FAILURE OF THE HOPF-OLEINIK LEMMA FOR A LINEAR ELLIPTIC PROBLEM WITH SINGULAR CONVECTION OF NON-NEGATIVE DIVERGENCE

In this article we study the existence, uniqueness, and integrability of solutions to the Dirichlet problem -div(M(x)del u)=-div(E(x)u)+f-div(M(x)del mu)=-div(E(x)mu)+f in a bounded domain of R-N with N >= 3. We are particularly interested in singular E with divE >= 0. We start by recalling known existence results when |E|is an element of LN that do not rely on the sign of divE. Then, under the assumption that divE >= 0 distributionally, we extend the existence theory to |E|is an element of L-2. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of E singular at one point as Ax/|x|(2), or towards the boundary as divE similar to dist(x,partial derivative Omega)(-2-alpha). In these cases the singularity of E leads to u vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. partial derivative u/partial derivative n<0, fails in the presence of such singular drift terms E.