Trace on the boundary for solutions of nonlinear differential equations
Eugene B. Dynkin, С. Е. Кузнецов · Transactions of the American Mathematical Society · 1998
Let L L be a second order elliptic differential operator in R d \mathbb {R}^{d} with no zero order terms and let E E be a bounded domain in R d \mathbb {R}^{d} with smooth boundary ∂ E \partial E . We say that a function h h is L L -harmonic if L h = 0 Lh=0 in E E . Every positive L L -harmonic function has a unique representation h ( x ) = ∫ ∂ E k ( x , y ) ν ( d y ) , \begin{equation*}h(x)=\int _{\partial E} k(x,y) u (dy), \end{equation*} where k k is the Poisson kernel for L L and ν u