The Dirichlet problem for double divergence form elliptic equations with measures as boundary conditions

Abstract: We introduce and study the Dirichlet problem for double divergence form elliptic equations with coefficients of low regularity and boundary conditions given by general Borel measures. Under broad assumptions we establish the solvability of this problem. It is also shown that a solution to a double divergence form equation on a domain serves as a solution to the Dirichlet problem on inner subdomains. The obtained results are applied to the study of properties of solutions to stationary Fokker--Planck--Kolmogorov equations.

• The paper establishes well-posedness and uniqueness of weak solutions for double divergence elliptic equations with measure boundary data under minimal regularity assumptions.
• It introduces a novel smooth approximation strategy that uses mollification to ensure convergence of solutions while preserving ellipticity and Dini-type regularity.
• The study corrects previous inconsistencies by validating Harnack inequalities, maximum principles, and diffusion process applications through rigorous boundary trace analysis.

The Dirichlet Problem for Double Divergence Form Elliptic Equations with Measure Boundary Data

Problem Statement and Functional Analytic Framework

This paper introduces and systematically analyzes the Dirichlet problem for second-order elliptic PDEs in double divergence form:

$${\rm div}^2(\varrho A) - {\rm div}(\varrho b) = {\rm div}^2 G - {\rm div} h,$$

where $A$, $G$ are symmetric matrix-valued functions, $b$, $h$ are vector fields with low regularity, and all coefficients are merely Borel or weakly regular. The boundary condition is prescribed via a general Borel measure $\eta$ on the boundary $\partial D$ of a subdomain $D \subset \Omega$:

$$\left\{ \begin{array}{lc} {\rm div}^2 (\varrho A) - {\rm div}(\varrho b) = {\rm div}^2 G - {\rm div}\,h & \text{in}~D, \ \varrho|_{\partial D} = \eta + \kappa \sigma_{d-1}, \end{array} \right.$$

with $\kappa$ given by a quotient involving $A$0, $A$1, and the normal vector. The weak solution concept is formulated in duality with test functions from relevant Sobolev or smooth function spaces. The extension of boundary data to general Borel measures, as opposed to classical $A$2 or continuous functions, is a primary technical advance.

Main Technical Results

Well-Posedness and Representation of Solutions

A rigorous existence and uniqueness theorem is obtained: Under standard uniform ellipticity conditions on $A$3 ((H2)), modulus of continuity assumptions ((H1)), and integrability constraints on lower-order terms ((H3)), the Dirichlet boundary value problem with measure data is shown to be well-posed in $A$4 spaces. The solution is unique in this class (Theorem 2) and satisfies robust a priori estimates depending linearly on the boundary measure total variation, the $A$5-norms of inhomogeneous terms, and the Sobolev norms of the coefficients.

A key result demonstrates that any nonnegative solution in the weak sense on a domain $A$6 restricts to a solution of a corresponding Dirichlet problem with boundary measures on arbitrary inner subdomains (Theorems 1 and 11). Importantly, under mild regularity or nonnegativity assumptions, boundary traces are identified in a weak sense as measures, even for solutions without Sobolev or pointwise traces.

Smooth Approximation and Weak Stability

The solution theory is robust under smooth mollification: Solutions to approximating problems with smooth coefficients and smooth boundary data converge weakly to the original solution as coefficients and boundary data converge (Theorem 3). In particular, for nonnegative solutions and vanishing lower-order data ($A$7), the approximants can be chosen nonnegative, and convergence is strong in $A$8-norm. This provides a constructive approach for existence and allows, in applications, to circumvent regularity assumptions present in the literature for Harnack-type and maximum principle results.

Continuity, Regularity, and Correction of Prior Claims

The analysis provides positive answers to previously open problems regarding the regularity of solutions and the structure of weak Dirichlet problems for double divergence elliptic equations with singular data. Notably, the results clarify, correct, and complete arguments in DK17 and follow-up works, where continuity of weak solutions was claimed under less restrictive classes than was justified. This is achieved through a careful passage to the boundary via mollification and the compactness of associated measure traces, sidestepping the a priori need for $A$9 bounds. For nonnegative solutions and under Dini-type oscillation assumptions on $G$0 (possibly $G$1), solutions admit continuous versions without boundedness assumptions. This restores correctness of Harnack inequalities, maximum principles, and boundary regularity results relying on these findings.

Implications for Elliptic Theory and Related Equations

The results bridge the gap between direct and divergence form equations and the previously less developed double divergence context. This unifies the treatment of Kolmogorov or stationary Fokker-Planck equations—prominent in infinite-dimensional analysis, SPDEs, and stochastic processes—with general boundary behavior and minimal regularity requirements. The well-posedness with measure-valued boundary data extends classical and recent theory applicable only for $G$2 or continuous data, see [Esc],[Gu19],[Gu24],[Sjorg1],[Sjorg2].

Further, the approach allows removal of a priori regularity assumptions in several downstream applications. For instance, all nonnegative solutions (weak or distributional sense) to the (double) divergence form elliptic equations in the presence of Dini-regular coefficients have a continuous representative. This ensures validity of Harnack inequalities, maximum principles, and regularity results previously established under stronger assumptions [BRS23],[BSumn],[DK17],[DEK18],[GKim]. Additionally, solution representations and continuity for stationary measures and densities in diffusion and Fokker-Planck-Kolmogorov contexts are guaranteed under optimal integrability and regularity.

Methodological Innovations

The central innovations include:

• Extension of Dirichlet problem theory to double divergence form elliptic equations with measure boundary data, not previously addressed in the literature.
• Construction of weak traces via mollification and compactness methods, even for solutions outside classical Sobolev or continuous classes.
• Smooth approximation (via mollifiers and compatible coefficient smoothing) that is shown to preserve ellipticity, integrability, and Dini-type regularity, facilitating convergence to weak solutions and the strong maximum principle.
• Correction and clarification of misapplications or gaps in prior literature, especially concerning the need for local boundedness assumptions.

Future Directions

The established framework naturally extends several research programs:

• Extension to parabolic analogs, notably double divergence form parabolic equations, where the interplay of stochastic calculus, time-dependent function spaces, and measure-theoretic boundary data is largely unexplored.
• Potential theory and probabilistic applications in the existence and uniqueness of invariant measures for diffusion processes with singular drift and boundary reflection, relevant to infinite-dimensional SDEs/SPDEs.
• Nonlinear generalizations and quasilinear PDEs where measure data arise in variational and geometric contexts.

Conclusion

The paper systematically develops the solvability, regularity, and boundary trace theory for double divergence form elliptic equations with measures as boundary conditions. It removes superfluous regularity assumptions in prior work, establishes weak well-posedness and smooth approximation, and provides definitive results on continuity and uniqueness of solutions for broad classes of equations, notably those arising in diffusion and stochastic dynamics (2604.17096). These advances both clarify foundational aspects and enable new analytical and probabilistic developments in PDE theory.