theorem A.135 Converse of the mean-value property

open in the book · appendices/A-long-proofs.tex:7544 · p. 2860

Rests on

depends_on definition A.134 Mean-value property ¶
- depends_on definition 6.2 Open set ¶
  - depends_on definition 6.1 Topological space ¶
    - depends_on definition 3.31 Empty set ¶
    - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.8 Conjunction ¶
      - depends_on definition 3.9 Disjunction ¶
      - depends_on definition 3.7 Negation ¶
    - depends_on definition 3.30 Subset ¶
    - depends_on equation 3.51 eq:set-indexed ¶
- depends_on equation 10.54 eq:pde-spherical-mean ¶
depends_on definition 10.66 Harmonic function ¶
- depends_on definition 7.98 Functions of class $C^{1}$ ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
- depends_on equation 10.24 eq:pde-poisson-equation ¶
depends_on theorem 10.69 Mean-value property ¶
- depends_on definition 10.66 Harmonic function ¶ ↺
- depends_on lemma 10.58 Darboux's equation for spherical means ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
  - depends_on theorem 7.133 Gauss ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-79 ¶
  - proves proof ch:08-pdes@proof-22 ¶
- proves proof ch:08-pdes@proof-28 ¶
- proves proof ch:08-pdes@prooflink-4 ¶
proves proof app:A-long-proofs@proof-88 ¶

Supports

depends_on corollary A.146 Locally uniform limits of harmonic functions ¶
depends_on corollary A.145 Harmonic functions are smooth ¶

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`	→	Mean-value property	declared	appendices/A-long-proofs.tex:7552
`depends_on`	→	Harmonic function	declared	appendices/A-long-proofs.tex:7552
`depends_on`	→	Mean-value property	declared	appendices/A-long-proofs.tex:7552
`depends_on`	←	Locally uniform limits of harmonic functions	declared	appendices/A-long-proofs.tex:7973
`depends_on`	←	Harmonic functions are smooth	declared	appendices/A-long-proofs.tex:7959
`proves`	←	app:A-long-proofs@proof-88	declared	appendices/A-long-proofs.tex:7913