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

• * 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.
• 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.

 Citation:

• 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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