Applied analysis · Graduate course
Sobolev Spaces and PDEs
The right spaces make hard equations solvable.
Sobolev spaces are the functional-analytic setting of modern partial differential equations. By replacing the classical derivative with the weak derivative, they build complete function spaces in which existence proofs actually work. Building on functional analysis, measure theory, and distributions, the course develops weak derivatives and the spaces \(W^{k,p}\), approximation and density, extension and trace theorems, the Sobolev and Morrey embeddings, the compact Rellich-Kondrachov embedding, the weak formulation of elliptic equations through Lax-Milgram, and existence and regularity for elliptic boundary-value problems.
What the course builds
Three capabilities the course builds
Weaken to solve
Replace pointwise derivatives with weak ones and recast a differential equation as an identity on a function space.
Control functions by their derivatives
Use embeddings, traces, and compactness to convert integrability of derivatives into continuity and convergence.
Prove PDEs have solutions
Apply Lax-Milgram and Rellich-Kondrachov to establish existence and regularity for elliptic boundary-value problems.
Before you start
Prerequisites
Sobolev theory sits at the meeting point of three courses. Functional analysis supplies Hilbert and Banach spaces, weak topologies, and Lax-Milgram; measure theory supplies the \(L^p\) spaces; and distribution theory supplies the weak derivative itself. Familiarity with the Lebesgue integral is assumed throughout.
The syllabus
Eight chapters, built in sequence
Each chapter will open with visual investigations, state every definition and theorem with complete proofs, and close with graded exercises carrying progressive hints and full corrections. Chapters are being published in sequence.
Integration by parts turned into a definition, the weak derivative, and why classical derivatives are too rigid for equations.
Open chapterDefinition and norms, completeness, and the Hilbert space \(H^k\) as the natural energy space for second-order problems.
Open chapterSmooth approximation, the Meyers-Serrin theorem that H equals W, and mollification as the basic tool.
Open chapterExtension operators across the boundary and the trace theorem giving meaning to boundary values of Sobolev functions.
Open chapterThe Gagliardo-Nirenberg-Sobolev inequality and Morrey's theorem: integrability of derivatives forces continuity.
Open chapterThe Rellich-Kondrachov theorem and the role of compactness in existence and eigenvalue problems.
Open chapterBilinear forms, coercivity, and the Lax-Milgram theorem producing unique weak solutions of elliptic equations.
Open chapterExistence for elliptic boundary-value problems and elliptic regularity lifting weak solutions to classical ones.
Open chapter