Historical Background about sobolev space

     以下摘自Luc Tartar"An Introduction to Sobolev Spaces and Interpolation Space"  

    In the mid 1930s, Sergei SOBOLEV introduced some functional spaces which have been very important in the development of partial differential equations, mostly those related to continuum mechanics or physics. They are known as Sobolev spaces, but others have mentioned having defined similar spaces, like FICHERA and FRIEDRICHS. A similar idea was used a little after by Jean LERAY in his study of weak solutions of the Navier–Stokes equation, and he thought that the lack of regularity is related to turbulent flows, but although nobody really understands at a mathematical level what turbulence is, it is quite clear from a continuum mechanics point of view that Jean LERAY’s ideas do not correspond to it; the ideas introduced later by KOLMOGOROV have received more attention, but have some defects which are not emphasized enough.

    The basic idea for defining a Sobolev space consists in using weak derivatives, as Sergei SOBOLEV or Jean LERAY did in the mid 1930s; it consists in giving a precise meaning to the statement that a function u from the Lebesgue space Lp(Ω) (for a nonempty open set Ω in RN) has all its partial derivatives also in Lp(Ω). However, they did not define partial derivatives for every function in Lp(Ω), but only said that some of these functions have partial derivatives belonging also to Lp(Ω), and the important step of defining more general mathematical objects, which permit one to define as many derivatives as one may want for any locally integrable function, was performed by Laurent SCHWARTZ, who called his objects distributions. Laurent SCHWARTZ went further than the theory developed by Sergei SOBOLEV, which he did not know about, and he points out that BOCHNER had also obtained some partial results, which he also only learnt about later. Laurent SCHWARTZ told me that some people quote GEL’FAND for developing the theory of distributions,but that what GEL’FAND did was mostly to popularize the theory. Someone pointed out to me that WEYL should be quoted for the theorytoo, but I have not checked that, and Laurent SCHWARTZ was not aware of his work when I last saw him.