theorem 10.77 Strong maximum principle for harmonic functions

open in the book · parts/02-mathematical-methods/08-pdes.tex:2096 · p. 362

Rests on

depends_on definition 6.13 Connected space ¶
- depends_on definition 3.36 Disjoint sets ¶
- 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 theorem 10.69 Mean-value property ¶
- 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 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 5.15 Basis ¶
  - depends_on equation 10.24 eq:pde-poisson-equation ¶
- 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 ch:08-pdes@proof-32 ¶

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type	direction	node	provenance	where
`depends_on`	→	Connected space	declared	parts/02-mathematical-methods/08-pdes.tex:2101
`depends_on`	→	Mean-value property	declared	parts/02-mathematical-methods/08-pdes.tex:2101
`proves`	←	ch:08-pdes@proof-32	declared	parts/02-mathematical-methods/08-pdes.tex:2104