theorem 10.75 Weak maximum principle

open in the book · parts/02-mathematical-methods/08-pdes.tex:2033 · p. 361

Rests on

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 6.11 Heine–Borel on $\R$ ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on equation 3.51 eq:set-indexed ¶
- proves proof ch:04-topology@proof-2 ¶
proves proof ch:08-pdes@proof-30 ¶

Supports

depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
depends_on theorem A.525 Helmholtz decomposition: uniqueness ¶
- depends_on remark A.527 The hypotheses, and what happens without them ¶
depends_on theorem 10.80 Parabolic maximum principle ¶
- depends_on corollary 10.81 Uniqueness for the heat equation ¶
- depends_on proposition A.163 The viscous problem produces the entropy inequality ¶

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`	→	Harmonic function	declared	parts/02-mathematical-methods/08-pdes.tex:2042
`depends_on`	→	Heine–Borel on $\R$	declared	parts/02-mathematical-methods/08-pdes.tex:2042
`depends_on`	←	Uniqueness and stability for the Dirichlet problem	declared	parts/02-mathematical-methods/08-pdes.tex:2084
`depends_on`	←	Helmholtz decomposition: uniqueness	declared	appendices/A-long-proofs.tex:25662
`depends_on`	←	Parabolic maximum principle	declared	parts/02-mathematical-methods/08-pdes.tex:2169
`proves`	←	ch:08-pdes@proof-30	declared	parts/02-mathematical-methods/08-pdes.tex:2045