Hopf maximum principle

Preliminaries

Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018

Maximum principleThe Hopf maximum principle for elliptic partial differential equations and differential inequalities generalizes the classical maximum-minimum principle for harmonic functions. That principle, in turn, can be thought of as generalizing the following simple fact from calculus: if a continuous function y defined on a closed interval [a,b] satisfies y′′=0 on the open interval (a,b), then y cannot achieve a local maximum or local minimum value at a point in (a,b) unless y is identically constant. This follows because y is a linear function on [a,b]. Corresponding results with significant consequences hold for solutions y to certain ordinary differential equations and differential inequalities. Two of those results are presented here, in adequate generality for our purposes. The pioneering work of Eberhard Hopf is far deeper than what is suggested here.

Constant vorticity equatorial flows beneath surface solitary waves with centripetal forces

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Journal Information

Published in Applicable Analysis, 2023

Yanjuan Yang

Moreover, using the symmetry of the free surface, we obtain Note that is bounded and harmonic in the interior of the domain, by the Hopf Maximum Principle, we obtain that must vanish identically in the fluid domain. Therefore, ψ is symmetric. Clearly, from (14), we know that u is symmetric and w is antisymmetric.

Hopf maximum principle

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Preliminaries

Constant vorticity equatorial flows beneath surface solitary waves with centripetal forces