Semismall perturbations in the Martin theory for elliptic equations (Q1376040)

The author considers the elliptic operator on a domain \(D\) in \(\mathbb{R}^n\) of the form \[ L=- \sum^n_{i,j= 1}\partial_i(a_{ij}(x) \partial_j)- \sum^n_{j=1} bj(x)\partial_j+ V(x), \] with given conditions on the coefficients. Among the conditions assumed in this paper that \((L,D)\) is subcritical, and \(\bigcup^\infty_{j=1} D_j= D\), where the closure \(\overline D_j\) of \(D_j\) is a compact subset of \(D_{j+1}\). Also it assumed that the Green function \(G^D_L\) for \((L,D)\) is equal to \(\underset{j\to\infty}{\text{Lim}} G_j\), where \(G_j\) is the Green function for \(L\) in \(D_j\) with zero Dirichlet boundary condition. The author defines the notion of \(G^D_L\)-semismallness at infinity, and proves that Martin boundaries are stable under perturbations which are \(G^D_L\)-semismall at infinity. He also presents many examples demonstrating his point. Several results related to the \(G^D_L\)-semismallness, namely boundedness, is given, too. The paper is rich with literature on this subject.