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Analysis of linear elliptic equations with general drifts and $L^1$-zero-order terms

Haesung Lee

TL;DR

The paper establishes well-posedness and quantitative regularity for linear elliptic equations with rough drifts and low-integrability zero-order terms. By transforming general drift fields into weakly divergence-free forms, it reduces divergence-form problems to more tractable settings and obtains explicit, computable constants in the estimates, which is valuable for error analysis. It provides existence and uniqueness of bounded weak solutions for divergence-form problems with $L^p$ drifts ($p\in(d,\infty)$) and nonnegative $L^1$ zero-order terms, and extends to strong solutions for non-divergence problems under $VMO$-type coefficient regularity with $L^s$-zero-order terms. In addition, it delivers detailed regularity results in several data regimes across dimensions $d\ge 2$, including contraction-type estimates, and covers both measurable and $VMO$ coefficient scenarios with explicit constants that facilitate rigorous error control in numerical methods and physics-informed modeling.

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 $p \in (d, \infty)$, and non-negative $L^1$-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 $s \in (1, \infty)$, 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.

Analysis of linear elliptic equations with general drifts and $L^1$-zero-order terms

TL;DR

The paper establishes well-posedness and quantitative regularity for linear elliptic equations with rough drifts and low-integrability zero-order terms. By transforming general drift fields into weakly divergence-free forms, it reduces divergence-form problems to more tractable settings and obtains explicit, computable constants in the estimates, which is valuable for error analysis. It provides existence and uniqueness of bounded weak solutions for divergence-form problems with drifts () and nonnegative zero-order terms, and extends to strong solutions for non-divergence problems under -type coefficient regularity with -zero-order terms. In addition, it delivers detailed regularity results in several data regimes across dimensions , including contraction-type estimates, and covers both measurable and coefficient scenarios with explicit constants that facilitate rigorous error control in numerical methods and physics-informed modeling.

Abstract

This paper provides a detailed analysis of the Dirichlet boundary value problem for linear elliptic equations in divergence form with -general drifts, where , and non-negative -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 condition and mild differentiability on the diffusion coefficients and assuming an -zero-order terms with , 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.
Paper Structure (9 sections, 20 theorems, 138 equations)

This paper contains 9 sections, 20 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Divergence type equations with measurable $A$
  4. The case of $d \geq 2$, $c \in L^1(U)$, $f \in L^{q}(U)$ with $q \in (\frac{d}{2}, \infty)$ and $\bold{F}= 0$
  5. The case of $d=2$, $c \in L^1(U)$, $f \in L^{q}(U)$, $\bold{F} \in L^{2q}(U, \mathbb{R}^d)$ with $q \in (1, \infty)$
  6. The case of $d=3$, $c \in L^1(U)$, $f \in L^{q}(U)$, $\bold{F} \in L^{\theta}(U, \mathbb{R}^d)$ with $\theta \in (6, \infty)$, $q \in (\frac{3}{2}, \frac{2\theta}{2+\theta}]$
  7. Divergence type equations with $VMO$ coefficients of $A$
  8. Non-divergence type equations with $VMO$ coefficients of $A$
  9. Auxiliary results

Key Result

Theorem 1.1

Assume that (S) holds, $c \in L^1(U)$ with $c \geq 0$ in $U$ and $f \in L^q(U)$ with $q \in (\frac{d}{2}, \infty)$. Then, the following hold:

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 19 more