Mathematics / advanced
Weak derivatives and Sobolev spaces
Build the function-space language of PDEs through weak differentiability, traces, embeddings, and compactness.
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Inside this module
Module 01 advanced
Weak derivatives and Sobolev spaces
Build the function-space language of PDEs through weak differentiability, traces, embeddings, and compactness.
- 01
Weak derivatives and distributional identities
Recover differentiation from integration against test functions and distinguish corners from genuine jumps.
21 min - 02
Sobolev norms and completeness
Measure functions and weak derivatives together, and prove that limits preserve the derivative identities.
21 min - 03
Mollification and density
Approximate weakly differentiable functions by smooth ones without confusing interior smoothing with boundary conditions.
22 min - 04
Extension operators and domain geometry
Move Sobolev functions to the whole space while preserving norm control, and see why zero extension needs boundary compatibility.
22 min - 05
Trace theorems and boundary data
Give boundary values to Sobolev equivalence classes and impose Dirichlet data through a bounded trace map.
23 min - 06
Poincaré and Friedrichs inequalities
Remove the constant kernel of differentiation and identify the geometric and boundary assumptions behind energy control.
22 min - 07
Sobolev embedding and critical exponents
Derive the Sobolev conjugate by scaling and distinguish subcritical, critical, and supercritical regularity claims.
24 min - 08
Morrey inequality and Hölder regularity
Turn sufficiently integrable gradients into quantitative continuity and distinguish a Sobolev class from its continuous representative.
23 min - 09
Rellich–Kondrachov compactness
Obtain strong subsequences from bounded Sobolev families and diagnose loss of compactness through concentration and translation.
24 min - 10
Fractional Sobolev spaces and interpolation
Measure nonlocal differences at intermediate smoothness and connect the Gagliardo seminorm with Fourier and interpolation viewpoints.
25 min