Uniform null controllability of the heat equation with rapidly oscillating periodic density

We consider the heat equation with fast oscillating periodic density, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform controllability result is the first in the multidimensional setting for the heat equation with oscillating density. Finally, we prove that the sequence of null controls converges to the optimal null control of the limit equation when the period tends to zero. To cite this article: L. Tebou, C. R. Acad. Sci. Paris, Ser. I 347 (2009).