We investigate the problem of optimizing the shape and

location of actuators or sensors for evolution systems

driven by a partial differential equation, like for

instance a wave equation, a Schrödinger equation, or a

parabolic system, on an arbitrary domain Omega, in

arbitrary dimension, with boundary conditions if there

is a boundary, which can be of Dirichlet, Neumann,

mixed or Robin. This kind of problem is frequently

encountered in applications where one aims, for

instance, at maximizing the quality of reconstruction

of the solution, using only a partial observation. From

the mathematical point of view, using probabilistic

considerations we model this problem as the problem of

maximizing what we call a randomized observability

constant, over all possible subdomains of Omega having

a prescribed measure. The spectral analysis of this

problem reveals intimate connections with the theory of

quantum chaos. More precisely, if the domain Omega

satisfies some quantum ergodic assumptions then we

provide a solution to this problem.



These works are in collaboration with Emmanuel Trélat

(Univ. Paris 6) and Enrique Zuazua (BCAM Bilbao, Spain).

In the presence of a translation space-like Killing field

the 3 + 1 vacuum Einstein equations reduce to the 2 + 1

Einstein equations with a scalar field. We work in

generalised wave coordinates. In this gauge Einstein

equations can be written as a system of quaslinear

quadratic wave equations. The main difficulty is due to

the weak decay of free solutions to the wave equation in 2

dimensions. To prove long time existence of solutions, we

have to rely on the particular structure of Einstein

equations in wave coordinates. We also have to carefully

choose the behaviour of our metric in the exterior region

to enforce convergence to Minkowski space-time at

time-like infinity.

The generation of functional interfaces such as superconducting and ferroelectric twin boundaries requires new ways to nucleate as many interfaces as possible in bulk materials and thin films. Materials with high densities of twin boundaries are often ferroelastics and martensites. Here we show that the nucleation and propagation of twin boundaries depend sensitively on temperature and system size. The geometrical mechanisms for the evolution of the ferroelastic microstructure under strain deformation remain similar in all thermal regimes, whereas their thermodynamic behavior differs dramatically: on heating, from power-law statistics via the Kohlrausch law to a Vogel-Fulcher law.We find that the complexity of the pattern can be well characterized by the number of junctions between twin boundaries. Materials with soft bulk moduli have much higher junction densities than those with hard bulk moduli. Soft materials also show an increase in the junction density with diminishing sample size. The change of the complexity and the number density of twin boundaries represents an important step forward in the development of ‘domain boundary engineering’, where the functionality of the materials is directly linked to the domain pattern.

We shall discuss the problem of the 'trend to equilibrium' for a 

degenerate kinetic linear Fokker-Planck equation. The linear equation is 

assumed to be degenerate on a subregion of non-zero Lebesgue measure in the 

physical space (i.e., the equation is just a transport equation with a 

Hamiltonian structure in the subregion). We shall give necessary and 

sufficient geometric condition on the region of degeneracy which guarantees 

the exponential decay of the semigroup generated by the degenerate kinetic 

equation towards a global Maxwellian equilibrium in a weighted Hilbert 

space. The approach is strongly influenced by C. Villani's strategy of 

'Hypocoercivity' from Kinetic theory and the 'Bardos-Lebeau-Rauch' 

geometric condition from Control theory. This is a joint work with Frederic 

Herau and Clement Mouhot.