Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary
Eugene B. Dynkin, С. Е. Кузнецов · Transactions of the American Mathematical Society · 1998
Let L L be a second order elliptic differential operator on a Riemannian manifold E E with no zero order terms. We say that a function h h is L L -harmonic if L h = 0 Lh=0 . Every positive L L -harmonic function has a unique representation h ( x ) = ∫ E ′ k ( x , y ) ν ( d y ) , \begin{equation*}h(x)=\int _{E’} k(x,y) u (dy), \end{equation*} where k k is the Martin kernel, E ′ E’ is the Martin boundary and ν u is a finite measure on E ′ E’ concentrated on the minimal part E ∗ E^{*} of E ′ E’ . We call ν u the trace of