In this paper, we prove that there exists a unique, bounded continuous weak solution to the Dirichlet boundary value problem for a general class of second-order elliptic operators with singular coefficients, which does not necessarily have the maximum principle. Our method is probabilistic. The time reversal of symmetric Markov processes and the theory of Dirichlet forms play a crucial role in our approach.
"Time-reversal and elliptic boundary value problems." Ann. Probab. 37 (3) 1008 - 1043, May 2009. https://doi.org/10.1214/08-AOP427