Solvability of divergence equation in Lipschitz spaces

Abstract: We study the solvability of the divergence equation $\operatorname{div} \u = f$ in bounded $C<sup>2$ domains under homogeneous Dirichlet boundary conditions for data $f\in C<sup>{0,α}(Ω)$ satisfying the compatibility condition $\int_Ωf =0.$ We construct a solution $\u$ such that for every $0<β<α$ $\u\in C^{{1,β}(Ω)ⁿ}$ satisfies $$|\u|<em>{C^{1,β}(Ω)} \le C|f|</em>{C^{0,α}(Ω)}.$$ The proof combines localization techniques with a boundary flattening procedure reducing the problem to a model half-cube.

• The paper establishes that for every 0 < β < α, the divergence equation admits a solution u in C^(1,β) with a quantified norm estimate from C^(0,α) data.
• It combines interior Calderón–Zygmund estimates with boundary correction techniques via local flattening and partition of unity.
• The work highlights a sharp regularity loss at the boundary and sets a framework for exploring solvability in less regular domains.

Solvability of the Divergence Equation in Lipschitz Spaces: An Expert Summary

Introduction and Functional Framework

The divergence equation $\operatorname{div} \mathbf{u} = f$ in a bounded $C^2$ domain $\Omega\subset\mathbb{R}^n$ under homogeneous Dirichlet boundary conditions is fundamental in the analysis of PDEs, notably for incompressible fluid models such as the Stokes and Navier–Stokes systems. The construction of right inverses for the divergence is also closely linked to Korn-type inequalities and to regularity theory for various elliptic systems.

It is well-established, via Bogovskiĭ's construction, that for $1 < p < \infty$ and $f \in L^p_0(\Omega)$ (mean-zero functions), this equation admits solutions $\mathbf{u} \in W^{1,p}_0(\Omega)^n$ with norm control. However, endpoint cases $p = 1$ and $p = \infty$ necessitate alternative function spaces due to lack of solvability in $W^{1,p}$ for those exponents. In this context, Lipschitz or Hölder space results have recently attracted attention, as these spaces provide a natural setting for continuity and better pointwise regularity.

The paper establishes the following: for $f\in C^{0,\alpha}(\Omega)$ with $C^2$0 and mean zero, there exists a solution $C^2$1 satisfying homogeneous Dirichlet conditions for any $C^2$2, together with the quantitative estimate

$C^2$3

where $C^2$4 depends on $C^2$5, $C^2$6, $C^2$7, and $C^2$8. This result demonstrates a loss of regularity at the boundary, i.e., generally one cannot obtain $C^2$9 norm control for $\Omega\subset\mathbb{R}^n$0 from a $\Omega\subset\mathbb{R}^n$1 datum.

Main Results and Technical Claims

The central theorem formalizes solvability of the divergence equation in the class $\Omega\subset\mathbb{R}^n$2 for each $\Omega\subset\mathbb{R}^n$3. The proof combines several key analytic and geometric ingredients:

• Interior regularity is derived from previous compactly supported results for the whole space, exploiting the boundedness of singular integrals in $\Omega\subset\mathbb{R}^n$4 spaces.
• Boundary regularity is achieved through a careful boundary flattening and a model local construction in half-cube domains, followed by patching via localization and partition of unity.

A notable technical point is the demonstration that the solution $\Omega\subset\mathbb{R}^n$5, although influenced by a datum in $\Omega\subset\mathbb{R}^n$6, may lose a fraction of regularity via mollification procedures inherent to the boundary correction steps. Thus, sharp preservation of the Hölder exponent up to the boundary is not generally achievable with these methods.

Strong claims established:

• For every $\Omega\subset\mathbb{R}^n$7, the solution achieves $\Omega\subset\mathbb{R}^n$8 regularity (not $\Omega\subset\mathbb{R}^n$9) with homogeneous Dirichlet data.
• The proof is valid for bounded $1 < p < \infty$0 domains; whether this regularity can be weakened to, e.g., $1 < p < \infty$1 boundaries, remains unresolved.

Proof Strategy and Analytical Techniques

Interior Estimates

For subdomains strictly inside $1 < p < \infty$2, one utilizes the Bogovskiĭ operator applied to compactly supported data, yielding solutions with full $1 < p < \infty$3 regularity. Key estimates rely on integral kernels with singularities treated via classical Calderón–Zygmund theory, ensuring control of both the solution and its derivatives.

Half-Cube and Boundary Correction

For domains intersecting the boundary, the problem is mapped locally (via translation and flattening) to a half-cube $1 < p < \infty$4, where a symmetrization and extension technique is used to construct an initial approximate solution. However, the normal component naturally vanishes at the flat part of the boundary while the tangential components require a divergence-free correction $1 < p < \infty$5 to enforce the full homogeneous Dirichlet condition.

The construction and analysis of $1 < p < \infty$6 are crucial: a mollification argument is carefully quantified using a convolution lemma, ensuring that the regularity loss (from $1 < p < \infty$7 to $1 < p < \infty$8) is precisely controlled and cannot be avoided with this technique.

Globalization via Partition of Unity

The global solution is assembled from local solutions via a partition of unity subordinated to a finite open cover, with mean-zero corrections ensuring that the integral condition on $1 < p < \infty$9 is maintained in each patch. Composition with $f \in L^p_0(\Omega)$0 diffeomorphisms preserves the required regularity due to standard chain rules for Hölder spaces.

Implications and Open Questions

The theory developed here positions the $f \in L^p_0(\Omega)$1 spaces (and their vector-valued analogs) as a viable functional analytic setting for divergence operators in domains with boundary, extending the reach of classical Sobolev and $f \in L^p_0(\Omega)$2 theory. This is instrumental for studying regularity properties of solutions to elliptic and parabolic PDEs, especially those where sharp pointwise estimates matter (e.g., boundary layer analysis, free boundary problems).

Unresolved directions and open problems include:

• Whether the strict loss in regularity at the boundary (i.e., $f \in L^p_0(\Omega)$3) is a technical artifact or reflects a genuine structural barrier.
• Whether these results extend to domains with less regularity, particularly $f \in L^p_0(\Omega)$4 or Lipschitz boundaries, given that the boundary flattening procedure requires second derivatives of the boundary defining function.

Potential applications:

• Improved control in regularity-driven fluid PDE models where pressure is recovered as a function in Lipschitz-type spaces.
• Use in geometric analysis and elasticity, for Korn- and Poincaré-type inequalities, where domain regularity and precise control of divergence are critical.

Conclusion

This paper rigorously establishes the solvability of the divergence equation in $f \in L^p_0(\Omega)$5 spaces up to the boundary for bounded $f \in L^p_0(\Omega)$6 domains with mean-zero data in $f \in L^p_0(\Omega)$7. While full preservation of the Hölder exponent across the boundary is precluded by the employed method, the approach offers a constructive and quantifiable solution for PDE models requiring pointwise regularity and fits naturally in contemporary analysis of incompressible flows and related systems. The question of sharpness—both for regularity loss and minimal boundary smoothness—remains an important problem for future work.

Reference: "Solvability of divergence equation in Lipschitz spaces" (2607.06387)