definition 10.1 Partial differential equation; order

open in the book · parts/02-mathematical-methods/08-pdes.tex:52 · p. 339

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 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 ¶

Supports

depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
- depends_on definition A.103 First-order quasilinear system with zero data ¶
  - depends_on lemma A.104 The recursion ¶
    - depends_on lemma A.109 Domination ¶
      - depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶
    - depends_on lemma A.105 Formal reduction of a normal problem ¶
  - depends_on proposition A.110 Explicit solution of the majorant problem ¶
    - depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶ ↺
- depends_on lemma A.105 Formal reduction of a normal problem ¶ ↺
depends_on definition 10.2 Linear, semilinear, quasilinear ¶
- depends_on definition A.103 First-order quasilinear system with zero data ¶ ↺
- depends_on definition 10.93 Scalar conservation law ¶
  - depends_on definition A.150 Kruzhkov entropy pair ¶
    - depends_on lemma A.157 Two elementary identities ¶
      - depends_on proposition A.159 The doubled inequality ¶
      - depends_on proposition A.162 The limit $\delta\to0$ ¶
    - depends_on proposition A.156 The Kruzhkov inequality ¶
      - depends_on proposition A.159 The doubled inequality ¶ ↺
    - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
      - depends_on corollary A.152 Uniqueness ¶
  - depends_on definition 10.95 Weak solution of a conservation law ¶
    - depends_on definition 10.99 Entropy condition ¶
      - depends_on definition 10.101 Entropy pair; entropy inequality ¶
    - depends_on definition 10.101 Entropy pair; entropy inequality ¶ ↺
    - depends_on example 10.98 Weak solutions are not unique ¶
    - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
    - depends_on theorem 10.103 Uniqueness in the entropy class ¶
    - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
      - depends_on definition 10.99 Entropy condition ¶ ↺
      - depends_on example 10.97 Burgers shock ¶
      - depends_on example 10.98 Weak solutions are not unique ¶ ↺
      - depends_on proposition 10.102 Jump form of the entropy inequality ¶
  - depends_on example 10.97 Burgers shock ¶ ↺
  - depends_on proposition 10.94 Gradient catastrophe ¶
- depends_on definition 10.3 Principal part and principal symbol ¶
  - depends_on definition 10.10 Characteristic surface ¶
    - depends_on proposition 10.13 Canonical form in two variables ¶
      - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
      - depends_on theorem 10.55 d'Alembert's formula ¶
    - depends_on proposition 10.11 What a characteristic surface is ¶
      - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶ ↺
      - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
  - depends_on definition 10.4 The second-order operator ¶
    - depends_on definition 10.10 Characteristic surface ¶ ↺
    - depends_on definition 10.42 Green's function ¶
      - depends_on definition 10.50 Retarded and advanced Green's functions ¶
      - depends_on example 10.49 The method of images ¶
      - depends_on proposition 28.24 Impulse response of the damped oscillator ¶
      - depends_on proposition 10.47 Bilinear expansion ¶
      - depends_on proposition 10.43 Representation of the solution ¶
      - depends_on theorem 10.45 Symmetry of the Green's function ¶
    - depends_on definition 10.6 Type of a second-order operator ¶
      - depends_on definition 10.79 Parabolic boundary ¶
      - depends_on example 10.9 ex:pde-three-types ¶
      - depends_on proposition 10.25 The backward heat problem is ill posed ¶
      - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶ ↺
      - depends_on theorem 10.8 The type is a coordinate invariant ¶
  - depends_on definition 10.83 Weak solution ¶
    - depends_on definition 10.87 Weak form of an elliptic problem ¶
      - depends_on corollary 10.89 Existence for the elliptic problem ¶
      - depends_on proposition 10.91 Dirichlet principle ¶
      - depends_on theorem 10.88 Lax–Milgram ¶
    - depends_on definition 10.85 Sobolev space ¶
      - depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
      - depends_on definition 10.87 Weak form of an elliptic problem ¶ ↺
    - depends_on definition 10.95 Weak solution of a conservation law ¶ ↺
    - depends_on proposition 10.84 Consistency ¶
- depends_on lemma 10.28 The separation constant ¶
  - depends_on definition A.118 Simple separation of the Helmholtz equation ¶
    - depends_on theorem A.120 Stäckel and Robertson conditions ¶
      - depends_on example A.122 Cartesian ¶
      - depends_on example A.123 Circular cylindrical ¶
      - depends_on example A.124 Spherical ¶
      - depends_on proposition A.128 The ellipsoidal system separates ¶
  - depends_on example 10.31 Cartesian separation ¶
    - depends_on example A.122 Cartesian ¶ ↺
  - depends_on example 10.32 Cylindrical separation ¶
    - depends_on example A.123 Circular cylindrical ¶ ↺
  - depends_on example 10.33 Spherical separation ¶
    - depends_on example A.124 Spherical ¶ ↺
  - depends_on proposition 23.9 Separation of the time ¶
    - depends_on proposition 23.10 The characteristic function is the abbreviated action ¶
      - depends_on remark 23.11 Whose principle, and where it fails ¶
    - depends_on remark 23.12 Two functions, two roles ¶
  - depends_on proposition 10.29 Time separation reduces the canonical equations to Helmholtz ¶
    - depends_on example 10.38 The vibrating string ¶
    - depends_on phenomenon 28.53 Chladni's figures ¶
    - depends_on theorem 10.36 Solution of the heat problem on an interval ¶
  - depends_on theorem 10.34 Separable systems for the Helmholtz operator ¶
    - depends_on remark 23.19 The same conditions govern the Helmholtz equation ¶
- depends_on proposition 28.44 Superposition ¶
- depends_on proposition 10.15 Method of characteristics, first order ¶
  - depends_on definition 10.93 Scalar conservation law ¶ ↺
  - depends_on proposition 10.16 Cauchy's characteristic strips ¶
  - depends_on proposition 10.94 Gradient catastrophe ¶ ↺
  - depends_on remark 31.90 Why a compression wave must steepen ¶
  - depends_on remark 23.3 What Part II owes this chapter ¶
- depends_on proposition 10.43 Representation of the solution ¶ ↺
- depends_on theorem 10.22 Cauchy–Kovalevskaya ¶ ↺
- depends_on theorem 10.52 Duhamel's principle ¶
  - depends_on corollary 10.53 Duhamel for the wave equation ¶
depends_on definition 10.24 Well-posed problem ¶
- 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 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

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Edges

type	direction	node	provenance	where
`depends_on`	→	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/08-pdes.tex:72
`depends_on`	→	Open set	declared	parts/02-mathematical-methods/08-pdes.tex:72
`depends_on`	←	Normal Cauchy problem of order $k$	declared	appendices/A-long-proofs.tex:6121
`depends_on`	←	Linear, semilinear, quasilinear	declared	parts/02-mathematical-methods/08-pdes.tex:90
`depends_on`	←	Well-posed problem	declared	parts/02-mathematical-methods/08-pdes.tex:661