Divergence-Free Transformation Method

• The paper introduces a positive invariant density that transforms the drift into a zero weak divergence form, overcoming coercivity issues.
• It establishes existence, uniqueness, and a priori bounds for weak solutions, even when the zero‐order coefficient is only in L¹.
• The method employs interpolation between endpoint integrability cases, extending well-posedness to a broad range of Dirichlet problems in elliptic PDEs.

Searching arXiv for the specified paper and closely related work on divergence-free transformations. The divergence-free transformation method, in the sense developed for linear elliptic equations, is a weighted reformulation of a second-order divergence-form Dirichlet problem in which the original drift is replaced by a transformed drift having zero weak divergence with respect to a positive density. In the formulation studied in "Remarks on well-posedness for linear elliptic equations via divergence-free transformation" (Lee, 27 Jan 2026), this device is used to establish existence, uniqueness, and a priori bounds for weak solutions when the zero-order coefficient has very low integrability, including the endpoint case $c \in L^1(U)$, where classical bilinear-form arguments encounter intrinsic limitations.

1. Elliptic model and weak formulation

The method is formulated for the Dirichlet problem on a bounded open set $U \subset \mathbb{R}^d$, $d \ge 3$,

$$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$

Here $A=(a_{ij})$ is measurable, possibly non-symmetric, uniformly elliptic, and bounded in the sense that

$$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$

for almost every $x$ and all $\xi\in\mathbb{R}^d$. The drift satisfies $\mathbf H\in L^p(U,\mathbb R^d)$ for some $p>d$, the zero-order coefficient satisfies $U \subset \mathbb{R}^d$0, and the data $U \subset \mathbb{R}^d$1 lies in the corresponding Lebesgue spaces appearing in the well-posedness statements (Lee, 27 Jan 2026).

The natural energy space is $U \subset \mathbb{R}^d$2, identified on bounded $U \subset \mathbb{R}^d$3 with the seminorm $U \subset \mathbb{R}^d$4 by Poincaré’s inequality. A function $U \subset \mathbb{R}^d$5 is a weak solution if

$U \subset \mathbb{R}^d$6

for all $U \subset \mathbb{R}^d$7, with the usual density extension to $U \subset \mathbb{R}^d$8 when the terms are integrable. The standing structural hypothesis used by the method is denoted $U \subset \mathbb{R}^d$9: $d \ge 3$0 is bounded, $d \ge 3$1, $d \ge 3$2 satisfies the ellipticity and boundedness conditions above, and $d \ge 3$3 for some $d \ge 3$4 (Lee, 27 Jan 2026).

2. Invariant density and zero-divergence drift

The defining step of the method is the construction of a positive weight $d \ge 3$5, described in the paper as an “infinitesimally invariant density.” Under $d \ge 3$6, there exists

$d \ge 3$7

positive on $d \ge 3$8, such that

$d \ge 3$9

Moreover, $$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$0 enjoys a Harnack-type bound on $$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$1 with a constant $$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$2 depending only on $$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$3, and $$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$4 (Lee, 27 Jan 2026).

The transformed drift is then defined by

$$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$5

This yields $$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$6 and

$$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$7

so the weighted drift has zero weak divergence.

The original and transformed equations are equivalent. If $$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$8, $$-\operatorname{div}\big(A(x)\nabla u(x)\big) + \mathbf{H}(x)\cdot \nabla u(x) + c(x)\,u(x) = f(x)\quad \text{in } U,\qquad u=0 \quad \text{on } \partial U.$$9 with $A=(a_{ij})$0, and $A=(a_{ij})$1, then $A=(a_{ij})$2 is a weak solution of

$A=(a_{ij})$3

if and only if it is a weak solution of

$A=(a_{ij})$4

The label “divergence-free” refers precisely to the fact that $A=(a_{ij})$5 has zero weak divergence. In the transformed weak formulation, this allows testing procedures in which the drift term does not introduce a coercivity-destroying sign, and the principal part remains uniformly elliptic because $A=(a_{ij})$6 is uniformly comparable to a positive constant on $A=(a_{ij})$7 through the $A=(a_{ij})$8-bound (Lee, 27 Jan 2026).

3. Well-posedness for low-integrability zero-order terms

The principal results are stated under $A=(a_{ij})$9 and the sign condition $$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$0. At the endpoint $$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$1, the paper proves uniqueness in the homogeneous problem and existence for nonhomogeneous data. More precisely, if

$$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$2

and

$$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$3

then $$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$4. If $$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$5 for some $$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$6, then there exists a unique weak solution

$$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$7

satisfying

$$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$8

The constants depend only on $$\langle A(x)\xi,\xi\rangle \ge \lambda \|\xi\|^2,\qquad \max_{i,j}|a_{ij}(x)|\le M,$$9, and $x$0 (Lee, 27 Jan 2026).

For the higher-integrability endpoint

$x$1

there exists a unique solution $x$2 with

$x$3

Between these endpoints, the method uses the Riesz–Thorin interpolation theorem. With $x$4, $x$5, and

$x$6

interpolation between the mappings

$x$7

produces existence and uniqueness results for intermediate integrability classes of $x$8, covering

$x$9

together with $\xi\in\mathbb{R}^d$0-bounds for $\xi\in\mathbb{R}^d$1. The resulting solution satisfies

$\xi\in\mathbb{R}^d$2

and

$\xi\in\mathbb{R}^d$3

The paper presents this interpolation step as the mechanism extending well-posedness from the endpoint cases to the full interval $\xi\in\mathbb{R}^d$4, $\xi\in\mathbb{R}^d$5 (Lee, 27 Jan 2026).

4. Comparison with classical bilinear-form theory

A central theme of the 2026 paper is the contrast between the divergence-free transformation and the classical bilinear-form method. The standard bilinear form is

$\xi\in\mathbb{R}^d$6

For the zero-order term, the usual estimate is

$\xi\in\mathbb{R}^d$7

This shows that the classical $\xi\in\mathbb{R}^d$8-based boundedness argument naturally requires $\xi\in\mathbb{R}^d$9. When $\mathbf H\in L^p(U,\mathbb R^d)$0, the required Sobolev control of $\mathbf H\in L^p(U,\mathbb R^d)$1 fails, the bilinear form need not be bounded on $\mathbf H\in L^p(U,\mathbb R^d)$2, and the Lax–Milgram framework breaks down. The same section notes that standard treatments of the drift term use $\mathbf H\in L^p(U,\mathbb R^d)$3 for boundedness, and coercivity may require perturbations of the form $\mathbf H\in L^p(U,\mathbb R^d)$4 with a non-explicit $\mathbf H\in L^p(U,\mathbb R^d)$5 depending on truncations of $\mathbf H\in L^p(U,\mathbb R^d)$6 (Lee, 27 Jan 2026).

The weighted reformulation avoids this obstruction. After transformation,

$\mathbf H\in L^p(U,\mathbb R^d)$7

the drift term is handled using

$\mathbf H\in L^p(U,\mathbb R^d)$8

This keeps the drift from producing a coercivity loss in energy testing. The zero-order term remains manageable at the $\mathbf H\in L^p(U,\mathbb R^d)$9-level once $p>d$0, and the uniform comparability of $p>d$1 controls the weighted ellipticity. The paper therefore presents the method not merely as an alternative proof strategy, but as a structural substitute for bilinear-form boundedness in the regime of minimal integrability for $p>d$2 (Lee, 27 Jan 2026).

5. Proof structure and canonical example

The proof architecture has four recurring steps. First, construct $p>d$3 and the transformed drift $p>d$4 from the invariant-density identity. Second, prove the equivalence between the original weak formulation and the weighted one for test functions in $p>d$5, using product rules with $p>d$6 and $p>d$7, density of smooth functions, and truncation. Third, test the transformed problem with $p>d$8 to obtain the principal energy bound

$p>d$9

hence

$U \subset \mathbb{R}^d$00

Sobolev embedding then gives

$U \subset \mathbb{R}^d$01

Fourth, treat the endpoint $U \subset \mathbb{R}^d$02 by duality, truncation, and approximation $U \subset \mathbb{R}^d$03, together with uniform $U \subset \mathbb{R}^d$04-bounds for the approximating problems (Lee, 27 Jan 2026).

The model example

$U \subset \mathbb{R}^d$05

makes the transformation especially transparent. Setting

$U \subset \mathbb{R}^d$06

one gets

$U \subset \mathbb{R}^d$07

hence

$U \subset \mathbb{R}^d$08

The equation

$U \subset \mathbb{R}^d$09

is therefore rewritten as

$U \subset \mathbb{R}^d$10

When $U \subset \mathbb{R}^d$11, $U \subset \mathbb{R}^d$12, and $U \subset \mathbb{R}^d$13, the same method yields

$U \subset \mathbb{R}^d$14

with

$U \subset \mathbb{R}^d$15

In this potential-drift case, the transformed drift vanishes identically, so the weighted structure is reduced to a purely elliptic divergence-form equation with a weighted zero-order term (Lee, 27 Jan 2026).

6. Scope, limitations, and terminological breadth

The framework is stated for $U \subset \mathbb{R}^d$16, bounded domains, Dirichlet boundary conditions, divergence-form operators, and drifts satisfying $U \subset \mathbb{R}^d$17 with $U \subset \mathbb{R}^d$18. The matrix $U \subset \mathbb{R}^d$19 may be non-symmetric; only boundedness and uniform ellipticity are required. The paper explicitly states that the borderline case $U \subset \mathbb{R}^d$20 remains open in this framework, because the Harnack-type control needed for $U \subset \mathbb{R}^d$21 may fail. It also states that Neumann or Robin boundary conditions, non-divergence-form operators, systems, and nonlinear equations are not treated (Lee, 27 Jan 2026).

A common source of ambiguity is terminological. In the elliptic theory of (Lee, 27 Jan 2026), the transformation is the passage to a weighted operator whose drift has zero weak divergence. In other parts of numerical analysis, the same label refers to materially different mechanisms. On curved domains, Piola-transform constructions preserve exact divergence-free structure for Scott–Vogelius, BDM, or related finite element pairs (Neilan et al., 2020, Durst et al., 2024, Li et al., 18 Dec 2025). In nonconforming VEM and HDG settings, divergence-free basis constructions transform mixed saddle systems into SPD velocity-only systems (Kwak et al., 2021, Dean et al., 2023, John et al., 5 Dec 2025). In MHD, constrained-transport, $U \subset \mathbb{R}^d$22-based local/global corrections, DG reconstruction, and meshless modified-gradient projections are used to enforce $U \subset \mathbb{R}^d$23 exactly or to round-off precision (Rossmanith, 2013, Cai et al., 2017, Chandrashekar et al., 2020, Liu et al., 12 Jan 2025, Tu et al., 4 Mar 2026). On surfaces, contravariant Piola maps enforce exactly tangential and pointwise surface-divergence-free velocities (Lederer et al., 2019).

This broader usage suggests that “divergence-free transformation method” is not a single canonical construction across PDE theory. In the specific elliptic setting of (Lee, 27 Jan 2026), however, it has a precise meaning: the introduction of a positive invariant density $U \subset \mathbb{R}^d$24 that converts

$U \subset \mathbb{R}^d$25

into a weighted divergence-form operator with transformed drift $U \subset \mathbb{R}^d$26 of zero weak divergence. Within that setting, its significance is the extension of well-posedness to the low-integrability regime $U \subset \mathbb{R}^d$27, $U \subset \mathbb{R}^d$28, including the endpoint $U \subset \mathbb{R}^d$29, where the classical bilinear-form method does not supply a satisfactory boundedness theory (Lee, 27 Jan 2026).