The Fundamental Solution of an Elliptic Equation with Singular Drift

Abstract

For $n\ge 3$, we study the existence and asymptotic properties of the fundamental solution for elliptic operators in nondivergence form, $\mathcal{L}(x,{\partial }_x)=a_{ij}(x){\partial }_i{\partial }_j+b_k(x){\partial }_k$, where the $a_{ij}$ have modulus of continuity $\omega (r)$ satisfying the square-Dini condition and the $b_k$ are allowed mild singularities of order $r^{-1}\omega (r)$. A singular integral is introduced that controls the existence of the fundamental solution. We give examples that show the singular drift $b_k{\partial }_k$ may act as a perturbation that does not dramatically change the fundamental solution of ${\mathcal{L}}^o=a_{ij}{\partial }_i{\partial }_j$, or it could change an operator ${\mathcal{L}}^o$ that does not have a fundamental solution to one that does.