Examples of capacity for some elliptic operators

Abstract:

We study $L$-capacities for uniformly elliptic operators of nondivergence form \[ L = \sum \limits _{i,j} {{a_{ij}}(x)\frac {{{\partial ^2}}} {{\partial {x_i}\partial {x_j}}} + } \sum \limits _j {{a_j}(x)\frac {\partial } {{\partial {x_j}}};} \] and construct examples of large sets having zero $L$-capacity for some $L$ , and small sets having positive $L$-capacity. The relations between ellipticity constants of the coefficients and the sizes of these sets are also considered.