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
- Integer-order Sobolev spaces and their norms
- Integer-order Sobolev extension from a half-space
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Open ball, closed ball and sphere in a metric space
- Bounded Cᵏ domains and boundary charts
- In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones
- Distinct points of a metric space have disjoint balls around them
- Independent generalized reflected-ray and GCC convention
- Bounded collar regularity for an exterior weak Dirichlet function
- Locally integrable functions as regular distributions
- Distributional derivative
- Weak Leibniz rule with a smooth factor
- Test function cutoffs and euclidean localization
- Spectral cutoff cores and unitary identifications of completed scales
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Banach space
- Topological vector spaces over the real and complex fields
- The dual space X^* of a normed space and its dual norm
- Every nonzero vector has a norming functional
- Hadamard three-lines theorem
- Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions
- Bochner spaces and bounded inclusion of value spaces
- Distributional time derivatives with values in a Banach space
- The essential supremum of a measurable function with respect to a measure
- The essential supremum is attained as the least essential bound
- Holder's inequality for integrals, including the endpoint cases
- Strongly measurable Banach-valued function
- The Axiom of Countable Choice (AC_ω)
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- Cᵏ boundary flattening preserves local W^k,p
- One flat reflection with simultaneous Lq and W1q bounds
- Compactly supported Sobolev functions extend by zero in every integer order
- A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions
- A C¹ diffeomorphism maps Lebesgue null sets to Lebesgue null sets
- A C¹ diffeomorphism maps Lebesgue measurable sets to Lebesgue measurable 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
One second-order flat reflection on L² and H²
Definition
Exterior and truncated obstacle domains
Lemma
Zero extension of a free exterior tail by distribution tests
Lemma
Admissible spectral strip profiles and the exact interpolation upper bound
Lemma
Finite-time bounds for Banach-valued derivatives
Lemma