definition 10.4 The second-order operator

open in the book · parts/02-mathematical-methods/08-pdes.tex:122 · p. 340

ground object -- no derivation owed

Rests on

depends_on definition 10.3 Principal part and principal symbol ¶
- 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 7.97 Partial derivative; gradient ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
- depends_on equation 10.2 eq:pde-multi-index ¶
depends_on proposition 7.105 Clairaut–Schwarz ¶
- depends_on definition 7.20 Continuity at a point ¶ ↺
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on proposition 7.29 Linearity ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:05-real-analysis@proof-64 ¶

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 10.42 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 ¶ ↺

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`	→	Principal part and principal symbol	declared	parts/02-mathematical-methods/08-pdes.tex:130
`depends_on`	→	Clairaut–Schwarz	declared	parts/02-mathematical-methods/08-pdes.tex:130
`depends_on`	←	Characteristic surface	declared	parts/02-mathematical-methods/08-pdes.tex:235
`depends_on`	←	Green's function	declared	parts/02-mathematical-methods/08-pdes.tex:1159
`depends_on`	←	Type of a second-order operator	declared	parts/02-mathematical-methods/08-pdes.tex:160