Misplaced Pages

In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres ${\displaystyle (S^{n},*)\to (S^{n},*)}$ or equivalently to boundary-sphere-preserving continuous maps between balls ${\displaystyle (B^{n},S^{n-1})\to (B^{n},S^{n-1})}$ to boundary-sphere-preserving maps between balls in a Banach space ${\displaystyle f:(B(V),S(V))\to (B(V),S(V))}$, assuming that the map is of the form ${\displaystyle f=id-C}$ where ${\displaystyle id}$ is the identity map and ${\displaystyle C}$ is some compact map (i.e. mapping bounded sets to sets whose closure is compact).

The degree was invented by Jean Leray and Juliusz Schauder to prove existence results for partial differential equations.

References

1. Leray, Jean; Schauder, Jules (1934). "Topologie et équations fonctionnelles". Annales scientifiques de l'École normale supérieure. 51: 45–78. doi:10.24033/asens.836. ISSN 0012-9593.
2. Mawhin, Jean (1999). "Leray-Schauder degree: a half century of extensions and applications". Topological Methods in Nonlinear Analysis. 14: 195–228. Retrieved 2022-04-19.
3. Mawhin, J. (2018). A tribute to Juliusz Schauder. Antiquitates Mathematicae, 12.

This topology-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Topology
• Topology stubs