definition 10.85 Sobolev space

open in the book · parts/02-mathematical-methods/08-pdes.tex:2283 · p. 364

ground object -- no derivation owed

Rests on

depends_on definition 12.2 Hilbert space ¶
- depends_on definition 5.18 Inner product ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on equation 5.44 eq:lin-norm-assoc ¶
depends_on definition 10.83 Weak solution ¶
- depends_on definition 17.38 Schwartz space and tempered distributions ¶
- 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 6.2 Open set ¶
  - depends_on equation 10.2 eq:pde-multi-index ¶

Supports

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.464 $H_{E}$ is a Hilbert space ¶
    - depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
      - depends_on definition A.467 Green function of the shifted problem ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶ ↺
- 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 theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
- depends_on lemma A.452 Young and Hölder ¶
  - depends_on lemma A.489 Courant–Lebesgue ¶
    - depends_on theorem A.490 Equicontinuity of the normalised class ¶
  - 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 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 ¶ ↺

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`	→	Hilbert space	declared	parts/02-mathematical-methods/08-pdes.tex:2295
`depends_on`	→	Weak solution	declared	parts/02-mathematical-methods/08-pdes.tex:2295
`depends_on`	←	The space $W^{1,r}(a,b)$	declared	appendices/A-long-proofs.tex:22181
`depends_on`	←	Weak form of an elliptic problem	declared	parts/02-mathematical-methods/08-pdes.tex:2329