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.

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椭圆PDE子解的量化Hopf型最大原理

假设$ u（x）$是有界域$ D $中椭圆方程的正子解，其中$ C ^ 2 $光滑边界$ \ partial D $。我们证明了Hopf最大值原理的定量形式，可以用以下公式表示：存在一个常数$ \ gamma> 0 $，使得$ \ partial _ {\ bf n} u（\ tilde x）$ –位于最高点$ \ tilde x \ in \ partial D $（根据强大的最大原则，必须位于$ \ partial D $）–满足$ \ partial _ {\ bf n} u（\ tilde x）> \ gamma u（假设零阶项的系数$ c（x）$满足$ \ sup_ {x \ in D} c（x）= -c _ * <0 $。常数$ \ gamma $仅取决于$ D $的几何形状，均匀椭圆度边界，系数的$ L ^ \ infty $边界以及$ c_ * $。