Existence and uniqueness of weak solution in W^1,2+ε for elliptic equation with drifts in weak-Lⁿ spaces

2020/11/15 by Hyunwoo Kwon, Kwon, Hyunwoo · 1 citation
Computer Science · Mathematics · #35B65 #35J25 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2011.07524

openalex publication_date 2020/11/15 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/28

Abstract

We consider the following Dirichlet problems for elliptic equations with singular drift b: (a) -div(A ∇ u)+div(ub)=f, (b) -div(A^T ∇ v)-b ⋅ ∇ v =g in Ω, where Ω is a bounded Lipschitz domain in ℝⁿ, n≥ 2. Assuming that b∈ L^n,∞(Ω)ⁿ has non-negative weak divergence in Ω, we establish existence and uniqueness of weak solution in W^1,2+ε₀(Ω) of the problem (b) when A is bounded and uniformly elliptic. As an application, we prove unique solvability of weak solution u in \bigcap_q<2 W^1,q₀(Ω) for the problem (a) for every f∈ \bigcap_q<2 W^-1,q(Ω).

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