definition 10.3 Principal part and principal symbol

open in the book · parts/02-mathematical-methods/08-pdes.tex:101 · p. 340

ground object -- no derivation owed

Rests on

depends_on definition 10.2 Linear, semilinear, quasilinear ¶
- 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 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 5.15 Basis ¶
  - 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 equation 10.2 eq:pde-multi-index ¶

Supports

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 corollary 10.60 Huygens' principle in three space dimensions ¶
- 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.104 The recursion ¶
      - depends_on proposition A.110 Explicit solution of the majorant problem ¶
    - 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 corollary 10.81 Uniqueness for the heat equation ¶
      - depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
  - 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 lemma A.91 Smallness after rescaling ¶
    - 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.463 Poincaré inequality; $B_{K}$ is an inner product ¶
    - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
    - depends_on lemma A.452 Young and Hölder ¶
      - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶ ↺
      - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶ ↺
  - 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 A.150 Kruzhkov entropy pair ¶
      - depends_on proposition A.156 The Kruzhkov inequality ¶
      - depends_on proposition A.163 The viscous problem produces the entropy inequality ¶ ↺
      - depends_on proposition 10.102 Jump form of the entropy inequality ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
      - depends_on theorem 10.103 Uniqueness in the entropy class ¶
  - 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 proposition 10.84 Consistency ¶

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`	→	Linear, semilinear, quasilinear	declared	parts/02-mathematical-methods/08-pdes.tex:111
`depends_on`	→	eq:pde-multi-index	declared	parts/02-mathematical-methods/08-pdes.tex:111
`depends_on`	←	Characteristic surface	declared	parts/02-mathematical-methods/08-pdes.tex:235
`depends_on`	←	The second-order operator	declared	parts/02-mathematical-methods/08-pdes.tex:130
`depends_on`	←	Weak solution	declared	parts/02-mathematical-methods/08-pdes.tex:2256