In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.
Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces, rather than in spaces of continuous functions and with the derivatives understood in the classical sense.
Other articles related to "sobolev space, sobolev, space, spaces":
... It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable ... This idea is generalized and made precise in the Sobolev embedding theorem ...
... in 1850, is a rigidity theorem about conformal mappings in Euclidean space ... mappings in R3 and higher-dimensional spaces ... of this system is defined to be an element ƒ of the Sobolev space W1,n loc(Ω,Rn) with non-negative Jacobian determinant almost everywhere, such that the Cauchy–Riemann system holds ...
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