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

* Corresponding author: Adam Bobrowski

Dedicated to Gisèle Ruiz Goldstein

Received  September 2019 Published  December 2020 Early access  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 and Continuous Dynamical Systems - S, 2020, 13 (12) : 3495-3502. doi: 10.3934/dcdss.2020248
References:
[1]

H. BerestyckiL. 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.

[2]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001.

[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.

[4]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.

[5]

D. H. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, 233. Springer-Verlag, Berlin-New York, 1979.

show all references

References:
[1]

H. BerestyckiL. 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.

[2]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001.

[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.

[4]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.

[5]

D. H. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, 233. Springer-Verlag, Berlin-New York, 1979.

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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