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In [[mathematics]], the '''Leray–Schauder degree''' is an extension of the degree of a [[base point]] preserving [[continuous map]] between [[sphere]]s <math> (S^n, *) \to (S^n , *)</math> or equivalently to boundary-sphere-preserving continuous maps between balls <math>(B^n, S^{n-1}) \to (B^n, S^{n-1})</math> to boundary |
In [[mathematics]], the '''Leray–Schauder degree''' is an extension of the degree of a [[base point]] preserving [[continuous map]] between [[sphere]]s <math> (S^n, *) \to (S^n , *)</math> or equivalently to boundary-sphere-preserving continuous maps between balls <math>(B^n, S^{n-1}) \to (B^n, S^{n-1})</math> to boundary-sphere-preserving maps between balls in a [[Banach space]] <math> f: (B(V), S(V)) \to (B(V), S(V))</math>, assuming that the map is of the form <math>f = id - C</math> where <math>id</math> is the [[identity map]] and <math>C</math> is some compact map (i.e. mapping bounded sets to sets whose closure is [[Compact (mathematics)|compact]]).<ref>{{Cite journal |last=Leray |first=Jean |last2=Schauder |first2=Jules |date=1934 |title=Topologie et équations fonctionnelles |url=http://www.numdam.org/item?id=ASENS_1934_3_51__45_0 |journal=Annales scientifiques de l'École normale supérieure |volume=51 |pages=45–78 |doi=10.24033/asens.836 |issn=0012-9593|doi-access=free }}</ref> |
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The degree was invented by [[Jean Leray]] and [[Juliusz Schauder]] to prove existence results for partial differential equations.<ref>{{cite journal |last1=Mawhin |first1=Jean |title=Leray-Schauder degree: a half century of extensions and applications |journal=Topological Methods in Nonlinear Analysis |date=1999 |volume=14 |pages=195–228 |access-date=2022-04-19| url=https://projecteuclid.org/journals/topological-methods-in-nonlinear-analysis/volume-14/issue-2/Leray-Schauder-degree--a-half-century-of-extensions-and/tmna/1475179840.full}}</ref><ref>Mawhin, J. (2018). A tribute to Juliusz Schauder. ''[[Antiquitates Mathematicae]]'', ''12''.</ref> |
The degree was invented by [[Jean Leray]] and [[Juliusz Schauder]] to prove existence results for partial differential equations.<ref>{{cite journal |last1=Mawhin |first1=Jean |title=Leray-Schauder degree: a half century of extensions and applications |journal=Topological Methods in Nonlinear Analysis |date=1999 |volume=14 |pages=195–228 |access-date=2022-04-19| url=https://projecteuclid.org/journals/topological-methods-in-nonlinear-analysis/volume-14/issue-2/Leray-Schauder-degree--a-half-century-of-extensions-and/tmna/1475179840.full}}</ref><ref>Mawhin, J. (2018). A tribute to Juliusz Schauder. ''[[Antiquitates Mathematicae]]'', ''12''.</ref> |
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Latest revision as of 13:20, 18 September 2024
In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres or equivalently to boundary-sphere-preserving continuous maps between balls to boundary-sphere-preserving maps between balls in a Banach space , assuming that the map is of the form where is the identity map and is some compact map (i.e. mapping bounded sets to sets whose closure is compact).[1]
The degree was invented by Jean Leray and Juliusz Schauder to prove existence results for partial differential equations.[2][3]
References
[edit]- ^ 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.
- ^ 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.
- ^ Mawhin, J. (2018). A tribute to Juliusz Schauder. Antiquitates Mathematicae, 12.
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