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
- One second-order flat reflection on L² and H²
- Bounded Cᵏ domains admit integer-order Sobolev extension
- Test function cutoffs and euclidean localization
- Cᵏ boundary flattening preserves local W^k,p
- Weak Leibniz rule with a smooth factor
- Compactly supported Sobolev functions extend by zero in every integer order
- Integer-order W^k,2 and Hᵏ agree with equivalent norms
- Bounded Cᵏ domains and boundary charts
- Integer-order Sobolev spaces and their norms
- Bounded restriction and cutoff localisation in Sobolev spaces
- Real-order Bessel-potential completion Hˢ
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- Exterior and truncated obstacle domains
- Independent generalized reflected-ray and GCC convention
- Compatible compact Banach triples
- Bochner spaces and bounded inclusion of value spaces
- Distributional time derivatives with values in a Banach space
- Finite-time bounds for Banach-valued derivatives
- The L2 pivot injects into the local negative Sobolev space
- The Axiom of Countable Choice (AC_ω)
- Restriction Bessel-potential spaces on polyhedral domains
- Every real-order Bessel-potential completion is Hilbert
- A quotient of a Banach space by a closed subspace is Banach
- L² with the integral pairing is a Hilbert space
- Jacques Simon, Compact sets in the space Lᵖ(0,T;B), Annali di Matematica Pura ed Applicata 146 (1987), 65–96 · §8, embedding assumption (8.1), printed p.84, and Corollary 4 first clause, printed p.85; scanned PDF leaf 11 (zero-based page 10)
- Weak shifted Dirichlet form and energy bound on an open set
- Bounded collar regularity for an exterior weak Dirichlet function
- Zero extension of a free exterior tail by distribution tests
- Whole-space H² regularity from an L² distributional Laplacian
- Zero-boundary Sobolev space as a norm closure
- Admissible spectral strip profiles and the exact interpolation upper bound
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Hadamard three-lines theorem
- Hilbert space
- Symmetric, self-adjoint and essentially self-adjoint operators
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Uniform endpoint localization in a smooth boundary chart
- Finite ambient partitions near compact sets
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
A compactly supported common L²/H² extension from a bounded smooth domain
Definition
Non-trapping obstacle
Theorem
General Banach-triple Aubin–Lions–Simon compactness
Lemma
Collar and tail assembly of the exterior graph domain
Lemma
Spectral cutoff dual tests and the exact interpolation lower bound
Lemma
A single collar-reflection and free-tail exterior extension
Lemma