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

Abstract

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

arXiv comment7 p