Dirichlet problems for second order linear elliptic equations with L¹-data

2022/09/09 by Kim, Hyunseok, Oh, Jisu · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.04414

Abstract

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain Ω in ℝⁿ, n ≥ 2: -∑_i,j=1ⁿ a^ijD_ij u + b ⋅ D u + cu = f in Ω and u=0 on ∂ Ω and - \rm div ( A D u ) + \rm div(ub) + cu = \rm div F in Ω and u=0 on ∂ Ω , where A=[a^ij] is symmetric, uniformly elliptic, and of vanishing mean oscillation (VMO). The main purposes of this paper is to study unique solvability for both problems with L¹-data. We prove that if Ω is of class C¹, \rm div A + b∈ L^n,1(Ω;ℝⁿ), c∈ L^(n)/(2),1(Ω) ∩ L^s(Ω) for some 1

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• Optimal regularity results in Sobolev-Lorentz spaces for linear elliptic equations with L¹- or measure data

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