equation 10.2 eq:pde-multi-index

Rests on

No declared or derived dependency edges point away from this node yet.

Supports

• depends_on definition A.107 Majorant ¶
  • depends_on lemma A.109 Domination ¶
    • depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶
  • depends_on lemma A.108 The standard majorant ¶
    • depends_on proposition A.110 Explicit solution of the majorant problem ¶
      • depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶ ↺
    • depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶ ↺
• 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 corollary 10.56 Domain of dependence and influence ¶
        • depends_on theorem 10.59 Kirchhoff'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 corollary 10.60 Huygens' principle in three space dimensions ¶
      • 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.105 Formal reduction of a normal problem ¶
      • 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 theorem 10.80 Parabolic maximum principle ¶
      • 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 lemma A.90 Affine normalization ¶
        • depends_on proposition 10.13 Canonical form in two variables ¶ ↺
        • depends_on proposition 10.11 What a characteristic surface is ¶ ↺
  • 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 corollary 10.89 Existence for the elliptic problem ¶ ↺
        • depends_on proposition 10.91 Dirichlet principle ¶ ↺
    • depends_on definition 10.85 Sobolev space ¶
      • depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
        • depends_on definition A.462 The weighted space and the energy space ¶
        • depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
        • depends_on lemma A.452 Young and Hölder ¶
      • depends_on definition 10.87 Weak form of an elliptic problem ¶ ↺
    • 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 corollary A.152 Uniqueness ¶
      • 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 proposition 10.84 Consistency ¶
• depends_on lemma A.104 The recursion ¶
  • depends_on lemma A.109 Domination ¶ ↺
  • depends_on lemma A.105 Formal reduction of a normal problem ¶ ↺