SYSTEMATIC MATHEMATICS

Nonlinear Elliptic & Parabolic Regularity

Why can an energy solution become Hölder continuous without having classical derivatives, why does the p-Laplacian change scale when its gradient changes size, why do viscosity solutions need touching tests instead of integration by parts, and how can a smooth equation still develop a singular set or a moving free boundary? This Stage 5 doctoral-specialization core answers those questions in eighty-one visible bilingual chapters across twelve content-sized units of lengths 6, 8, 7, 8, 7, 5, 7, 8, 6, 7, 6, and 6. It begins by separating existence, uniqueness, weak formulation, regularity, boundary behavior, and numerical appearance; defining structural constants and critical scaling; and deriving energy, comparison, Caccioppoli, compactness, De Giorgi, Moser, Harnack, Schauder, and Calderón–Zygmund mechanisms. It then develops quasilinear divergence equations, monotone operators, p-Laplacian and V_p geometry, reverse Hölder self-improvement, harmonic approximation, nonstandard growth, viscosity touching, Pucci extremals, comparison by doubled variables, ABP, Krylov–Safonov, Evans–Krylov, Bellman and Isaacs boundaries, quasiconvex variational integrals, partial regularity, capacity, renormalized solutions, Wolff potentials, removability, and nonlinear Wiener tests. The parabolic half constructs time-weak solutions, energy and Harnack theory, intrinsic cylinders, expansion of positivity, porous-medium and fast-diffusion boundaries, doubly nonlinear flows, nonlinear gradient excess, fully nonlinear parabolic comparison and stability, control equations, and monotone schemes. The final units study obstacle complementarity, blow-ups, Bernoulli and Stefan interfaces, Allen–Cahn limits, elliptic systems, harmonic maps, minimal surfaces, mean-curvature flow, the unresolved three-dimensional Navier–Stokes smoothness boundary, and six independently reproducible research dossiers. Every chapter defines its symbols in words, derives one mechanism step by step, solves a concrete example, names the nearest failure, and distinguishes a theorem from a formal calculation or plot. This course provides doctoral research preparation in selected nonlinear regularity theories; it does not claim a universal postdoctoral syllabus or complete coverage of every nonlinear PDE, system, boundary geometry, singular flow, stochastic equation, or open regularity problem.

Before this course: Completed Sobolev Spaces, Distributions & Weak PDE; Measure & Lebesgue Integration; Functional Analysis; Calculus of Variations; Partial Differential Equations; Harmonic Analysis & Wavelets; and Optimization & Convex Analysis, together with the supporting real analysis, topology, and differential geometry routes. Learners must already be able to work with weak derivatives, Sobolev embedding and compactness, dual spaces, monotone operators, measure convergence, distributional integration by parts, classical elliptic/parabolic equations, and basic manifolds. No prior unified course in De Giorgi–Nash–Moser theory, viscosity solutions, fully nonlinear equations, nonlinear potentials, intrinsic parabolic geometry, or free boundaries is assumed.

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

What a regularity theorem asks and what it does not ask

Objective: Why must a theorem distinguish Ω′⊂⊂Ω from Ω itself?

A weak solution may exist in an energy space while lacking pointwise derivatives. A regularity theorem starts from a precisely stated weak class and proves membership in a smaller, smoother class on a declared region. Before estimating a solution, identify the domain, equation type, solution language, coefficient structure, data space, scaling, boundary or initial conditions, and the smaller region on which the conclusion is claimed.

Read the notation through the definition rather than guessing from the formula: For Ω⊂R^n, an interior estimate has the form ‖u‖_Y(Ω′)≤C(‖u‖_X(Ω)+‖f‖_Z(Ω)) for every Ω′⊂⊂Ω. Here X is the assumed solution space, Y is the improved space, Z measures the data, and C may depend on dist(Ω′,∂Ω) and structural constants. The exact conclusion established in this chapter is: A statement of interior Hölder regularity neither supplies boundary continuity nor proves existence or uniqueness unless those conclusions are separately established.

The proof route is auditable: 1. Identify the input class X, improved class Y, data class Z, and the nested domains Ω′⊂⊂Ω. 2. Track every constant through cutoff derivatives; this exposes dependence on the distance from Ω′ to the boundary. 3. Compare the proved estimate with the desired global claim and list the missing boundary geometry, trace data, or compatibility conditions. At every passage to a limit, record the uniform estimate, the topology of convergence, the nonlinear term that survives, and the hypothesis that prevents concentration or loss of boundary data.

Reconstruct “On Ω=(−1,1), let u(x)=|x|. State its membership in W^{1,∞}(Ω), C^{0,1}(Ω), and C^1(Ω), and explain which assertion is a regularity gain over W^{1,2}.” before using the general theorem. Then test the nearest failure: An interior estimate can blow up as Ω′ approaches ∂Ω. Smooth coefficients do not repair a reentrant corner, incompatible boundary data, or a solution branch that has not been shown to exist. This keeps regularity, existence, uniqueness, and numerical appearance as four separate claims.

A statement of interior Hölder regularity neither supplies boundary continuity nor proves existence or uniqueness unless those conclusions are separately established.

Identify the input class X, improved class Y, data class Z, and the nested domains Ω′⊂⊂Ω.

Track every constant through cutoff derivatives; this exposes dependence on the distance from Ω′ to the boundary.

Compare the proved estimate with the desired global claim and list the missing boundary geometry, trace data, or compatibility conditions.

On Ω=(−1,1), let u(x)=|x|. State its membership in W^{1,∞}(Ω), C^{0,1}(Ω), and C^1(Ω), and explain which assertion is a regularity gain over W^{1,2}.

  1. The weak derivative is sign(x) almost everywhere, bounded by one.
  2. The reverse triangle inequality gives ||x|−|y||≤|x−y|, so u is Lipschitz.
  3. The one-sided derivatives at zero are −1 and 1, so u is not C^1.

Result: The function lies in W^{1,∞} and C^{0,1}, hence also W^{1,2}, but not C^1; a Lipschitz conclusion is a genuine gain over an energy-only W^{1,2} assumption.