# American Institute of Mathematical Sciences

December  2020, 13(12): 3495-3502. doi: 10.3934/dcdss.2020248

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

 1 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656, Warszawa, Poland 2 Lublin University of Technology, Nadbystrzycka 38A, 20–618 Lublin, Poland

Dedicated to Gisèle Ruiz Goldstein

Received  September 2019 Published  January 2020

Fund Project: 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.

Citation: Tomasz Komorowski, Adam Bobrowski. A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3495-3502. doi: 10.3934/dcdss.2020248
##### References:
 [1] H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.  Google Scholar [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar [3] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, , Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [4] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.  Google Scholar [5] D. H. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, 233. Springer-Verlag, Berlin-New York, 1979.  Google Scholar

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##### References:
 [1] H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.  Google Scholar [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar [3] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, , Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [4] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.  Google Scholar [5] D. H. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, 233. Springer-Verlag, Berlin-New York, 1979.  Google Scholar
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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