definition 10.66 Harmonic function

open in the book · parts/02-mathematical-methods/08-pdes.tex:1783 · p. 359

ground object -- no derivation owed

Rests on

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 definition 3.43 Map ¶
  - 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
depends_on equation 10.24 eq:pde-poisson-equation ¶

Supports

depends_on theorem A.135 Converse of the mean-value property ¶
- depends_on corollary A.146 Locally uniform limits of harmonic functions ¶
- depends_on corollary A.145 Harmonic functions are smooth ¶
depends_on theorem A.491 Dirichlet problem for the disc; quoted ¶
depends_on theorem 10.69 Mean-value property ¶
- depends_on corollary A.145 Harmonic functions are smooth ¶ ↺
- depends_on theorem A.135 Converse of the mean-value property ¶ ↺
- depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
depends_on theorem 10.75 Weak maximum principle ¶
- 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`	→	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/08-pdes.tex:1787
`depends_on`	→	eq:pde-poisson-equation	declared	parts/02-mathematical-methods/08-pdes.tex:1787
`depends_on`	←	Converse of the mean-value property	declared	appendices/A-long-proofs.tex:7552
`depends_on`	←	Dirichlet problem for the disc; quoted	declared	appendices/A-long-proofs.tex:24110
`depends_on`	←	Mean-value property	declared	parts/02-mathematical-methods/08-pdes.tex:1857
`depends_on`	←	Weak maximum principle	declared	parts/02-mathematical-methods/08-pdes.tex:2042