On some properties of weak solutions to elliptic equations with divergence-free drifts
Nikolai Dmitrievich Filonov, Timofey Shilkin · Contemporary mathematics - American Mathematical Society · 2018
We discuss the local properties of weak solutions to the equation − Δ u + b ⋅ ∇ u = 0 -\Delta u + b\cdot abla u=0 . The Dirichlet boundary value problem for this equation is studied as well. The corresponding theory is well-known in the case b ∈ L n b\in L_n , where n n is the dimension of the space. Our main interest is focused on the case b ∈ L 2 b\in L_2 . In this case the structure assumption div b = 0 \operatorname {div} b=0 turns out to be crucial.