Séminaire Analyse non linéaire et EDP

I will present recent joint work with Charles Smart, in which we prove a Harnack inequality for fully nonlinear elliptic equations in dimension $d$ with possibly unbounded ellipticity-- provided the ellipticity belongs to $L^d$. This yields a stochastic homogenization result for such equations, with applications to random diffusions in (strictly elliptic) random environments.

We prove a blow-up criterion in terms of the upper bound ofthe density for the strong solution to the 3-D compressible Navier-Stokesequations. The initial vacuum is allowed. The main ingredient of theproof is a priori estimate for an important quantity under the assumptionthat the density is upper bounded, whose divergence can be viewed asthe effective viscous flux.

Some twenty years ago Berenger introduced theremarkable method of perfectly matchedlayers for truncating to a rectangle, the computation ofsolutions of Maxwell's equations in 1+2 and 1+3 dimensionalspace time. Only recently have some of the fundamentalquestions concerning this method been resolved.For example the stability of the original methodand its perfection. We discuss the analysis of thisand related methods that are constructed to performbetter in variable coefficient settings where the perfectionof Berenger no longer holds. Research donewith Laurence Haplern, Sabrina Petit, and LudovicMetivier.