$\mathcal W^{\alpha, p}$ and $C^{0,\gamma}$ regularity of solutions to $(\mu - \Delta + b \cdot \nabla)u=f$ with form-bounded vector fields

TL;DR

This paper investigates the regularity of solutions to elliptic equations involving the operator $- abla ext{·} abla + b ext{·} abla$ with form-bounded vector fields, establishing how solution smoothness depends on the vector field's form-bound.

Contribution

It characterizes the quantitative dependence of Sobolev and Hölder regularity of solutions on the form-bound of the vector field in the operator.

Findings

Established $ ext{W}^{1+rac{2}{q},p}$ regularity dependence on form-bound.

Derived $C^{0,eta}$ regularity estimates based on form-bound.

Applicable to vector fields with critical singularities.

Abstract

We consider the operator $-\Delta +b \cdot \nabla$ with $b:\mathbb R^d \rightarrow \mathbb R^d$ ($d \geq 3$) in the class of form-bounded vector fields (containing vector fields having critical-order singularities), and characterize quantitative dependence of the $\mathcal W^{1+\frac{2}{q},p}$ ($2 \leq p < q$) and the $C^{0,\gamma}$ regularity of solutions to the corresponding elliptic equation in $L^p$ on the value of the form-bound of $b$.