Publication: Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence
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2022-11-21
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In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem −div(M(x)∇u)=−div(E(x)u)+f in a bounded domain of RN with N≥3. We are particularly interested in singular E with divE≥0. We start by recalling known existence results when |E|∈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|∈L2. 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∼dist(x,∂Ω)−2−α. In these cases the singularity of E leads to u vanishing to a certain order. In particular, this shows that the strong maximum principle fails in the presence of such singular drift terms E.
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