SYSTEMATIC MATHEMATICS

Sobolev Spaces, Distributions & Weak Partial Differential Equations

Why can |x| have a derivative when it has a corner, why does a jump create a Dirac mass, what does H^{-1} mean, how can a boundary value exist for an almost-everywhere equivalence class, and why does weak existence not automatically make a PDE solution smooth or unique? This course answers those questions in sixty-four visible bilingual chapters across eleven content-sized units of lengths 6, 6, 5, 6, 5, 8, 5, 6, 5, 7, and 5. It begins with almost-everywhere equality, local integrability, bump tests, distributions, Dirac mass, distributional derivatives, jumps, convolution, mollifiers, fundamental solutions, tempered Fourier calculus, and negative-order spaces. It then builds weak derivatives, W^{k,p}, H^k, reflexivity, approximation, W_0^{1,p}, extension, Poincaré, Sobolev, critical, Morrey, interpolation, Rellich, traces, fractional boundary data, H(div), capacity, weak Poisson equations, Lax–Milgram, Neumann compatibility, rough elliptic coefficients, Caccioppoli estimates, difference-quotient regularity, De Giorgi–Nash–Moser, compact resolvent, Galerkin and Céa, monotone operators, p-Laplacian, Minty limits, Bochner evolution spaces, heat flow, entropy shocks, and the Leray–Hopf boundary. Every symbol is tied to a domain, dimension, measure, exponent, representative, test class, trace, dual pairing, convergence mode, and theorem hypothesis. The course proves the foundational distribution, Sobolev, embedding, trace, weak elliptic, monotone, and energy mechanisms it uses while stating structured proof routes for the deepest regularity and evolution results. It does not claim to replace full courses in Besov and Triebel–Lizorkin spaces, microlocal or pseudodifferential analysis, nonlinear regularity, geometric measure theory, free boundaries, kinetic equations, compensated compactness, stochastic PDE, advanced Navier–Stokes theory, or research-level numerical analysis.

Before this course: Completed Real Analysis, Measure & Lebesgue Integration, Functional Analysis, Partial Differential Equations, Calculus of Variations, and Numerical Analysis. Learners should know Lebesgue L^p spaces, Hölder and Minkowski inequalities, weak convergence in Banach and Hilbert spaces, compactness, classical integration by parts, divergence-form PDE models, variational weak equations, and finite-dimensional Galerkin systems. No prior complete course in distributions, Sobolev spaces, trace and extension theory, elliptic regularity, monotone operators, Bochner spaces, entropy solutions, or Leray–Hopf theory is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Almost-everywhere equality changes the object being studied

Objective: What additional theorem would make pointwise language legitimate for a Sobolev class?

Lebesgue spaces do not remember the value of a function at an isolated point or any null set. Their elements are equivalence classes of representatives. This is essential rather than cosmetic: weak derivatives, norms, and PDE solutions are statements about the class unless a later regularity theorem selects a continuous representative. Before calculating, name the open domain Ω, dimension n, measure, function or distribution space, exponent, test class, boundary condition, and the meaning of equality. The exact conclusion proved here is: If f=g a.e., then they have the same L^p norm, the same integral against every bounded compactly supported measurable function where the product is integrable, and the same distribution induced by integration. Point evaluation f(x_0) is not well defined on a general L^p class.

Read every new symbol from its definition: On a measurable set Ω, write f=g almost everywhere, abbreviated f=g a.e., when {x∈Ω:f(x)≠g(x)} has measure zero. The space L^p(Ω), 1≤p≤∞, consists of equivalence classes under this relation whose p-norm is finite. A weak or distributional statement is an identity against declared test functions, not a claim that an unproved pointwise derivative exists.

Reconstruct the worked problem “On Ω=(−1,1), let f(x)=0 for every x and let g(x)=1 at x=0 but g(x)=0 otherwise. Compare their L^2 distance, integrals against ψ(x)=1−x², and point values.” by following the pairing, integral, norm, or energy line by line. Then test the first nearby failure: A statement such as u(0)=0 is meaningless for a bare L^2 class. It becomes meaningful only after choosing a representative with proved continuity, defining an averaged or quasi-continuous value, or interpreting boundary values through a trace operator. “Equal a.e.” also does not mean equal at every point.

If f=g a.e., then they have the same L^p norm, the same integral against every bounded compactly supported measurable function where the product is integrable, and the same distribution induced by integration. Point evaluation f(x_0) is not well defined on a general L^p class.

The difference f−g vanishes outside a null set N. Integrating |f−g|^p over Ω gives zero, so their L^p distance is zero.

If ψ is bounded with compact support and the products are integrable, then (f−g)ψ also vanishes outside N, hence ∫(f−g)ψ=0.

Changing one value f(x_0) leaves the equivalence class unchanged. Therefore any rule assigning the class a point value would give two different answers to the same element and cannot be well defined.

On Ω=(−1,1), let f(x)=0 for every x and let g(x)=1 at x=0 but g(x)=0 otherwise. Compare their L^2 distance, integrals against ψ(x)=1−x², and point values.

  1. The disagreement set is {0}, which has Lebesgue measure zero, so f=g a.e.
  2. Hence ||f−g||_2²=∫_{−1}^1|f−g|²dx=0 and ∫fψdx=∫gψdx=0.
  3. Nevertheless f(0)=0 and g(0)=1. The two formulas are different representatives of the same L^2 element.

Result: Their L^2 distance and both tested integrals are zero, but their chosen point values differ; L^2 identifies them.