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August 21, 2025

Regularity of elliptic equations in double divergence form and applications to Green’s function estimates

Key Points

Solutions to elliptic equations in double divergence form are shown to be differentiable on the zero level set, demonstrating regularity.
Key evidence includes derived pointwise bounds for derivatives, enhancing estimated outcomes for elliptic operators.
The approach utilizes the Dini mean oscillation condition for coefficients in the proof provided, reinforcing the methodology's strength.
Implications may significantly extend analytic techniques for uniformly elliptic operators under broader conditions.

Abstract

We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfy the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a pointwise bound for the derivative. As an application, we establish global pointwise estimates for the Green’s function of second-order uniformly elliptic operators in non-divergence form, assuming Dini mean oscillation coefficients in bounded C 1, α C^1, domains.

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