definition 10.24 Well-posed problem

open in the book · parts/02-mathematical-methods/08-pdes.tex:649 · p. 346

ground object -- no derivation owed

Rests on

depends_on definition 10.1 Partial differential equation; order ¶
- 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 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 definition 6.24 Metric ¶
- depends_on definition 3.43 Map ¶
  - depends_on definition 3.24 Quantifiers ¶
    - depends_on definition 3.23 Predicate ¶
      - depends_on definition 3.1 Logical proposition ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on definition 3.28 Set ¶
- depends_on definition 3.38 Ordered pair and Cartesian product ¶ ↺

Supports

depends_on corollary A.152 Uniqueness ¶
depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
depends_on corollary 10.81 Uniqueness for the heat equation ¶
depends_on definition 30.22 Boundary conditions of elastostatics ¶
- depends_on proposition 30.50 Love waves need a slow surface layer ¶
- depends_on proposition 30.48 Rayleigh's secular equation ¶
  - depends_on proposition 30.49 The Rayleigh speed of a Poisson solid ¶
    - depends_on phenomenon 30.47 The three seismic phases ¶
- depends_on proposition 30.23 Virtual work; the weak form ¶
  - depends_on phenomenon 30.70 Hertzian contact ¶
    - depends_on definition A.698 The Hertz contact problem ¶
      - depends_on lemma A.700 The displacement produced by an elliptic pressure ¶
      - depends_on proposition A.701 The two-body compliance rule ¶
      - depends_on remark A.699 The third condition is not decoration ¶
      - depends_on theorem A.702 Hertz's solution ¶
    - depends_on example A.703 The numbers of the chapter, recomputed ¶
    - depends_on example 30.71 Contact stresses are enormous ¶
  - depends_on proposition 30.34 Minimum of the potential energy ¶
    - depends_on example 30.35 From single crystals to the isotropic aggregate ¶
  - depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶
depends_on proposition 10.25 The backward heat problem is ill posed ¶
depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
- depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶

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`	→	Partial differential equation; order	declared	parts/02-mathematical-methods/08-pdes.tex:661
`depends_on`	→	Metric	declared	parts/02-mathematical-methods/08-pdes.tex:661
`depends_on`	←	Uniqueness	declared	appendices/A-long-proofs.tex:8123
`depends_on`	←	Uniqueness and stability for the Dirichlet problem	declared	parts/02-mathematical-methods/08-pdes.tex:2084
`depends_on`	←	Uniqueness for the heat equation	declared	parts/02-mathematical-methods/08-pdes.tex:2203
`depends_on`	←	Boundary conditions of elastostatics	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:703
`depends_on`	←	The backward heat problem is ill posed	declared	parts/02-mathematical-methods/08-pdes.tex:680
`depends_on`	←	Local existence and uniqueness; Choquet-Bruhat; imported	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1908