Comparison theorem

From HandWiki

In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry.

Differential equations

In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof), provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property.[1][2]

  • Chaplygin inequality[3]
  • Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations.
  • Sturm comparison theorem
  • Aronson and Weinberger used a comparison theorem to characterize solutions to Fisher's equation, a reaction--diffusion equation.
  • Hille-Wintner comparison theorem

Riemannian geometry

In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. [4]

Other

References

  1. "Comparison theorem - Encyclopedia of Mathematics". https://www.encyclopediaofmath.org/index.php/Comparison_theorem. 
  2. See also: Lyapunov comparison principle
  3. "Differential inequality - Encyclopedia of Mathematics". https://www.encyclopediaofmath.org/index.php/Differential_inequality. 
  4. Jeff Cheeger and David Gregory Ebin: Comparison theorems in Riemannian Geometry, North Holland 1975.
  5. M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712
  6. Weisstein, Eric W.. "Berger-Kazdan Comparison Theorem". http://mathworld.wolfram.com/Berger-KazdanComparisonTheorem.html. 
  7. F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356
  8. R.L. Bishop & R. Crittenden, Geometry of manifolds