Counting dimensions of L-harmonic functions
  In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially of degree d. More precisely, in terms of a rectangular coordinate system {x_1,...,x_n}, a second order uniformly elliptic operator of divergence form, L, acting on a function f in H^1_loc(R^n) is given by   Lf = sum_{ij} d/dx_i (a^{ij}(x) df/dx_j)   where (a^{ij}(x)) is an n x n symmetric matrix satisfying the ellipticity bounds   \lambda I <= (a^{ij}) <= Lambda I   for some constants 0 < lambda <= Lambda < \infty. Other than the ellipticity bounds, we only assume that the coefficients (a_{ij}) are merely measurable functions. 
Peter Li, Jiaping Wang