Research interests: harmonic analysis and partial differential equations
Harmonic analysis: to study the boundedness in various function spaces for the linear/multilinear operators related to the Fourier restriction problem and PDEs, function spaces, space-time estimates for dispersive equations.
Nonlinear partial differential equation: to study the nonlinear evolutionary PDEs arising in mathematical physics such as nonlinear dispersive equations, Navier-Stokes equation. Main concerns are the low-regularity local/global well-posedness theory, long-time/blow-up behavior of the solution, stability of the equation, playing tools from many other areas including harmonic analysis, functional analysis, ODE and probability, etc.
Harmonic analysis and PDEs
The questions I am interested in mix geometry with partial differential equations.
Geometric evolution equations: here we deform geometric objects in a smooth way to help us understand their shape. This is a very powerful recent technique in geometric analysis, and has been crucial in solving many open geometric questions, including the Poincare conjecture (proved by Perelman).
Eigenvalue estimates: Every geometric object has a set of numbers attached to it, called the spectrum. These are like the resonant frequencies of a drum. I am interested in how the shape of the object affects the spectrum.
Capillary surfaces and the calculus of variations: Consider a meniscus--- the interface between the water in a glass and the air above it. The special shape of the meniscus is due to an interplay between the energy used in the water/glass interface and the air/water interface, and the gravitational energy--- the shape we see will minimise the sum of these energies. It will also dependent on the shape of the glass and the volume of the water. This is a typical problem in the calculus of variations, where minimizing a physical quantity can give rise to interesting geometric shapes.
Yann Bernard’s research lies at the intersection of analysis, geometry, and the study of partial differential equations. His main area of focus deals with higher-order elliptic problems appearing in differential geometry, in particular pertaining to the Willmore energy as well as to minimal surfaces and related concepts.