definition A.467 Green function of the shifted problem

open in the book · appendices/A-long-proofs.tex:22946 · p. 3022

ground object -- no derivation owed

Rests on

depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
- depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
  - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
    - depends_on definition 9.2 Linear equation; homogeneity ¶
      - depends_on definition 9.1 Ordinary differential equation ¶
    - depends_on remark 9.3 Normal form and first-order systems ¶
    - depends_on theorem 9.8 Picard–Lindelöf ¶
      - depends_on definition 9.4 Lipschitz condition ¶
      - depends_on equation 9.3 eq:slt-ode-system ¶
      - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-3 ¶
  - depends_on definition A.438 Momentum variable and the accessory system ¶
    - depends_on equation 16.49 eq:calcvar-jacobi ¶
    - depends_on equation 9.3 eq:slt-ode-system ¶ ↺
  - proves proof app:A-long-proofs@proof-261 ¶
- depends_on lemma 9.56 Lagrange identity ¶
  - depends_on definition 9.54 Sturm–Liouville differential operator ¶
  - proves proof ch:07-odes-sturm-liouville@proof-28 ¶
- proves proof app:A-long-proofs@proof-264 ¶
depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
- 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 theorem 7.42 Fundamental theorem of calculus, I ¶
    - 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 theorem 7.34 Rolle ¶
      - 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 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 $L^{2}_{r}$ ¶ ↺

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`	→	The Wronskian of the accessory equation is constant	declared	appendices/A-long-proofs.tex:22970
`depends_on`	→	The homogeneous Dirichlet problem is trivial	declared	appendices/A-long-proofs.tex:22970
`depends_on`	←	The kernel is well defined, symmetric and Lipschitz	declared	appendices/A-long-proofs.tex:22986
`depends_on`	←	The Green operator inverts $L_{K}$	declared	appendices/A-long-proofs.tex:23039