A class of second-order linear elliptic equations with drift: renormalized solutions, uniqueness and homogenization
Résumé
In this paper a class of $N$-dimensional second-order linear elliptic equations with a drift is studied.
When the drift belongs to $L^2$ the existence of a renormalized solution is proved. There is also uniqueness in the class of the renormalized solutions modulo~$L^\infty$, but the uniqueness is violated when the drift equation is regarded in the distributions sense. Then, considering a sequence of oscillating drifts which converges weakly in $L^2$ to a limit drift in $L^q$, with $q>N$, the homogenization process makes appear an extra zero-order term involving a non-negative Radon measure which does not load the zero capacity sets. This extends the homogenization result obtained in \cite{BrGe} by relaxing the equi-integrability of the drifts in $L^2$.