Francois Genoud

### Dr François Genoud

I am a Research Associate in the Department of Mathematics of Heriot-Watt University, where I arrived in August 2010. I have obtained my PhD in Mathematics from the Ecole Polytechnique Federale de Lausanne (EPFL), Switzerland, in 2008, and then spent one and a half year as a Swiss NSF Postdoctoral Fellow at OxPDE, University of Oxford. As an undergraduate, I studied Physics at EPFL and ETH Zurich.

### Research

In this brief summary I present my most important results.

#### Nonlinear Schrödinger equations

The research work for my PhD thesis and its further developments pertain to nonlinear Schrödinger equations (NLS) with nonlinearities having a nontrivial spatial dependence, sometimes referred to as `inhomogeneous' NLS. Physically speaking, these describe propagation phenomena in inhomogeneous nonlinear media. I have proved results of bifurcation and stability of standing waves for such equations. A first series of papers [1-5] deal with equations with a power-type nonlinearity. More recently I have studied NLS with asymptotically linear nonlinearities, see [7,11,16].

The bifurcation analysis for the stationary NLS provides a great deal of information about the stability of the standing waves. In the one-dimensional case, I have obtained global smooth curves of solutions, bifurcating from the essential spectrum of the linearization. Furthermore, the standing waves can be proved to be orbitally stable along the global curves. In higher dimensions, the analysis is much more involved, local results hold [1-3], but many aspects of the global analysis are still open. I am currently working on these problems.

The inhomogeneous NLS arises in the physics of nonlinear waveguides. I have studied existence and stability of travelling waves in self-focusing waveguides. In [5] I have established bifurcation and stability results for the one-dimensional NLS, and applied them to Kerr media, modelled by a cubic NLS. Other materials of interest have a saturable refractive index and are rather modelled by an asymptotically linear NLS. I proved asymptotic bifurcation of solutions for such equations in [7], and what I believe to be the first rigorous stability result for the asymptotically linear NLS in [16].

#### Boundary value problems

Part of this is joint work with Bryan Rynne, here at Heriot-Watt. We study both semilinear and quasilinear boundary value problems.

Semilinear problems:

In [8] we have characterized the spectrum of linear second order ODE problems with multi-point boundary conditions. Using this characterization, we have proved the existence of nodal solutions of related nonlinear problems. Those are the first results in this area dealing with multi-point boundary conditions together with varying coefficients.

We have also obtained similar results for half-linear problems, see [12], where the classical notion of spectrum is replaced by that of half-eigenvalues, or, in the case of constant coefficients, the Fucik spectrum.

Quasilinear problems:

In [15] we have obtained Landesman-Lazer conditions for the existence of solutions to a p-Laplacian Dirichlet problem on an interval, at resonance with respect to half-eigenvalues. In the case of constant coefficients, this substantially extends previous results for problems at resonance with respect to the Fucik spectrum.

In [14] I have considered a Dirichlet p-Laplacian problem with radial symmetry, in a ball. Using purely analytical arguments, I have obtained very detailed information about smooth branches of positive solutions bifurcating from the first eigenvalue of the p-Laplacian. I am currently working on extensions of these bifurcation results to both unbounded domains and sign-changing solutions.

In collaboration with Ann Derlet, from Universite Toulouse 1 Capitole, we use variational methods to study sign-changing solutions of quasilinear equations in $R^N$. We have already got some new existence results, and are currently working on multiplicity results.

#### Liquid crystals

Another area of interest is the mathematical description of liquid crystals, in the framework of statistical mechanics. A long term goal is to obtain continuum limits describing liquid crystals at the macroscopic level, starting from the Hamiltonian at the microscopic level. This is an ambitious programme, that will involve a variety of methods from statistical mechanics, analysis and nonlinear PDE's.

In collaboration with Sven Bachmann, from UC Davis, we have already obtained some promising partial results, based on an infinite hierarchy of integro-differential equations satisfied by the k-particle distribution functions. We are currenty preparing a paper on the mean-field regime using this method.

### Publications

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.

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

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

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

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

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

7. 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.

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

10. (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.

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

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

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

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

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

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

17. Some bifurcation results for quasilinear Dirichlet boundary value problems, under review.

18. Monotonicity of bifurcating branches for the radial p-Laplacian, under review.

#### In preparation

(with S. Bachmann) Rigorous mean-field limit for anisotropic particles in the continuum.

(with A. Derlet) Nodal solutions of quasilinear problems in $R^N$.

Last updated 16 May 2013.