spectrum and minimal surfaces

Peter Topping: Ricci flow and Ricci limit space

Paul Yang: CR Geometry in three dimensions

1

List of Seminars – Sergio Cruz Blázquez

Liouville-Type problems on compact surfaces: a variational approach. Aleks Jevnikar,

University of Rome ‘Tor Vergata’. Place: Dipartimento di Matematica (Pisa). Date:

25/10/2017.

The saga of a fish: from a survival guide to closing lemmas for dynamical systems. Eugene

Stepanov, St. Petersbusg State University. Place: Dipartimento di Matematica (Pisa). Date:

25/10/2017.

Monocity and symmetry of solutions with uniform limits for some elliptic systems. Nicola

Soave, Politecnico di Milano. Place: Scuola Normale Superiore (Pisa). Date: 30/10/2017.

A Hele-Shaw tumor growth model as a gradient flow. Simone di Marino, Scuola Normale

Superiore. Place: Dipartimento di Matematica (Pisa). Date: 8/11/2017.

Branched transportation: stability and new models. Andrea Marchese, Universität Zürich.

Place: Dipartimento di Matematica (Pisa). Date: 15/11/2017.

Asymptotic planar N-bubble. Giacomo Del Nin, Università di Pisa. Place: Dipartimento di

Matematica (Pisa). Date: 22/11/2017.

Functions of fractal bounded variation. Roger Züst, Universität Bern. Place: Dipartimento di

Matematica (Pisa). Date: 22/11/2017.

Stable constant-mean-curvature hypersurfaces: regularity and compactness, Costante

Belletteni, University College London. Place: Scuola Normale Superiore (Pisa). Date:

24/11/2017.

Variational approximations and relaxations of the Steiner problem. Mauro Bonafini, Università

di Trento. Place: Dipartimento di Matematica (Pisa). Date: 06/12/2017.

Weak universality of singular SPDEs. Massimiliano Gubinelli, Hausdorff Center for

Mathematics, Bonn, Germany. Place: Scuola Normale Superiore. Date: 07/12/2017.

Optimal Partitions. Edouard Oudet. Univ. de Savoie, Chambery. Place: Dipartimento di

Matematica (Pisa). Date: 13/12/2017.

A minimizing problem with two constraint conditions. Norihisa Ikoma, University of Tokyo.

Place: Scuola Normale Superiore (Pisa). Date: 14/12/2017

Un approccio variazionale alla regolarità delle frontier libere singolari, Bozhidar Velichkov,

Université Grenoble Alpes. Place: Dipartamento di Matematica (Pisa). Date: 07/02/2018

A Feynman-Kac Formula on Manifolds with Boundary and Geometric Applications, Levi

Lopes de Lima, Universidade Federal do Ceará. Place: Scuola Normale Superiore. Date:

08/02/2018

Il funzionale di Willmore e il metodo diretto dell’approccio varifold, Marco Pozzeta, Università

di Pisa. Place: Dipartimento di Matematica (Pisa). Date: 27/02/2018.

Duality for the W_\infty Wasserstein distance, Luigi de Pascale, Dipartimento di Matematica

ed Informatica “U. Dini” (Firenze). Place: Dipartimento di Matematica (Pisa). Date:

07/03/2018.