Alphabeta MathPersonal research
Supporting prerequisites · 01 — The Dirichlet Laplacian generates an analytic heat semigroup
Presents 15 supporting prerequisites, covering published library prerequisites and dependency levels 0–1.
1 · Prerequisites
- The L² operator associated with a symmetric elliptic form
- Zero-boundary Sobolev space as a norm closure
- Hilbert space
- The associated elliptic operator is densely defined, symmetric and lower bounded
- The symmetric elliptic form operator is self-adjoint with compact resolvent
- Adjoint of a densely defined operator
- The adjoint is well defined, closed, and reverses inclusions
- Self-adjoint nonpositive operators generate bounded analytic semigroups
- Sectorial operator with the semigroup sign convention
- Complex sector and bounded analytic semigroup
- Semigroup sign and generator conventions
- Quadratic spectral bounds control a self-adjoint parabolic semigroup
- The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction
- The Poincare constant is the reciprocal square root of the first Dirichlet eigenvalue
- The Rayleigh principle for the first Dirichlet eigenvalue
- Discrete spectrum of a symmetric elliptic Dirichlet operator
- The Axiom of Countable Choice (AC_ω)
- The sharp Dirichlet Poincare inequality on an interval
- Global H² Dirichlet regularity
- Higher-order boundary regularity for Dirichlet problems
- Integer-order Sobolev spaces and their norms
- Smoothing estimates for the semigroup generated by a sectorial operator
- The generator of the contour semigroup is the sectorial operator
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Divergence on a bounded C1 Euclidean domain
- Divergence for finite piecewise C1 presentations
- Classical normal derivative
- 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
- The space Lᵖ(μ) as the quotient by null functions
- Weak derivative of a locally integrable function
- Higher-order Sobolev embedding
- Meyers–Serrin density on an arbitrary open set
- Holder's inequality for integrals, including the endpoint cases
- Integer-order Sobolev spaces are Banach
- Weak partial derivatives lower the Sobolev order
- Bounded Cᵏ domains admit integer-order Sobolev extension
- Classical derivatives agree with weak derivatives
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Harmonic functions are real analytic
- Choice function
- Complex norm and complex normed space
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Complete metric space: every Cauchy sequence converges in the space
- Realification preserves a complex norm and its metric
- Strongly measurable Banach-valued function
- The Banach-valued simple integral is well defined
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Real and complex inner product spaces, with the inner product linear in the first argument
- Linear map between vector spaces over the same field
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Finite, countably infinite, countable, uncountable
- Open cover, subcover, compact metric space, and compact subset of a metric space
- The essential supremum of a measurable function with respect to a measure
- The class L¹(μ) of integrable functions
- Real and imaginary parts, complex conjugation, and modulus
- The spaces C_c(ℝⁿ) and C_c^∞(ℝⁿ)
- The space C₀(ℝⁿ) of continuous functions vanishing at infinity
- Arithmetic and lattice operations preserve measurability whenever they are defined
- Threshold characterisations of real-valued and extended-real-valued measurability
- The Dirac set function at a point
- A Dirac set function is a probability measure
- Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0
- Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable
- 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
- Integrable real and complex functions, and their integrals
- Linearity, locality, and commutation of weak derivatives
- Uniqueness of a weak derivative as an almost-everywhere class
- The Sobolev norm descends to equivalence classes
- Normed subspace
- Independent generalized reflected-ray and GCC convention
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on Y^X and on C(X,Y)
- A sequence in a metric space has at most one limit
- Continuity of a map of topological spaces at a point and globally
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
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
Corollary
The Dirichlet Laplacian generates an analytic heat semigroup
Corollary
First Green identity
Corollary
The Sobolev space W^k,p is an algebra above the critical index
Corollary
Unique continuation for harmonic functions
Definition
The Axiom of Choice
Definition
Banach space
Definition
Bochner-integrable function
Definition
Bounded Cᵏ domains and boundary charts
Definition
Bounded C1 domains and their outward normals
Definition
A bounded linear operator between normed spaces
Definition
Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
Definition
Complex Lp classes and Euclidean test-function conventions
Definition
Sobolev extension domains and extension operators
Lemma
Riesz lemma
Lemma