A Class of Second-Order Linear Elliptic Equations with Drift: Renormalized Solutions, Uniqueness and Homogenization

Abstract

In this paper a class of N-dimensional second-order linear elliptic equations with a drift is studied. When the drift belongs to L2 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 L2 to a limit drift in Lq, 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 [3] by relaxing the equi-integrability of the drifts in L2.