Lyapunov-type inequality to general second-order elliptic equations

We establish Lyapunov-type inequality for equations concerning general class of second-order non-symmetric elliptic operators with singular coefficients. Our approach is based on the probabilistic representation of solutions and stochastic calculus. We also discuss a Lyapunov-type inequality for equations pertaining to second-order symmetric operator with some regularity assumptions on the coefficients and a nonlinear Neumann boundary condition.