nLab: Sobolev Space
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nLab wiki entry defining Sobolev spaces as spaces of weakly differentiable functions with integrability conditions, with pointers to the standard literature. A concise definitional reference for readers who already know measure theory and basic functional analysis.
More resources on Sobolev Spaces
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Terry Tao's Notes on Sobolev Spaces
Terence Tao's blog posts and UCLA graduate lecture notes tagged Sobolev spaces, covering Sobolev embedding, weak derivatives, fractional Sobolev norms and their role in harmonic analysis. Readers learn the estimates and heuristics behind modern nonlinear PDE arguments.
Functional Analysis, Sobolev Spaces and Partial Differential Equations
Graduate textbook developing functional analysis (Hahn–Banach, weak topologies, Lp spaces, compact operators, spectral theory) and applying it to Sobolev spaces and the variational theory of elliptic, parabolic and hyperbolic PDEs. Includes extensive exercises, many with solutions.
Sobolev Spaces
Standard reference monograph on Sobolev spaces, covering weak derivatives, embedding and compactness theorems, interpolation, extension operators, fractional-order spaces and Orlicz–Sobolev spaces. Readers gain the function-space toolkit used in elliptic PDE and the calculus of variations.