Equivalence of the Green’s functions for diffusion operators in 𝑅ⁿ: a counterexample

Source abstract

In a smooth domain in R n {{\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.