A Banach (or Hilbert) space of equivalence classes of functions on a domain whose weak derivatives up to a specified order belong to an Lp space; commonly denoted W^{k,p} or H^{k} for p=2.

Principle

Organize function regularity by integrability of weak derivatives rather than pointwise differentiability, trading classical smoothness for control of average variation.

Demonstration

H1(Ω) is the closure of C∞_c(Ω) under the norm ||u||_{H1}= (||u||_{L2}^2+||∇u||_{L2}^2)^{1/2}; solutions of variational formulations of elliptic boundary-value problems naturally lie in H1(Ω).

Misapplication

Treating a weak derivative as if it were a pointwise derivative everywhere and imposing classical boundary traces without verifying trace theorems or additional regularity.

Consequence

Enables existence and compactness results (e.g., Rellich–Kondrachov embeddings) and a priori estimates for PDEs by controlling Lp norms of derivatives rather than pointwise behavior.

Reversal

The classical C^k spaces require pointwise continuous derivatives of order k; reversing the idea replaces integrability-based regularity by uniform pointwise smoothness.

Boundary

Applies to measurable functions on domains with Lebesgue measure and to orders and p for which weak derivatives exist; excludes distributions that are not representable by Lp functions and settings where only fractional or Besov-type regularity is appropriate.

Semantic Tension

Often contrasted with Hölder spaces: both quantify regularity, but Sobolev spaces emphasize integrability of derivatives while Hölder spaces emphasize uniform modulus of continuity.

Synthesis

A Sobolev space is the space of functions whose weak derivatives up to a given order are Lp-integrable, providing an integrability-based framework for analysing existence, stability and approximation of PDE solutions.