Sobolev inequality

A family of inequalities that bounds a function’s norm using norms of its derivatives. These bounds yield embeddings between Sobolev spaces and other function spaces.

Sobolev embedding theorem: A theorem asserting continuous inclusions between Sobolev spaces and other function spaces under specified conditions. The inequality supplies the norm estimate behind these continuous inclusions.
Sobolev space: A function space defined by integrability conditions on functions and their weak derivatives. The inequality compares norms that define these spaces.
Sergei Sobolev: A Russian mathematician who founded the modern theory of distributions and introduced Sobolev spaces. His work established the framework and inequalities bearing his name.
Elliptic partial differential equation: A partial differential equation whose highest-order terms satisfy an ellipticity condition. Energy estimates use Sobolev inequalities to control solutions and improve their regularity.
Hardy inequality: An inequality controlling a weighted integral of a function by an integral involving its derivative. It emphasizes singular weights rather than the integrability gain typical of Sobolev bounds.
Critical Sobolev exponent: The exponent marking the limiting integrability in a Sobolev embedding for a given dimension and derivative order. It identifies the target exponent at which the usual embedding changes character.
Weak derivative: A derivative defined through integration against smooth test functions rather than pointwise differentiation. Sobolev inequalities apply even when classical derivatives do not exist.
Laurent Schwartz: A French mathematician who developed the theory of distributions in the twentieth century. Distribution theory later clarified the weak derivatives central to Sobolev analysis.
Navier–Stokes equations: The nonlinear partial differential equations governing the motion of viscous fluids. Sobolev estimates control nonlinear terms in existence and regularity arguments.
Trudinger–Moser inequality: An endpoint inequality giving exponential integrability when a limiting Sobolev embedding no longer yields a finite power exponent. It replaces the failed critical-power bound at the two-dimensional endpoint.

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