Research interests

The context: calculus of variations and partial differential equations

My research aims to analyse mathematically problems in the calculus of variations and partial differential equations.

Ordinary differential equations establish a relationship between a function and its derivatives. The solution to a differential equation is a function rather than a number. Ordinary differential equations have been ubiquitous in physical models starting with Isaac Newton’s theory of gravitation, which prescribes a relationship between the position as a function of time and the acceleration, which amounts mathematically to a second order derivative of the position. Other ordinary differential equations models include Pierre-François Verhulst's logistic equation for population and electric circuit equations.

Partial differential equations deal with functions of several variables. For instance, the heat equation models the temperature as a function of an instant in time and a position in space through a relationship between the time derivative and the second-order derivative in the spatial variable. Partial differential equations appear in most models involving some time and spatial variables. Examples include Fourier’s heat equation, the Navier-Stokes equations in fluid mechanics, which is used for weather forecasting, climate modelling and the design of aerodynamical efficient structure, Maxwell’s equations of electromagnetism, which are used to build a motor and wireless transmissions, the Schrödinger equation in quantum physics, the Einstein field equations in general relativity, reaction-diffusion equations in the Turing instability mechanism in morphogenesis and the Black-Scholes equation for pricing of derivatives in finance.

Calculus of variations involves finding functions that minimise or maximise a given quantity. For example, the isoperimetric problem asks which curve of a given length will enclose the largest maximal area, with the obvious candidate for a solution being a circle. The brachistochrone problem consists in finding a curve along which a bead would travel from a given point to another in shortest time–such a curve would be dangerously interesting on children playgrounds. Other practical problems include minimising the air resistance of a shape, maximising the heat insulation with a given amount of material. From a theoretical perspective, many physical differential equation models can be reformulated as a minimisation problem known as least action principles. For instance, Fermat’s principle reformulates Snell’s law for the refraction as light travelling along the path that takes the least time.

The calculus of variations and differential equations are also an important tool for answering geometrical questions. Geodesics, which are defined as shortest paths on curved surfaces, and their higher-dimensional generalisations known as harmonic maps, are solutions to problems in the calculus of variation problems and differential equation. Perelman solution’s to the Poincaré conjecture relied on the Ricci flow which is a partial differential equation that describes the diffusion by curvature on a manifold.

As well as their more serious applications to mathematics, science and technology, calculus of variations and differential equations are used in computer graphics for video games and films.

Mathematical questions

Unlike the algebraic equations for which we learned formulas at school, most problems in calculus of variations and partial differential equations have no formulas for solution–this can be proved as a theorem in differential Galois theory. Algorithms that can compute approximate solutions. The mathematical analysis of these problems studies the existence of solutions, their uniqueness, their dependence on the parameters of the problem and their qualitative properties such as symmetry and convexity.

Although mathematical analysis seems to be concerned of infinite precision, it proceeds essentially through controlling the errors of approximation. This requires to develop inequalities for functions involving integrals and derivatives such as Sobolev inequalities.

Problems in the calculus of variations problems and partial differential equations can be viewed as counterparts of one-variable optimisation and equation solving on infinite-dimensional spaces of functions. Similarly to how the set of rational numbers misses import numbers such as \(\sqrt{2}\) and \(\pi\), classical spaces of differentiable functions are ill-suited for many problems. Sobolev spaces and their generalisations provide well-behaved spaces at the price of including wilder functions. Thus, it is necessary to understand the structure of these function spaces and the properties of the individual functions functions that they contain.

Some contributions

Harmonic and Sobolev mappings

According to the classical heat equation, the equilibrium configuration of temperature is a harmonic function. If we want instead to model the magnetisation by unit vector with the same approach, we will end up harmonic mappings into the sphere. In general, harmonic mappings into surfaces or manifolds are delicate objects, because of the presence of a nonlinear constraint.

My contributions to the theory of Sobolev spaces of mappings include the strong approximation of smooth maps in higher-order Sobolev spaces (with P. Bousquet and A. Ponce), the detection of topological singularities with Fuglede detectors (also with P. Bousquet and A. Ponce), the lifting of fractional Sobolev mappings with a compact covering space (with P. Mironescu), the extension of traces of Sobolev mappings and the analytical obstruction to the weak approximation of Sobolev mappings (with A. Detaille).

Some superconductivity models and computer graphics problems lead to some harmonic map problem that do not have a solution in the natural Sobolev space. However, one can still define families of relaxed problem whose solution should converge to the desired harmonic map. This approach was first developped by F. Bethuel, H. Brezis and F. Hélein for the Ginzburg-Landau relaxation. I have described the asymptotics for a general compact target on two-dimensional domains for Ginzburg-Landau relaxations (with A. Monteil and R. Rodiac) and for \(p\)-harmonic relaxations (with B. Van Vaerenbergh) as well as for targets with a finite fundamental group for \(p\)-harmonic relaxations (with B. Bulanyi and B. Van Vaerenbergh).

Fluid vortices

Some two-dimensional fluid models such as the Euler equations and the lake equations have point vortices as generalised solutions. Due to their singularities, they cannot be considered as classical equations. One way to justify these solutions is to show that they are limit of classical solutions. I have achieved this in the stationary case of the two-dimensional Euler equation (with D. Smets) and of the lake equations (with S. De Valeriola), and for the evolution problem of the lake equations (with J. Dekeyser).

Semi-linear elliptic problems

I have been working on several semilinear elliptic problems, including the nonlinear Schrödinger equation (with D. Bonheure and V. Moroz) and the magnetic nonlinear Schrödinger equation (with D. Bonheure, M. Nys, S. Fournais, L. Le Treust, N. Raymond and J. Di Cosmo). I have also worked on the Choquard equation also known as the Hartree equation or Schrödinger equation, with V. Moroz and other collaborators. This equation features a nonlocal nonlinearity, which decouples the superquadraticity of the nonlinearity from the superlinearity of the equation, which creates some Brezis-Nirenberg type problem at infinity and which modifies the behaviour at infinity observed in the local nonlinear Schrödinger equation. Our work has triggered a tremendous level of subsequent activity.

Bourgain-Brezis estimates

At the beginning of the 2000s, J. Bourgain and H. Brezis have obtained several new, unexpected endpoint Sobolev estimates. I have provided proofs of some weaker versions of their inequalities that are significantly shorter. I have also introduced a class of cancelling operators that characterises the differential operators with Sobolev estimates at the endpoint \(p=1\).

The Brezis-Van Schaftingen-Yung formula

In 2021, H. Brezis, P.-L. Yung and myself have obtained a new characterisation of Sobolev spaces in terms of weak-type quantities. Our estimates and the associated function spaces have developed further in many directions including interpolation theory of function spaces and harmonic analysis.