Alphabeta MathPersonal research
Exterior geometry and function spaces
Specify the obstacle and generalized-ray conventions, Sobolev and time-space notation, Dirichlet realization and spectral scales; connect bounded fractional inputs to exterior localization, traces, compactness and continuation.
1 · Prerequisites
- The Axiom of Choice
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Integer-order Sobolev spaces and their norms
- The notation Hᵏ and the reserved zero-boundary symbol
- Local weak solutions of a divergence-form operator
- Real analytic germs in several variables
- Interior Hᵏ⁺² elliptic regularity
- Bounded Cᵏ domains admit integer-order Sobolev extension
- Higher-order Sobolev embedding
- A product of connected spaces is connected in the product topology, and that argument is a theorem of ZF; for an infinite index set it is the assertion that the product of nonempty spaces is nonempty that uses the Axiom of Choice
- Harmonic functions are real analytic
- Unique continuation for harmonic functions
- The Laplacian of a C² function and of a C² vector field
- A subset of ℝ is connected if and only if it is order-convex, that is, an interval
- Classical derivatives agree with weak derivatives
- The real exponential function and the number e by a power series
- The exponential function is smooth and (exp)'=exp
- The derivatives of sine and cosine are cosine and minus sine
- The chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f ∘ g is differentiable at c with (f ∘ g)'(c) = f'(g(c)) g'(c)
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- Cᵏ Euclidean maps are closed under componentwise algebra and composition
- Zero-boundary Sobolev space as a norm closure
- The Dirichlet Laplacian generates an analytic heat semigroup
- Semibounded closed-form representation and perturbation
- Integer-order Sobolev spaces are Banach
- Smooth compactly supported functions of an open set are dense in L²
- L² with the integral pairing is a Hilbert space
- Distributional derivative
- Weak derivatives are represented distributional derivatives
- AC supplies the countable and dependent choices used in Banach integration
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Closed sectorial form and its associated operator
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Unbounded Borel functional calculus: domains, products, spectral mapping
- The unbounded PVM integral is densely defined, closed and normal
- Spectral form domain and core of a semibounded operator
- Test function cutoffs and euclidean localization
- Weak Leibniz rule with a smooth factor
- Global H² Dirichlet regularity
- Locally integrable functions as regular distributions
- Bounded Cᵏ domains and boundary charts
- Weak Dirichlet solutions for a divergence-form operator
- Uniformly elliptic divergence-form operators and their sesquilinear forms
- Distributional derivatives are polynomial Fourier multipliers
- Plancherel theorem
- Integer-order W^k,2 and Hᵏ agree with equivalent norms
- Polynomial growth functions define tempered distributions
- Fourier transform agrees with l one and plancherel transforms
- Smooth polynomially bounded multipliers on schwartz space
- Locally integrable functions embed in distributions
- Weighted tempered-distribution characterization of Hˢ
- Tempered distributions embed continuously in distributions
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Lemma
Helmholtz continuation through a harmonic lift
Lemma
Weak shifted Dirichlet form and energy bound on an open set
Lemma
Bounded collar regularity for an exterior weak Dirichlet function
Lemma