Equivalence of the Green’s functions for diffusion operators in $\textbf {R}^{n}$: a counterexample

Abstract:

In a smooth domain in ${{\mathbf {R}}^n}$, the Green’s functions for second-order, uniformly elliptic operators in divergence form are all proportional to the Green’s function for the Laplacian [7]. In this paper we show that the above result fails for diffusion operators, that is, second-order, uniformly elliptic operators with continuous coefficients in nondivergence form. In fact, we give an example in which the Green’s function is locally unbounded away from the pole.