research activity - Analysis Group TU Delft

Research statement
My research interests lie in the rigorous mathematical analysis of nonlinear differential equations. I am familiar with a broad variety of techniques in nonlinear analysis and the calculus
of variations, to study partial differential equations (PDEs) and ordinary differential equation
(ODEs). An important part of my work revolves around bifurcation theory, which is a powerful
tool to understand the qualitative features of nonlinear differential equations. For instance, I have
applied bifurcation methods to obtain soliton-like solutions of nonlinear Schr¨odinger equations
(NLS), and to study their stability. This approach to NLS is fairly original and has proved very
successful, in particular to deal with problems where usual symmetries (such as space translation
or scaling invariance) are broken.
Although the core of my work is in nonlinear analysis for PDEs, my earlier education was in
physics, and I am always interested in the relations between physics and mathematical analysis.
For instance, there are important connections between NLS and nonlinear optics, or Bose–Einstein
condensates. Some of my recent interests in soliton1 theory are directly related to applications
in these areas. Another part of my research is concerned with continuum limits in the statistical
physics of liquid crystals, giving rise to challenging nonlinear integral equations. I have also
recently started to study water wave equations, for which bifurcation techniques also play an
important role.
I will now start by describing in more detail my research results. A short research plan follows.
Below, [1], [2], . . . refer to the bibliography ending the research statement, while [P1], [P2], . . .
refer to my list of publications.
Track record. My main research results can be classified in two, closely connected, subareas.
The first is directly concerned with NLS while the second covers various boundary value problems
for nonlinear ODEs and elliptic PDEs.
Nonlinear Schr¨
odinger equations. My work on NLS mostly deals with equations 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 localized standing waves (solitons) for inhomogeneous NLS. A first
series of papers [P1-P5] deal with equations with a power-type nonlinearity. More recently, I have
also studied more general nonlinearities, including asymptotically linear ones, see [P7, P12, P16].
A large part of this work is concerned with bifurcation for the associated stationary equations,
which are semilinear elliptic equations (see equation (2) below). This involves various tools from
nonlinear analysis, topological and variational methods. The strength of my approach is to provide
soliton curves, which are very useful to study the stability of solitons, for instance by means of
the general stability theory developed in [15].
The inhomogeneous NLS arises in various areas of mathematical physics, such as mean-field
limits of large quantum systems (e.g. Bose–Einstein condensates) and nonlinear optics. In fact, the
early days of NLS theory saw very close interactions between the developments of the mathematical
analysis and the physics of nonlinear waveguides. In this context, I have studied existence and
stability of spatial solitons in self-focusing planar waveguides. In [P5] I applied the analytical
results on NLS to Kerr media, modelled by a cubic NLS. Materials with a saturable refractive
index are rather modelled by the asymptotically linear NLS and have been considered in [P16].
1
I use the term ‘soliton’ in a loose sense — essentially meaning spatially localized solutions — outside of its
original scope in integrable systems.
F. Genoud
1
Boundary value problems. While I was at Heriot–Watt University I worked with Bryan Rynne
on diverse nonlinear boundary value problems for ODEs. This was a good opportunity for me
to get a deeper knowledge of various analytical and topological methods, for instance in degree
theory and in the spectral theory of non-selfadjoint operators.
In [P8] and [P13] we obtained nodal (sign-changing) solutions of nonlinear boundary value
problems on an interval, with multi-point boundary conditions, which make the linearized problem
non-selfadjoint. However, under appropriate assumptions, one can essentially recover the spectral
structure of the usual Sturm–Liouville theory. These spectral results, together with bifurcation
theory, allowed us to obtain nodal solutions of the nonlinear problems in [P8, P13], thereby
substantially improving previous results in this area.
In fact, the spectral theory in [P13] considers half-eigenvalues rather than eigenvalues. The
notion of half-eigenvalue occurs naturally when studying half-linear problems, e.g. problems involving operators of the form u 7→ u00 + au+ − bu− , where u± = max{±u, 0}, and a, b ∈ R are
given parameters (or given functions, in a more general setting). These half-linear operators arise
when looking at asymptotics of so-called jumping nonlinearities, modelling systems with a restoring force responding differently with respect to two opposite directions, e.g. in suspension bridges
[19]. In [P15] Rynne and I derived so-called Landesman–Lazer conditions for the existence of solutions to a Dirichlet problem involving the p-Laplacian2 on an interval, with jumping nonlinearities,
under a resonance condition. Our method enabled us to consider much more general equations
than those previously dealt with in the literature.
In [P14] I considered a Dirichlet p-Laplacian problem in arbitrary dimension, with radial symmetry. Using purely analytical arguments, I obtained solution curves bifurcating from the first
eigenvalue of the p-Laplacian. The results of [P14] shed new light on previous global bifurcation
results for quasilinear problems, mostly based on topological arguments.
Research plan. New challenging research directions have recently come to my attention, in
particular in the areas of soliton theory for NLS, global NLS dynamics, and nonlinear water
waves. Beside these, I plan to keep working on nonlinear boundary value problems and nonlinear
elliptic PDEs for their own sake, and also on continuum limits for liquid crystals.
Soliton theory for NLS. Consider an NLS equation of the form
i∂t ψ + ∆ψ + f (x, |ψ|2 )ψ = 0,
ψ = ψ(t, x) : I × RN → C,
(1)
where I ⊂ R is an interval.3 We call (spatial) soliton a standing wave solution of the form
ψ(t, x) = eiωt u(x), where u : RN → R is localized — typically u ∈ H 1 (RN ) and u(x) → 0
exponentially as |x| → ∞. Such a solution exists if and only if
∆u − ωu + f (x, u2 )u = 0.
(2)
The existence of positive solutions of (2) can be obtained under various hypotheses on f , the easiest
case being the pure power nonlinearity, f (x, u2 ) = |u|p−1 for some p > 1. It turns out that the
stability analysis of solitons greatly benefits from having solution curves ω 7→ uω . In fact, under
reasonable assumptions on the coefficient f , the standing wave ψω0 (t, x) = eiω0 t uω0 (x) is stable
if and only if the function ω 7→ kuω kL2 is continuous and strictly increasing in a neighbourhood
of ω = ω0 . A large part of my work on NLS has been devoted to obtaining smooth curves of
solutions of (2), and to using them to study the stability of the corresponding standing waves of
(1), for various coefficients f .
The case where f = f (|x|, u2 ) is positive, non-increasing in |x| and increasing in u > 0 is fairly
well understood and, under appropriate additional assumptions, curves of stable (or unstable)
2The p-Laplacian is defined by ∆ (u) = div(|∇u|p−2 u), i.e. ∆ (u) = (|u0 |p−2 u0 )0 in dimension one.
p
p
3The maximal interval I where the solution exists is called its lifespan ; e.g. I = R for the standing waves.
F. Genoud
2
solutions can be shown to exist, parametrized by the frequency ω as above. By contrast, very
little is known when f does not have these positivity/monotonicity properties. However, in a recent
collaboration [S1] with Boris Malomed (Tel Aviv University) and Rada Weish¨aupl (University of
Vienna), we proved stability of solitons for the one-dimensional NLS with a delta-function potential
and a combination of cubic focusing and quintic defocusing nonlinearities. More precisely, we
consider (1) with V (x) = δ(x), > 0, and f (x, |ψ|2 ) = 2|ψ|2 − |ψ|4 , where δ(x) is the Dirac mass
at x = 0. We obtained explicit formulas for all positive solutions of the stationary equation (2)
and we proved their stability using bifurcation and spectral methods. This problem belongs to
a family of models featuring NLS with multiple-power laws that has started to draw substantial
attention in recent years and presents interesting open problems, with strong connections with
applications in nonlinear waveguides (see the references in [S1]).
Applications of NLS to Bose–Einstein condensates are for instance pointed out in [7], where solitons of NLS with inhomogeneous defocusing nonlinearities are studied, challenging the widespread
belief that solitons only occur in self-focusing structures.4 Stable solutions of (1) are analyzed
numerically in [7] in some special cases, provided that |f (|x|, u2 )| is sufficiently rapidly increasing
in |x|. It is reasonable to expect that similar results may be obtained analytically for general
defocusing nonlinearities, which would be a significant breakthrough.
Global NLS dynamics. There is an abundant literature about the global dynamics of (1) —
see [4] for a fairly up-to-date account. However, the general focus has mainly been restricted to
the pure power case, i.e.
f (x, |ψ|2 ) = |ψ|p−1 , p > 1.
(3)
Two important issues about the asymptotic behaviour of solutions are finite time blow-up and
scattering. A pivotal role in the dynamics is played by the critical Sobolev space of the problem.
The notion of criticality is related to the invariance of (1) & (3) under the scaling
2
ψ(t, x) → ψλ (t, x) = λ p−1 ψ(λ2 t, λx)
(λ > 0).
We say that the Sobolev space H s (RN ) is critical if the homogeneous Sobolev norm k · kH˙ s (RN ) is
invariant by the scaling. When considering inhomogeneous nonlinearities, the scaling symmetry is
in general broken. In [P11] I could however exhibit finite time blow-up solutions for an L2 critical
inhomogeneous NLS. The blow-up analysis in the classic L2 critical case, i.e. for (1) & (3) with
p = 1 + 4/N , has been the subject of intensive study for the past 15 years (see [25] for a survey),
but inhomogeneous equations have only been scarcely investigated (see however [3]). In a recent
collaboration [S2] with Vianney Combet (University Lille 1), we were able to extend important
results from the pure power case (3) to the inhomogeneous equation considered in [P11].
I am also currently starting a project about scattering theory for a class of inhomogeneous NLS
in collaboration with Vladimir Georgiev (University of Pisa) and Jacopo Bellazzini (University
of Sassari). Considering f (x, |ψ|2 ) = V (x)|ψ|p−1 with V decaying at infinity, we have already
observed new phenomena, in striking contrast with the pure power case. A long-term goal of this
research would be to obtain a general classification of solutions such as that presented in [22]. For
inhomogeneous nonlinearities, the absence of symmetries and the ability to spatially modulate the
intensity of the nonlinearity lead to new interesting properties of the solutions.
Nonlinear water waves. I have recently started to work on the nonlinear theory of water
waves, in collaboration with Adrian Constantin, here at the University of Vienna. We consider
periodic gravity water waves with vorticity (i.e. not irrotational), described by the Euler equations.
Assuming that the waves are two-dimensional (i.e. the motion is identical in any direction parallel
to the crest of the wave), the main issues are related to the free boundary problem describing
the wave motion, the unknowns of which are the velocity field, and the water’s free surface. We
4For (1), ‘(de)focusing’ amounts to f (x, |ψ|2 ) being positive (negative) and increasing (decreasing) in |ψ|.
F. Genoud
3
consider both waves of finite depth over a flat bed and deep water waves, for which the fluid
domain extends to infinite depth.
Existence results for waves with vorticity have been obtained by global bifurcation methods,
both in the finite [10] and the infinite [18] depth cases. On the other hand, an explicit solution
to the deep water wave problem was exhibited by Gerstner in 1802,5 which remains the only
known explicit non-trivial solution (i.e. with a non-flat surface) to this day. It turns out that
Gerstner’s wave is not irrotational, and has therefore been regarded as a mere curiosity for a long
time. The importance of flows with vorticity has since been widely acknowledged, in particular
in the modelling of wave-current interactions — see the discussion at the end of Section 2.3 in [8]
— and there is some indication that Gerstner’s wave can actually be a realistic model of some
wave motions with strong underlying currents. A detailed account of Gerstner’s solution can be
found in [8]. Now the bifurcation analysis in [18] is fairly involved and it remains unclear whether
Gerstner’s explicit solution is captured by this approach. This is something we wish to investigate.
In the finite depth setting, we are studying the analytic continuation of the velocity field through
the free boundary. This has been carried out in deep water in [24], but the finite depth problem remains open. It is important for engineering applications, in particular in connection with
the problem of reconstructing the free surface from pressure measurements at the bottom. The
methods involved lie mostly within complex analysis and two-dimensional harmonic analysis.
The paper [P20] in collaboration with David Henry, from University College Cork, proves the
instability of an equatorially trapped solution of the geophysical water waves equations. These
equations model oceanic (large scale) nonlinear water waves, for which the Coriolis effect cannot
be neglected. Henry [17] previously found an explicit equatorially trapped (i.e. localized about
the equator) solution in the presence of an underlying current, and we show that this solution is
unstable. Oceanic waves are expected to be generically unstable, and there are several challenging
open problems in this area which we are willing to address, in collaboration with David Henry
and Adrian Constantin.
Continuum limits for liquid crystals. Liquid crystals are an intermediate state of matter,
between the crystalline solid and the isotropic liquid. They are made of anisotropic (e.g. rod-like)
molecules and are characterized [1, 12] by the existence of a phase transition between the isotropic
state, where both the position and the orientation of the molecules are randomly distributed, and
the so-called mesophase, where the molecules tend to align themselves along particular directions,
while the distribution of their positions remains random. In other words, the state is translationinvariant but the rotational invariance is broken.
As in the case of isotropic molecules, the intermolecular interaction is approximated by a twobody potential, with steric repulsion depending on the orientation of the molecules, and a long
range attractive potential. Mean-field theories, showing the existence of a mesophase, have been
developed by physicists since the 1940s, the two main contributions to this day being [23] (dealing
with steric repulsion only) and [21] (considering attractive forces only). A summary of these
models can be found in [13]. It is a challenging problem to obtain rigorous continuum limits
describing liquid crystals at the macroscopic level, starting directly from the Hamiltonian at the
microscopic level. Some recent efforts in this direction [2] rather assume the existence of the
mean-field model and try to connect it with a macroscopic theory.
In collaboration with Sven Bachmann, from the Ludwig Maximilian University of Munich, we
are currently working on the mean-field regime, for systems in the continuum. We have already
got some promising partial results in this direction, based on an infinite hierarchy of integrodifferential equations satisfied by the k-particle distribution functions, that we borrowed from the
5and studied rigorously in [9] and [16]
F. Genoud
4
‘molecular field theory’ in [20]. Similar arguments have proved successful in the study of bosonic
systems (see e.g. the review paper [26]) and we believe that our approach will provide a rigorous
derivation of the mean-field limit, for a large class of interaction potentials. This would put the
classical theories [21, 23] on firm mathematical grounds, and would bridge the gap between [2]
and the microscopic description.
Nonlinear boundary value problems. In a recent paper [P19] Ann Derlet and I used variational methods to prove the existence of sign-changing solutions of nonlinear problems involving
the p-Laplacian in RN . This initiative was motivated by a series of contributions about nodal solutions of nonlinear elliptic problems (see e.g. [5, 6]) and is based on the nodal Nehari set approach
used in Derlet’s PhD dissertation [11, Section 4.3]. This method has proved quite successful in
our work and we now plan to apply Lyusternik-Schnirelmann methods to study the multiplicity
of solutions. My PhD student, Elek Csobo, who will start at TU Delft in September 2015, will
work on this problem.
Having a solid expertise of bifurcation techniques for semilinear elliptic equations, I also wish to
investigate further bifurcation problems for quasilinear equations, e.g. involving the p-Laplacian.
For instance, it should be possible to handle sign-changing solutions with the analytical method
I developed in [P14], although this seems more difficult and would require structurally different
assumptions than in [P14] (where all solutions necessarily are of one sign). I also plan to consider
problems in more general (non-radial), possibly unbounded domains. So far, most bifurcation
results for quasilinear equations were obtained by topological methods, and stronger conclusions
for these problems would follow from the analytical approach.
References
[1] J.M. Ball, D´efauts dans les cristaux et dans les cristaux liquides, Cours d’´ecole doctorale, Paris
(2009), available at
http://people.maths.ox.ac.uk/ball/Teaching/parisox.pdf
[2] J.M. Ball, A. Majumdar, Nematic liquid crystals: from Maier-Saupe to a continuum theory, Mol.
Cryst. Liq. Cryst. 525 (2010), 1–11.
[3] V. Banica, R. Carles, T. Duyckaerts, Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation, Comm. P.D.E. 36 (2011), no. 3, 487–531.
[4] The Nonlinear Dispersive PDEs web page, maintained by J. Colliander, M. Keel, G. Staffilani, H.
Takaoka, T. Tao, available at
http://www.math.ucla.edu/~tao/Dispersive/
[5] T. Bartsch, Z. Liu, T. Weth, Nodal solutions of a p-Laplacian equation, Proc. London Math. Soc.
91 (2005), no. 1, 129–152.
[6] T. Bartsch, Z.-Q. Wang, M. Willem, The Dirichlet problem for superlinear elliptic equations, in
Stationary Partial Differential Equations, Vol. II, 1–55, Handb. Differ. Equ., Elsevier/North-Holland,
Amsterdam, 2005.
[7] O.V. Borovkova, Y.V. Kartashov, L. Torner, B.A. Malomed, Bright solitons from defocusing
nonlinearities, Phys. Rev. E 84, 035602(R) (2011).
[8] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and
Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.
[9] A. Constantin, On the deep water wave motion, J. Phys. A 34 (2001), no. 7, 1405–1417.
[10] A. Constantin, W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl.
Math. 57 (2004), no. 4, 481–527.
[11] A. Derlet, Eigenvalues of the p-Laplacian in Population Dynamics and Nodal Solutions of a Prescribed Mean Curvature Problem, PhD thesis, Universit´e Libre de Bruxelles, 2011.
[12] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, Oxford University Press, 1993.
[13] F. Genoud, Statistical theories of liquid crystals, Onsager, Maier-Saupe and beyond, OxPDE
Lunchtime Seminar (2010), available at
F. Genoud
5
http://people.maths.ox.ac.uk/zarnescu/OxPDE-F.Genoud.pdf
[14] B.V. Gisin, R. Driben, B.A. Malomed, Bistable guided solitons in the cubic-quintic medium, J.
Opt. B: Quantum Semiclass. Opt. 6 (2004) S259–S264.
[15] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal. 74 (1987), 160–197.
[16] D. Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008), suppl. 2, 87–95.
[17] D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur.
J. Mech. B Fluids 38 (2013), 18–21.
[18] V. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal. 37 (2006),
no. 5, 1482–1521.
[19] A.C. Lazer, P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new
connections with nonlinear analysis, SIAM Rev. 32 (1990), no. 4, 537–578.
[20] G.R. Luckhurst, G.W. Gray Eds., The Molecular Physics of Liquid Crystals, Academic Press
(1979).
[21] W. Maier, A. Saupe, Eine einfache molekular statistische theorie der nematischen kristallinfl¨
ussigen
phase. Teil I, Z. Naturforsch. 14 a (1959), 882–889.
[22] K. Nakanishi, W. Schlag, Global dynamics above the ground state energy for the cubic NLS
equation in 3D, Calc. Var. Partial Differential Equations 44 (2012), no. 1-2, 1–45.
[23] L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. N.Y. Acad. Sci. 51
(1949), 627–659.
[24] P.I. Plotnikov, J.F. Toland, The Fourier coefficients of Stokes’ waves, in Nonlinear Problems in
Mathematical Physics and Related Topics I, 303–315, Int. Math. Ser. 1, Kluwer/Plenum, New York,
2002.
¨l, Stability and blow up for the nonlinear Schr¨odinger equation, Lecture notes for the Clay
[25] P. Raphae
summer school on evolution equations, ETH, Zurich (2008), available at
http://claymathorg.ksoqpeue.ru/programs/summer_school/2008/raphael.pdf
[26] B. Schlein, Effective evolution equations in quantum physics, available at
http://arxiv.org/abs/1111.6995
F. Genoud
6