A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs

Suppose that $ u(x) $ is a positive subsolution to an elliptic equation in a bounded domain $ D $, with the $ C^2 $ smooth boundary $ \partial D $. We prove a quantitative version of the Hopf maximum principle that can be formulated as follows: there exists a constant $ \gamma>0 $ such that $ \partial_{\bf n}u(\tilde x) $ – the outward normal derivative at the maximum point $ \tilde x\in \partial D $ (necessary located at $ \partial D $, by the strong maximum principle) – satisfies $ \partial_{\bf n}u(\tilde x)>\gamma u(\tilde x) $, provided the coefficient $ c(x) $ by the zero order term satisfies $ \sup_{x\in D}c(x) = -c_*<0 $. The constant $ \gamma $ depends only on the geometry of $ D $, uniform ellipticity bound, $ L^\infty $ bounds on the coefficients, and $ c_* $. The key tool used is the Feynman–Kac representation of a subsolution to the elliptic equation.