Sobolev space

A function space whose norm measures a function and its weak derivatives through integrability. Sobolev spaces allow differential equations to be studied when classical derivatives do not exist.

Weak derivative: A derivative defined through integration against smooth test functions rather than pointwise limits. Sobolev spaces measure derivatives in this generalized sense.
Sobolev norm: A norm combining integrability measures of a function and its weak derivatives up to a specified order. It quantifies the function and derivatives that define membership in the space.
Weak solution: A solution of a differential equation defined by an integral identity rather than pointwise derivatives. Weak solutions often belong to Sobolev spaces even when classical solutions do not exist.
Classical differentiability: Differentiability defined by pointwise limits of difference quotients. Sobolev spaces require weak derivatives, not pointwise classical derivatives.
Lebesgue space: A function space defined by integrability with respect to a measure. The integrability conditions in Sobolev definitions use Lebesgue spaces.
Sobolev embedding theorem: A theorem stating conditions under which Sobolev-space membership implies stronger integrability or continuity. It converts derivative control into regularity and integrability estimates.
Variational method: A method for solving problems by finding extrema or stationary points of functionals. Energy minimizers are commonly sought in Sobolev spaces.
Hölder space: A function space whose norm controls boundedness and a Hölder continuity modulus. Hölder spaces describe pointwise regularity, unlike Sobolev spaces' integral control.
Distribution: A generalized function defined by its action on smooth test functions. Weak derivatives are defined within distribution theory.
Trace theorem: A theorem defining boundary values for functions with limited interior regularity. It explains how Sobolev functions can have meaningful boundary values.

Linked from 26 pages