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

  • * Corresponding author: Adam Bobrowski

    * Corresponding author: Adam Bobrowski

Dedicated to Gisèle Ruiz Goldstein

T.K. acknowledges the support of the National Science Centre: NCN grant 2016/23/B/ST1/00492.
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  • 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.

    Mathematics Subject Classification: Primary: 35B50, 35A23; Secondary: 35A09.


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  • Figure 1.  The solid curve $ \partial D $ separates $ D $ (below) from its complement $ D^\complement $ (above). The set $ \partial K( x,r/2)\cap K( y,r) $ forms an arc on which the centers of the small dotted circles, representing $ \partial K(z,\delta) $, lie.

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