Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains

2018/11/30 by Kim, Hyunseok, Kwon, Hyunwoo · 1 citation
#35J15 #35J25 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.12619

Abstract

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: -\triangle u +div(ub) =f and -\triangle v -b ⋅ ∇ v =g in a bounded Lipschitz domain Ω in ℝⁿ (n≥ 3), where b:Ω→ ℝⁿ is a given vector field. Under the assumption that b ∈ Lⁿ(Ω)ⁿ, we first establish existence and uniqueness of solutions in L_α^p(Ω) for the Dirichlet and Neumann problems. Here L_α^p(Ω) denotes the Sobolev space (or Bessel potential space) with the pair (α,p) satisfying certain conditions. These results extend the classical works of Jerison-Kenig [17] and Fabes-Mendez-Mitrea [12] for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in L²(∂Ω). Our results for the Dirichlet problems hold even for the case n=2.

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• Singular-degenerate parabolic systems with the conormal boundary condition on the upper half space

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