Analysis of linear elliptic equations with general drifts and L¹-zero-order terms

2024/08/22 by Haesung Lee, Lee, Haesung · 2 citations
Computer Science · Mathematics · #35B35 #35R05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Primary: 35J25 #Secondary: 35B65 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2408.12295

openalex publication_date 2024/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper provides a detailed analysis of the Dirichlet boundary value problem for linear elliptic equations in divergence form with L^p-general drifts, where p ∈ (d, ∞), and non-negative L¹-zero-order terms. Specifically, by transforming the general drifts into weak divergence-free drifts, we establish the existence and uniqueness of a bounded weak solution, showing that the zero-order term does not influence the quantity of the unique weak solution. Additionally, by imposing the VMO condition and mild differentiability on the diffusion coefficients and assuming an L^s-zero-order terms with s ∈ (1, ∞), we demonstrate the existence and uniqueness of a strong solution for the corresponding non-divergence type equations. An important feature of this paper is that, due to the weak divergence-free property of the drifts in the transformed equations, the constants appearing in our estimates can be explicitly calculated, which is expected to offer significant applications in error analysis.

Cited by

• Notes on L²-estimates in linear elliptic equations with general coefficients
• Optimal control of a two-dimensional elliptic equation with exponential nonlinearity and Dirac measure data

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