On some divergence-form singular elliptic equations with codimension-two boundary: $L^p$-estimates

TL;DR

The paper addresses global weighted $L^p$ estimates for the gradient of solutions to divergence-form elliptic equations with a codimension-two singular boundary, where coefficients lie in a weighted VMO framework and the weight $\rho^{-2\alpha}$ induces singular behavior. By translating the problem into weighted spaces with measures $\mathrm{d}\mu$ and $\mathrm{d}\mu_\sigma$, and employing a combination of barrier arguments, polar/cylindrical coordinate analysis, and Krylov-type methods, the authors obtain a global weighted $L^p$ gradient estimate under a small BMO/VMO perturbation $(\delta_0,R_0)$ and large parameter $\lambda$. The analysis proceeds from a foundational $L^2$ theory to constant-coefficient regularity results in Section 3, followed by a perturbation/VMO argument in Section 4 to handle general coefficients, yielding both global and local $L^p$ estimates and a boundary regularity framework. The results extend existing codimension-one studies to codimension-two singularities and provide a robust regularity theory applicable to problems arising in harmonic maps with prescribed singularities and related degenerate elliptic operators.

Abstract

We establish a global weighted $L^p$ estimate for the gradient of the solution to a divergence-form elliptic equations, where the coefficients are in a weighted VMO space and the equations have singularities on a co-dimension two boundary.

Theorems & Definitions (30)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3: Sect. 1.3.1 (iv) of Maz'ya M
• Lemma 2.4
• proof
• Lemma 3.1
• proof
• Proposition 3.2