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The Mathematical Intelligencer
Article . 1998 . Peer-reviewed
License: Springer TDM
Data sources: Crossref
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article . 1998
Data sources: zbMATH Open
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The third boundary condition—was it robin’s?

The third boundary condition -- was it Robin's?
Authors: Gustafson, Karl; Abe, Takehisa;

The third boundary condition—was it robin’s?

Abstract

A real-valued function \(u\) of \(n\) real variables is called harmonic in an \(n\)-dimensional domain if it satisfies Laplace's equation \(\nabla u=0\), where \(\nabla\) is the \(n\)-dimensional Laplace operator. Certain boundary conditions to be satisfied by \(u\) are familiar in the literature. One is Dirichlet's boundary condition, specifying the value of \(u\) on the boundary of the domain. Another is Neumann's boundary condition, specifying the normal derivative of \(u\) on the boundary. A third, less common, condition has become known in certain sources as Robin's condition: \(u\) should satisfy \(\partial u/\partial n+\alpha u=f(x)\) on the boundary of the domain, where \(\partial u/\partial n\) denotes the normal derivative of \(u\), \(\alpha\) is a positive constant, and \(f(x)\) is a given function. The purpose of this paper is to enquire into who this Robin was and also to see if there is any justification in attaching his name to this condition. Gustave Robin (1855-1897) was a professor of mathematical physics in Paris. He wrote his doctoral thesis under the guidance of Emile Picard. Its subject matter was the determination of an electrostatic potential on a convex surface and it required the solution of a Neumann-type boundary condition for a harmonic function. This seems to be the nearest connection to Robin's involvement in the third boundary condition. Robin published very little in his lifetime, although his Oeuvres scientifiques were published in three volumes shortly after his death, and his name has fallen into obscurity. The authors' main finding is that there is little reason to connect Robin's name with the third boundary condition, as he never explicitly worked on this problem. Russian literature from the 1920's mentions his name and contributions, but only in connection with the type of problem considered in his thesis. The authors also suggest that Stefan Bergman (1895-1977) was the first to attribute Robin's name to this third boundary condition, in 1948. A second part to this paper will examine Robin's work more closely.

Keywords

History of mathematics in the 20th century, mathematical physics, harmonic function, History of mathematics in the 19th century, History of partial differential equations, Gustave Robin, third boundary condition

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
78
Top 10%
Top 10%
Average
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