Dr François Genoud's home page

Contact details:

Delft Institute of Applied Mathematics
Delft University of Technology
Mekelweg 4
2628 CD Delft, The Netherlands

Email: S.F.Genoud@tudelft.nl
Room HB 04.030
Phone: +31 (0)1 527 858 08

Academic information:

Here is my CV.

Below are a list of publications and a list of collaborators.

Current role:

Assistant Professor (Tenure Track) in the Analysis group of TU Delft.

Research interests:

Partial differential equations, evolution equations, nonlinear analysis, bifurcation theory, calculus of variations, mathematical physics.

Here are a short research statement and a detailed list of publications.

Here is a series of lectures about bifurcation theory applied to nonlinear Schrödinger equations.

And here is a talk where a derivation of the nonlinear Schrödinger equation is given, in the context of the paraxial approximation for nonlinear optical waveguides.

Teaching:

2016-17: In the first semester I taught the Functional Analysis course in the Dutch Mastermath programme, plus service courses. I am currently teaching Spectral Theory at TU Delft, and some service courses.

Here are lecture notes for Spectral Theory, where the spectral theorem for general selfadjoint operators in Hilbert space is proved, with elementary applications to quantum mechanics.

2015-16: Functional Analysis in the Dutch Mastermath programme, and some service courses.

2014-15: Spectral Theory at TU Delft, and some service courses.

Research outline

The core of my research is in nonlinear partial differential equations (PDEs). I apply functional analytic methods to study PDEs in a rigorous mathematical framework. Since my early education in physics, I have always been fascinated by the diversity of wave motions in Nature. My research is now mainly focused on nonlinear PDEs describing wave phenomena in various contexts. I prove theorems about existence and qualitative properties of such waves.

My most important contributions are in nonlinear Schrödinger equations (NLS), which arise in the modelling of a variety of physical phenomena, including the propagation of light in nonlinear optical media (e.g. optical fibres), Langmuir waves in plasma, Bose-Einstein condensates, etc. I have gained deep insight into NLS using bifurcation methods, which is an original approach in this area. A special focus is on NLS with a nontrivial spatial dependence, which govern wave propagation in inhomogeneous media. I study dynamical properties (stability, blow-up, scattering) of nonlinear waves modelled by these equations.

Another part of my research is concerned with the statistical physics of liquid crystals. The study of phase transitions from the isotropic liquid to the liquid crystal state involves challenging nonlinear integro-differential equations describing the statistical distribution of molecules at the macroscopic level. Ongoing work concerns the derivation of macroscopic models (of Landau-de Gennes type) from microscopic principles.

I am also interested in the mathematical modelling of water waves. I have for instance studied stability properties of geophysical waves (large-scale oceanic waves), or the pressure field in solitary waves. Future work in this area will include the study of water waves with non-zero vorticity using bifurcation techniques.

Publications

(CP = conference proceedings)

26. Instability of an integrable nonlocal NLS,
C. R. Math. Acad. Sci. Paris 355 (2017), 299-303, arXiv.

25. Extrema of the dynamic pressure in a solitary wave,
Nonlinear Anal. 155 (2017), 65-71, arXiv.

24. (with S. Bachmann) Scaling limits and phase transitions for nematic liquid crystals in the continuum, submitted, under review, arXiv.

23. (with V. Combet) Classification of minimal mass blow-up solutions for an $L^2$ critical inhomogeneous NLS,
J. Evol. Equ. 16 (2016), 483-500, arXiv.

22. (with B. A. Malomed and R. M. Weishäupl) Stable NLS solitons in a cubic-quintic medium with a delta-function potential,
Nonlinear Anal. 133 (2016), 28-50, arXiv.

21. (with S. de Bièvre and S. Rota Nodari) Orbital stability: analysis meets geometry, in:
C. Besse, J. C. Garreau (eds.), Nonlinear Optical and Atomic Systems,
Lecture Notes in Mathematics 2146, Springer, 2015, pp. 147-273, ISBN 978-3-319-19015-0, arXiv.

20. (with A. Derlet) Existence of nodal solutions for quasilinear elliptic problems in $R^N$,
Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), 937-957, arXiv.

19. (with D. Henry) Instability of equatorial water waves with an underlying current,
J. Math. Fluid Mech. 16 (2014), 661-667, arXiv.

18. (CP) Monotonicity of bifurcating branches for the radial p-Laplacian,
Monografías Matemáticas García de Galdeano 39 (2014), 111-119.

17. (CP) Some bifurcation results for quasilinear Dirichlet boundary value problems,
Electron. J. Diff. Equ. Conference 21 (2014), 87-100.

16. Orbitally stable standing waves for the asymptotically linear one-dimensional NLS,
Evolution Equations and Control Theory 2 (2013), 81-100, arXiv.

15. (with B. P. Rynne) Landesman-Lazer conditions at half-eigenvalues of the p-Laplacian,
J. Differential Equations 254 (2013), 3461-3475, arXiv.

14. Global bifurcation for asymptotically linear Schrödinger equations,
NoDEA Nonlinear Differential Equations Appl. 20 (2013), 23-35, arXiv.

13. Bifurcation along curves for the p-Laplacian with radial symmetry,
Electron. J. Diff. Equ. 2012, no. 124, arXiv.

12. (with B. P. Rynne) Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems,
J. Differential Equations 252 (2012), 5076-5095, arXiv.

11. An inhomogeneous, $L^2$ critical, nonlinear Schrödinger equation,
Z. Anal. Anwend. 31 (2012), 283-290, arXiv.

10. (CP, with B. P. Rynne) Some recent results on the spectrum of multi-point eigenvalue problems for the p-Laplacian,
Commun. Appl. Anal. 15 (2011), 413-434.

9. Bifurcation from infinity for an asymptotically linear problem on the half-line,
Nonlinear Anal. 74 (2011), 4533-4543.

8. (with B. P. Rynne) Second order, multi-point problems with variable coefficients,
Nonlinear Anal. 74 (2011), 7269-7284, arXiv.

7. (CP) Nonlinear Schrödinger equations on $R$: global bifurcation, orbital stability and nonlinear waveguides,
Commun. Appl. Anal. 15 (2011), 395-412.

6. A uniqueness result for $\Delta u - \lambda u + V(|x|) u^p = 0$ on $R^2$,
Adv. Nonlinear Stud. 11 (2011), 483-491.

5. A smooth global branch of solutions for a semilinear elliptic equation on $R^N$,
Calc. Var. Partial Differential Equations 38 (2010), 207-232.

4. Bifurcation and stability of travelling waves in self-focusing planar waveguides,
Adv. Nonlinear Stud. 10 (2010), 357-400.

3. Existence and orbital stability of standing waves for some nonlinear Schrödinger equations, perturbation of a model case,
J. Differential Equations 246 (2009), 1921-1943.

2. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation,
Discrete Contin. Dyn. Syst. 25 (2009), 1229-1247.

1. (with C. A. Stuart) Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves,
Discrete Contin. Dyn. Syst. 21 (2008), 137-186.

Collaborators

Sven Bachmann, Ludwig Maximilian University of Munich (Germany)

Jacopo Bellazzini, University of Sassari (Italy)

Stephan de Bièvre, Université Lille 1 (France)

Vianney Combet, Université Lille 1 (France)

Ann Derlet, Haute Ecole Robert Schuman (Belgium)

Vladimir Georgiev, University of Pisa (Italy)

David Henry, University College Cork (Ireland)

Stefan Le Coz, Université Paul Sabatier, Toulouse (France)

Boris Malomed, Tel Aviv University (Israel)

Benedetta Pellacci, Parthenope University of Naples (Italy)

Simona Rota Nodari, Université de Bourgogne, Dijon (France)

Julien Royer, Université Paul Sabatier, Toulouse (France)

Bryan Rynne, Heriot-Watt University (Scotland)

Charles Stuart, EPFL (Switzerland)

Rada Maria Weishäupl, University of Vienna (Austria)

Last updated March 27, 2017.