theorem 10.80 Parabolic maximum principle

open in the book · parts/02-mathematical-methods/08-pdes.tex:2161 · p. 363

Rests on

depends_on definition 10.79 Parabolic boundary ¶
- depends_on definition 10.6 Type of a second-order operator ¶
  - depends_on definition 10.4 The second-order operator ¶
    - depends_on definition 10.3 Principal part and principal symbol ¶
      - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
      - depends_on equation 10.2 eq:pde-multi-index ¶
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-64 ¶
  - depends_on theorem 5.116 Sylvester's law of inertia ¶
    - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    - proves proof ch:03-linear-algebra-representations@proof-53 ¶
- depends_on definition 6.3 Closed 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 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 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
depends_on theorem 10.75 Weak maximum principle ¶
- 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 7.97 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 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 equation 3.51 eq:set-indexed ¶ ↺
  - proves proof ch:04-topology@proof-2 ¶
- proves proof ch:08-pdes@proof-30 ¶
proves proof ch:08-pdes@proof-33 ¶

Supports

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`	→	Parabolic boundary	declared	parts/02-mathematical-methods/08-pdes.tex:2169
`depends_on`	→	Weak maximum principle	declared	parts/02-mathematical-methods/08-pdes.tex:2169
`depends_on`	←	Uniqueness for the heat equation	declared	parts/02-mathematical-methods/08-pdes.tex:2203
`depends_on`	←	The viscous problem produces the entropy inequality	declared	appendices/A-long-proofs.tex:8677
`proves`	←	ch:08-pdes@proof-33	declared	parts/02-mathematical-methods/08-pdes.tex:2172