Well-posedness of linear elliptic equations with $L^d$-drifts under divergence-type conditions

Abstract: We establish the well-posedness of linear elliptic equations with critical-order drifts in $L^d$ and positive zero-order coefficients in $L^1$ or $L^{\frac{2d}{d+2}}$, where classical methods are often too restrictive. Our approach relies on a divergence-free transformation and a structural condition on the drift vector field, which admits a decomposition into a regular component and another whose weak divergence belongs to $L^{\tilde{q}}$ for some $\tilde{q} > \frac{d}{2}$. This condition is essential for constructing a suitable weight function $ρ$ via the weak maximum principle and the Harnack inequality. Within this framework, we prove the existence and uniqueness of weak solutions, significantly relaxing the regularity assumptions on the zero-order coefficients in $L^{\frac{d}{2}}$.

• The paper establishes existence and uniqueness of weak solutions for linear elliptic equations with minimal regularity assumptions on drift and zero-order terms.
• It employs a divergence-type decomposition and a weight transformation to recast the problem into a divergence-free formulation, enabling sharp energy and maximum estimates.
• Well-posedness is achieved even with drifts in L^d and zero-order terms in L^1 or L^(2d/(d+2)), challenging classical regularity barriers.

Well-Posedness of Linear Elliptic Equations with $L^d$-Drifts under Divergence-Type Conditions

Introduction

The paper addresses the existence and uniqueness of weak solutions for second-order linear elliptic equations in divergence form with drift vector fields in the critical space $L^d$ in dimension $d\geq 3$ and with minimal regularity assumptions on the zero-order coefficient $c$. The operator under consideration is

$$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$

where $A$ is a uniformly strictly elliptic bounded matrix-valued function, $\mathbf{H} \in L^d(U, \mathbb{R}^d)$, and $c$ is a non-negative function in $L^1$ or $L^{\frac{2d}{d+2}}$.

The work relaxes the regularity on both the drift and the zero-order term compared to the classical literature, where well-posedness typically requires $L^d$0 with $L^d$1 and $L^d$2. The principal technical advance is showing that under a natural divergence-type decomposition of $L^d$3, solutions can be constructed with significantly weaker assumptions on $L^d$4 and at the critical regularity for the drift.

Methodology and Analytical Techniques

The approach relies crucially on several structural and analytical ingredients:

1. Divergence-Type Decomposition: The drift $L^d$5 is decomposed as $L^d$6 where $L^d$7 ($L^d$8) and $L^d$9 with distributional divergence in $d\geq 3$0 for some $d\geq 3$1. This divergence-type condition is critical to enabling the subsequent weight construction.
2. Divergence-Free Transformation: Utilizing the decomposition, a strictly positive weight function $d\geq 3$2 is constructed, solving an adjoint equation with the modified drift. The solution $d\geq 3$3 is then used to recast the original elliptic equation to one where the new drift $d\geq 3$4 admits a divergence-free weighted form, allowing the use of tools from the theory of symmetric elliptic operators.
3. Sharp Function Spaces for $d\geq 3$5: The work proves that existence and uniqueness hold already under $d\geq 3$6 (for $d\geq 3$7, $d\geq 3$8) and even $d\geq 3$9 (for data in the same space). This challenges the classical $c$0 regularity barrier for the zero-order coefficient.
4. Compactness, Maximum Principle, and Harnack Inequality: Compactness and Sobolev embedding arguments, the weak maximum principle, and elliptic regularity, especially the Harnack inequality, are used to establish positivity, uniqueness, and estimate bounds for the weight $c$1 and the solution $c$2.
5. Fredholm Alternative: Non-coercive structures introduced by the low regularity of $c$3 and the drift are handled by suitable compactness and operator-theoretic constructs.

Main Theorems and Results

The core theorem is a well-posedness result for the equation above under the specified structure and regularity assumptions (see Theorem 1 in the paper).

Key assertions:

• Existence and Uniqueness: There exists a unique weak solution $c$4 for $c$5, $c$6, $c$7, with a-priori energy estimates

$c$8

• Contraction Property: If $c$9, then a contraction (maximum) estimate holds in all $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$0 spaces.
• Case $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$1: Well-posedness persists and estimates as above hold for right sides in the same space, again with uniqueness.

Notable features:

• No $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$2 or symmetry/higher regularity assumed for $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$3.
• The drift $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$4 is sharp (critical) for the techniques, and the additional divergence-type structure is essential and, as demonstrated by explicit examples in the text, necessary for local boundedness.
• The zero-order coefficient regularity is shown to be strictly weaker than the classical paradigm.

Analytical and Structural Implications

• Criticality and Sharpness: The work systematically explores the boundary of well-posedness as allowed by Sobolev embedding, compactness, and the structure of the equation. The requirement that $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$5 belongs to $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$6, $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$7, is shown to be necessary by explicit counterexamples.
• Generalization of Known Results: Classical results, such as those of Trudinger, Stampacchia, and Gilbarg-Trudinger, are strictly extended here, especially in the setting where $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$8 is in $$- \operatorname{div}(A \nabla u) + \langle \mathbf{H}, \nabla u \rangle + cu = f \quad \text{in } U, \qquad u=0 \text{ on } \partial U,$$9.
• Method Transferability: The strategy via reduction to weights and divergence-free forms relates to the invariant measure problem for Fokker-Planck equations and potential theory, suggesting connections to stochastic processes and Dirichlet forms.
• Limitations and Open Questions: The technique depends critically on the possibility of a divergence decomposition for $A$0 and control over the elliptic regularity of the associated weighted problems. Whether further regularity weakening of the drift or the extension to non-divergence form equations is possible remains open.

Numerical/Quantitative Highlights

• Explicit energy and $A$1 bounds are established with constants depending only on the structural parameters ($A$2, $A$3, $A$4), the domain, and Lebesgue exponents. For contraction, the dependence on the lower bound of $A$5 is explicit.
• Counterexamples exhibit that, in the absence of the divergence-type structure, solutions may fail to be locally bounded even though $A$6.

Outlook and Directions for Future Work

• Relaxation of Drift Regularity: The extension to cases with $A$7 or with more general divergence forms requires new ideas, possibly using additional probabilistic or variational arguments.
• Non-divergence Form Operators: As conjectured, the weakening of $A$8-regularity for non-divergence cases remains largely unresolved outside $A$9 settings, and advances here would have significant implications for regularity theory.
• Quantitative Estimates and Stability: Stability of the constructed constants under mollification and their dependence on structural norms are not completely explicit; addressing this would enhance the applicability of the theory to numerical analysis and homogenization.
• Stochastic Analysis Implications: The techniques and results have direct connections to the study of invariant measures and Fokker-Planck equations, which may have implications for ergodic theory and SDEs with irregular drifts.

Conclusion

The work provides a comprehensive analysis of linear divergence-form elliptic equations with drift at the optimal Lebesgue space scaling and with minimal assumptions on the lower-order term. By leveraging a divergence-type decomposition, tailored a-priori estimates, and regularity theory, it establishes strong well-posedness and stability results beyond the classical framework. The necessity of the structural assumption on the drift is made precise, and the findings open the pathway to further study of critical regularity phenomena and their consequences in both deterministic and probabilistic PDE theory.