Alphabeta MathPartial Differential Equations
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
6 pagesThis page constructs the real-order Bessel-potential spaces by completing Schwartz space in the weighted Fourier norm.
2 definitions, 3 lemmas, 2 theorems, 1 corollaryExamples & counterexamples →Surface integration is constructed from the Gram density of regular Euclidean charts and a finite ambient partition.
4 definitions, 7 lemmas, 2 theorems, 2 corollaries, 1 remarkExamples & counterexamples →This page develops the viscosity theory of first-order Hamilton--Jacobi equations from the test-function definition.
7 definitions, 10 lemmas, 2 propositions, 9 theorems, 4 corollaries, 1 remarkExamples & counterexamples →This page introduces PDE language in the scalar classical setting, isolates the principal symbol as the coordinate-stable part of the highest-order operator, and uses it to…
8 definitions, 4 lemmas, 6 theorems, 1 corollary, 2 remarksExamples & counterexamples →This page develops the theory of strongly continuous (C 0) semigroups on Banach spaces from the definition through the generation theorems, with the sign convention u'=Au and the resolvent normalised as R(λ,A)=(λ I-A)⁻¹.
6 definitions, 14 lemmas, 9 theorems, 3 corollaries, 1 remarkExamples & counterexamples →- Weak Derivatives and Sobolev Spaces28 results
This page develops weak differentiation of locally integrable functions on Euclidean domains.
5 definitions, 12 lemmas, 6 theorems, 4 corollaries, 1 remarkExamples & counterexamples →
Level 1
3 pagesAn analytic Cauchy problem has two separate tasks: its equation must determine the Taylor coefficients, and the resulting formal series must converge.
2 definitions, 9 lemmas, 3 theorems, 1 remarkExamples & counterexamples →- Harmonic Functions and Mean Values in Rn16 results
Local averages characterize harmonicity and lead, through mollification, to Weyl regularity on arbitrary open subsets of ℝⁿ.
3 definitions, 5 lemmas, 4 theorems, 3 corollaries, 1 remarkExamples & counterexamples → This page develops the approximation and extension theory of integer-order Sobolev spaces.
3 definitions, 4 lemmas, 5 theorems, 2 corollaries, 2 remarksExamples & counterexamples →
Level 2
3 pagesClassical maximum and comparison principles distinguish the Laplacian sign, connectedness and boundary assumptions.
2 definitions, 6 lemmas, 9 theorems, 8 corollaries, 1 remarkExamples & counterexamples →Starting with a quasilinear Cauchy problem, this page separates ODE solvability from the inverse-projection step that actually constructs a PDE graph.
5 definitions, 10 lemmas, 4 theorems, 1 corollary, 2 remarksExamples & counterexamples →- Sobolev Traces and Zero Boundary Values21 results
This page develops the first-order trace theory of Sobolev spaces on bounded C¹ domains and identifies the zero-boundary space as the kernel of the trace.
2 definitions, 10 lemmas, 7 theorems, 1 corollary, 1 remarkExamples & counterexamples →
Level 3
1 pageThis page fixes the sign convention -ΔΦ=δ 0 and the normalized kernel Φ(x)=|x|²⁻ⁿ/((n-2)ω n-1) for n≥3, Φ(x)=-(2π)⁻¹log|x| for n=2, proves the distributional Dirac identity…
5 definitions, 7 lemmas, 6 theorems, 3 corollaries, 1 remarkExamples & counterexamples →
Level 4
1 pageThis page solves the Dirichlet problem on balls and half-spaces by explicit Poisson kernels and then develops the interior regularity estimates that follow from the Poisson representation.
1 definition, 6 lemmas, 8 theorems, 5 corollaries, 1 remarkExamples & counterexamples →
Level 5
3 pages- Sobolev Poincare and Morrey Inequalities26 results
This page develops the first-order Sobolev inequalities and the Poincare and Morrey inequalities on Euclidean domains.
2 definitions, 7 lemmas, 12 theorems, 3 corollaries, 2 remarksExamples & counterexamples → - The Heat Kernel and the Cauchy Problem20 results
This page defines the heat operator ∂ t-Δ x, its classical solutions and the Cauchy problem with data on the time-zero slice, and then constructs the Gaussian heat kernel…
3 definitions, 6 lemmas, 7 theorems, 3 corollaries, 1 remarkExamples & counterexamples → - Wave Equation Representation Formulas28 results
This page develops the classical representation formulas for the Cauchy problem of the wave equation Box cu=∂ t²u-c²Δ u=0 in ℝⁿ.
2 definitions, 13 lemmas, 9 theorems, 3 corollaries, 1 remarkExamples & counterexamples →
Level 6
3 pagesThe page starts from the parabolic cylinder and its parabolic boundary, the class C^2,1(Q̄), and the two elementary facts behind every maximum argument: a negative…
4 definitions, 10 lemmas, 11 theorems, 4 corollaries, 2 remarksExamples & counterexamples →This page proves the compact embedding theorems behind existence theory for PDEs.
1 definition, 8 lemmas, 12 theorems, 5 corollaries, 1 remarkExamples & counterexamples →This page develops the energy method for the wave equation Box cu=∂ t²u-c²Δ u=f in ℝⁿ and uses it to prove uniqueness, continuous dependence, finite speed of propagation and…
3 definitions, 5 lemmas, 7 theorems, 3 corollaries, 1 remarkExamples & counterexamples →
Level 7
2 pages- Lax--Milgram and Weak Elliptic Solutions28 results
This page builds the abstract machinery of the Lax--Milgram theorem and applies it to weak elliptic Dirichlet problems.
4 definitions, 13 lemmas, 5 theorems, 5 corollaries, 1 remarkExamples & counterexamples → This page develops the theory of scalar conservation laws u t+div x f(u)=0 from the definitions through existence, uniqueness and wave structure, with the vanishing-viscosity method as its backbone.
6 definitions, 6 lemmas, 4 propositions, 9 theorems, 6 corollariesExamples & counterexamples →
Level 8
2 pagesThis page builds the elliptic Fredholm theory and the spectral theory of a symmetric divergence-form Dirichlet operator.
4 definitions, 9 lemmas, 9 theorems, 6 corollaries, 1 remarkExamples & counterexamples →- Schauder and Lᵖ Elliptic Estimates22 results
This page develops the two classical second-order regularity theories for uniformly elliptic equations in nondivergence form and their interaction: the Schauder scale C^2,α and the Sobolev scale W^2,p.
2 definitions, 7 lemmas, 10 theorems, 2 corollaries, 1 remarkExamples & counterexamples →
Level 9
1 pageThis page develops the Sobolev regularity theory of second-order divergence-form elliptic equations: interior H² and Hᵏ⁺² estimates, the boundary theory of the Dirichlet…
2 definitions, 13 lemmas, 7 theorems, 5 corollaries, 2 remarksExamples & counterexamples →
Level 10
3 pagesThis page develops analytic semigroups from the complex-analytic calculus that produces them through the PDE realisations that motivate them.
3 definitions, 12 lemmas, 6 theorems, 4 corollaries, 2 remarksExamples & counterexamples →This page develops the direct method of the calculus of variations and the Euler--Lagrange equation for integral functionals, first on an abstract normed space and then on the Sobolev spaces attached to a bounded C¹ domain.
3 definitions, 13 lemmas, 7 theorems, 3 corollaries, 1 remarkExamples & counterexamples →This page develops the weak maximum principle for coercive divergence-form equations with bounded measurable coefficients, the De Giorgi level-set iteration that proves it…
1 definition, 11 lemmas, 6 theorems, 2 corollaries, 2 remarksExamples & counterexamples →
Level 11
1 pageThis page develops variational problems whose admissible set is constrained in one of the two ways that dominate the calculus of variations: by a closed convex set, which…
1 definition, 12 lemmas, 10 theorems, 2 corollaries, 1 remarkExamples & counterexamples →