lemma A.466 The homogeneous Dirichlet problem is trivial

open in the book · appendices/A-long-proofs.tex:22928 · p. 3021

Rests on

depends_on equation A.764 eq:app-sl-completeness-operator ¶
depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
- depends_on definition A.462 The weighted space and the energy space ¶
  - depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
    - depends_on definition 10.85 Sobolev space ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition 10.83 Weak solution ¶
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-24 ¶
  - depends_on definition 16.75 Rayleigh quotient ¶
    - depends_on definition 9.54 Sturm–Liouville differential operator ¶
    - depends_on equation 16.42 eq:calcvar-sl-from-isoperimetric ¶
- depends_on lemma A.452 Young and Hölder ¶
  - depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶ ↺
  - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
    - depends_on proposition 7.30 Leibniz rule ¶
      - depends_on definition 7.12 Product of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-14 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-22 ¶
  - proves proof app:A-long-proofs@proof-268 ¶
- proves proof app:A-long-proofs@proof-274 ¶
proves proof app:A-long-proofs@proof-276 ¶

Supports

depends_on definition A.467 Green function of the shifted problem ¶
- depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
  - depends_on theorem A.470 $T$ is compact ¶
    - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
      - depends_on lemma A.472 The pairing identity ¶
- depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
  - depends_on lemma A.472 The pairing identity ¶ ↺
  - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶ ↺
depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:app-sl-completeness-operator	declared	appendices/A-long-proofs.tex:22932
`depends_on`	→	Poincaré inequality; $B_{K}$ is an inner product	declared	appendices/A-long-proofs.tex:22932
`depends_on`	←	Green function of the shifted problem	declared	appendices/A-long-proofs.tex:22970
`depends_on`	←	The Green operator inverts $L_{K}$	declared	appendices/A-long-proofs.tex:23039
`proves`	←	app:A-long-proofs@proof-276	declared	appendices/A-long-proofs.tex:22936