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# Benedetta Noris

INdAM-COFUND Outgoing fellow between 2012-11-01 and 2014-10-31

### Project information

#### Title

Analysis of BEC: Mathematical analysis of phase separation and vortex formation in Bose-Einstein condensates

#### Outgoing host organisation Return host organisation

Laboratoire de Mathématiques de Versailles Università degli studi di Milano-Bicocca

Avenue des Etats-Unis, 45 Piazza Ateneo Nuovo, 1

78035 Versailles cedex, France 20126 Milano, Italy

http://lmv.math.cnrs.fr http://home.matapp.unimib.it/

In the outgoing phase I plan to work at Laboratoire de Mathématiques de Versailles in collaboration with professors A. Aftalion and O. Kavian, in the return phase at Università degliStudi di Milano-Bicocca, in collaboration with professor S. Terracini.

My research project concerns the study of partial differential equations, and systems of such equations, which arise in the mathematical modeling of Bose-Einstein condensates. According to the different physical phenomena, I shall study different mathematical problems.

The phenomenon of phase separation in many-component condensates is modeled by a system of coupled Gross-Pitaevskii equations with diverging coupling coefficient. The components of the condensate tend to spatially separate as the coupling coefficient tends to infinity. The limit configuration solves a free boundary problem. I have been studying related questions starting from my Ph. D. thesis with Terracini.

Prof. Aftalion is an expert in the field of Bose-Einstein condensates and in particular in the analysis of vortex formation. Vortices are singularities of the wave function which appear in a condensate under rotation. My research plan includes the analysis of the interaction of these two singular phenomena, which will require to join our competences.

The problems and results described until now concern the time independent elliptic equations. In collaboration with professor Kavian, I shall investigate the associated time dependent systems. No results are known in this direction in the present literature.

### Fellow information

#### Research interests

My research interests focus on partial differential

equations and nonlinear analysis. I have studied semilinear and

quasilinear equations, systems of such equations, related free

boundary problems, and abstract variational methods. More precisely:

- Systems of nonlinear SchrÃ¶dinger equations

modeling Bose-Einstein condensation: existence and multiplicity of

solutions, qualitative properties. The case of strongly competitive

interaction: regularity of the solutions of the free boundary problem,

a priori estimates, Gamma convergence. Orbital stability of solitary

waves. Appearance of vortices in the condensate under rotation. - Aharonov-Bohm magnetic operator: gauge

invariance, optimal partitions, nodal lines. - Elliptic equations with Sobolev supercritical

growth: existence and multiplicity of positive radial solutions - Abstract variational methods. A generalization

of the method of the Nehari manifold, with applications to systems of

nonlinear SchrÃ¶dinger equations. A generalization of the method of

lower and upper solutions in invariant cones, with applications to

semilinear elliptic equations.

#### Previous positions, awards

*2012*: Ricercatore a tempo determinato di tipo (a), Facoltà di Scienze statistiche, Università degli Studi di Milano-Bicocca (declined)*2010-2012*: Assegno di Ricerca, Università degli Studi di Milano-Bicocca*2009*: Post-doc fellowship FCT Fundação para a Ciência e Tecnologia , Universidade de Lisboa (declined)*2006-2009*: Phd fellowship in Mathematics, Università degli Studi di Milano-Bicocca

#### Publications, preprints, other works

- B. Noris,
*On nonlinear Schrödinger systems with strong competition*, Ph.D. Thesis, UniversitÃ degli Studi di Milano-Bicocca (2009) - B. Noris, H. Tavares, S. Terracini, G. Verzini,
*Uniform Holder bounds for nonlinear Schrödinger systems with strong competition*, Comm. Pure Appl. Math. 63, 267-302 (2010) - B. Noris, M. Ramos,
*Existence and bounds of positive solutions for a nonlinear Schrödinger system*, Proc. Amer. Math. Soc. 138, 1681-1692 (2010) - B. Noris, S. Terracini,
*Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions*, Indiana Univ. Math. J. 59, 1361-1404 (2010) - D. Bonheure, B. Noris, T. Weth,
*Increasing radial solutions for Neumann problems without growth restrictions*, Ann. Inst. H. Poincaré Anal. Non LinÃ©aire 29, 573-588 (2012) - M. Grossi, B. Noris,
*Positive constrained minimizers for supercritical problems in the ball*, Proc. Amer. Math. Soc. 140, 2141-2154 (2012) - B. Noris, H. Tavares, S. Terracini, G. Verzini,
*Convergence of minimax and continuation of critical points for singularly perturbed systems*, J. Eur. Math. Soc. 14, 1245-1273 (2012) - B. Noris, G. Verzini,
*A remark on natural constraints in variational methods and an application*, J. Differential Equations 254, 1529-1547 (2013)

to superlinear SchrÃ¶dinger systems - M.-M. Boureanu, B. Noris, S. Terracini,
*Sub and supersolutions, invariant cones and multiplicity results for p-Laplace equations*, Contemporary Math. AMS volume 595, 91-119 (2013) - V. Bonnaillie-NoÃ«l, B. Noris, M. Nys, and S. Terracini,
*On the eigenvalues of Aharonov-Bohm operators with*, Analysis & PDE 7-6, 1365-1395 (2014); DOI:10.2140/apde.2014.7.1365

varying poles - A. Aftalion, B. Noris, and C. Sourdis,
*Thomas-Fermi approximation for coexisting two*, preprint (2014); arXiv:1403.4695, to appear on Comm. Math. Phys. 2015

component Bose-Einstein condensates and nonexistence of vortices for

small rotation - B. Noris, H. Tavares, and G. Verzini,
*Stable solitary waves with prescribed L2-mass for*, preprint (2014); arXiv:1405.5549, to appear on Discrete Contin. Dyn. Syst.-A 2015

the cubic SchrÃ¶dinger system with trapping potentials - B. Noris, M. Nys and S. Terracini,
*On the eigenvalues of Aharonov-Bohm operators with*, preprint (2014); arXiv:1411.5244

varying poles: pole approaching the boundary of the domain - B. Noris, H. Tavares, and G. Verzini,
*Existence and orbital stability of the ground*, preprint (2013); arXiv:1307.3981, To appear on Analysis & PDE

states with prescribed mass for the L2-critical and supercritical NLS

on bounded domains

#### Conferences, schools, other events

- Co-organizer of the special session
*Nonlinear Elliptic Systems and Applications*at Joint International Meeting of the American, European and Portuguese Mathematical Societies (AMS-EMS-SPM) http://aep-math2015.spm.pt

Porto, 10-13 June 2015 *Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation,*Talk delivered at*at The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications*Madrid, 7-11 July 2014- Co-organizer of the workshop
*Maths physics on cold atoms: topological defects and phase separation*

http://lmv.math.cnrs.fr/conferences-et-colloques/maths-physics-meeting-on-cold/

Institut Henri Poincaré, Paris, 13-14 Jan 2014 *Workshop Variational methods and partial differential equations,*Louvain-la-Neuve (Belgium), July 11-12, 2013*Conference Giornata INdAM 013,*Palermo (Italy), 6 June 2013*School 5th Women in Mathematics Summer School on Mathematical Theories towards Environmental Modelsâ*

ICTP, Trieste (Italy), 27 May – 1 June 2013*A theoretical approach to phase separation in multicomponent Bose-Einstein condensates,*Talk delivered at*Maths Physics Young Researchers Meeting,*Institut Henri Poincaré, Paris (France), 11-12 April 2013*A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems*

Talk delivered at*Singular limit problems in nonlinear PDEs,*CIRM Luminy, Marseille (France), 26-30 November 2012*Convergence of minimax and continuation of critical points for singularly perturbed systems*

Talk delivered at*The 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications*

Orlando (Florida), USA, 1-5 July 2012- Co-organizer of the workshop
*A week on fractional diffusion*, http://www2.mate.polimi.it/ocs/?cf=27

Dipartimento di Matematica, Politecnico di Milano, 17-20 Jan 2012 *Some remarks on the method of Nehari manifold and applications,*Talk delivered at*Workshop on nonlinear differential equations,*Pienza (Siena, Italy), 7-11 November 2011*A Free Boundary Problem Arising in the Context of Bose-Einstein Condensation,*Talk delivered at*congress “Connections for Women: Free Boundary Problems, Theory and Applications”,*MSRI, Berkeley (California), USA, 13-14 January 2011*Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions*

Talk delivered at*Conference in memory of Prof. Yadava,*TIFR, Bangalore (India), 7 Jan 2009