Notes on linear elliptic equations with L2-gradient perturbations and singular zero-order coefficients
Abstract
In this paper, we study the existence, uniqueness, and quantitative estimates for weak solutions to linear elliptic Dirichlet problems of the form −div(γ∇ u)+⟨ ∇ϕ+H,∇ u⟩+(c+α)u=f in U, u=0 on ∂ U, where U⊂ ℝd is bounded, γ∈[1,∞) is a constant, ϕ∈ H1,2(U)∩ L∞(U), H∈ Lp(U,ℝd) for some p ∈ (d, ∞), and c∈ L1(U) with c≥0. A key feature of this setting is that the drift contains the low-regularity term ∇ϕ, which is only assumed to belong to L2(U,ℝd), while the zero-order coefficient is merely integrable. We prove that, even under these rough assumptions, well-posedness and quantitative energy and L2 estimates remain valid. In addition, by using a previously established interpolation result, we characterize a trade-off between the integrability of the source term f and that of the zero-order coefficient c, and show that well-posedness together with the corresponding quantitative estimates hold under these interpolated assumptions.
Related
- Sharp maximum estimates for linear uniformly elliptic equations with measurable coefficients
- Existence of positive solutions to singular elliptic problems with gradient terms
- Geometric sets of elliptic equations with Dini coefficients
- Bounds of singular sets for elliptic equations in C1, Dini domains with singular potentials
- Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients
BibTeX
@misc{lee2026notes,
title = {{Notes on linear elliptic equations with $L^2$-gradient perturbations and singular zero-order coefficients}},
author = {Haesung Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06227},
eprint = {2609.06227},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06227},
}
RIS
TY - UNPB
TI - Notes on linear elliptic equations with L^2-gradient perturbations and singular zero-order coefficients
AU - Haesung Lee
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06227
UR - https://arxiv.org/abs/2609.06227
AB - In this paper, we study the existence, uniqueness, and quantitative estimates for weak solutions to linear elliptic Dirichlet problems of the form −div(γ∇ u)+⟨ ∇ϕ+H,∇ u⟩+(c+α)u=f in U, u=0 on ∂ U, where U⊂ ℝ^d is bounded, γ∈[1,∞) is a constant, ϕ∈ H^1,2(U)∩ L^∞(U), H∈ L^p(U,ℝ^d) for some p ∈ (d, ∞), and c∈ L^1(U) with c≥0. A key feature of this setting is that the drift contains the low-regularity term ∇ϕ, which is only assumed to belong to L^2(U,ℝ^d), while the zero-order coefficient is merely integrable. We prove that, even under these rough assumptions, well-posedness and quantitative energy and L^2 estimates remain valid. In addition, by using a previously established interpolation result, we characterize a trade-off between the integrability of the source term f and that of the zero-order coefficient c, and show that well-posedness together with the corresponding quantitative estimates hold under these interpolated assumptions.
ER -