Sobolev embedding at order k #
For Ω ⊆ ℝⁿ open and bounded with ∂Ω of class C¹, p ∈ [1, ∞), k ∈ ℕ and
u ∈ W^{k,p}(Ω),
- if
k < n/pthenu ∈ L^q(Ω)with1/q = 1/p - k/nand‖u‖_{L^q} ≤ C‖u‖_{W^{k,p}}; - if
k > n/pthenu ∈ C^{k-1-⌊n/p⌋,γ}(closure Ω)withγany value in(0,1)whenn/p ∈ ℕandγ = ⌊n/p⌋ - n/p + 1otherwise, and‖u‖_{C^{k-1-⌊n/p⌋,γ}} ≤ C‖u‖_{W^{k,p}}.
This file states both clauses at the exponents the theorem names. Theorem IV.2.3 of the lecture notes cited below is the form followed here.
Reading W^{k,p} as a family #
A member of W^{k,p}(Ω) is read here as a family F : ι → ℝⁿ → ℝ closed under weak
differentiation, nxt i k naming a weak k-derivative of F i, together with a depth function
dep recording how far an index sits above the root. The order k of the theorem is the supply
m, the uniform bound M on the L^p seminorms of the members of depth at most m is
‖u‖_{W^{k,p}(Ω)}, and D^α u for |α| ≤ k is the member at depth |α|. That is the
vocabulary the interior statements of this development already use, and it asks nothing of a
multi-index calculus.
Two clauses #
Clause (i) is EllipticPdes.Embedding.exists_const_memLp_of_gradClosed_domain_ideal: the strict
rung condition p k < n is exactly positivity of the landing reciprocal 1/p - k/n, and the
ladder lands on the exponent naming it.
Clause (ii) splits in two. When n/p ∉ ℕ the rung count is ⌊n/p⌋, the landing exponent
P satisfies p k < n < p (k+1), and Morrey's exponent 1 - n/P is ⌊n/p⌋ - n/p + 1; that is
exists_const_holderOnWith_domain_ideal. When n/p ∈ ℕ the rung count n/p sends the landing
reciprocal to zero, the ladder reaches every finite exponent, and the Hölder exponent is free in
(0,1); that is exists_const_holderOnWith_domain_free.
Main declarations #
EllipticPdes.Embedding.exists_const_memLp_of_gradClosed_domain_ideal: clause (i).EllipticPdes.Embedding.exists_const_holderOnWith_domain_ideal: clause (ii) atn/p ∉ ℕ.EllipticPdes.Embedding.exists_const_holderOnWith_domain_free: clause (ii) atn/p ∈ ℕ.
References #
James Guo, Partial Differential Equations (Course Lecture Notes), Theorem IV.2.3 (pp. 32-33); L. C. Evans, Partial Differential Equations (2nd ed.), §5.6.3 Theorem 6.
Clause (i) #
Clause (i) of the Sobolev embedding. Under the strict rung condition p₀ s < d,
which is k < n/p, every member of depth at most m - s lies in L^q(Ω) at the exponent
1/q = 1/p₀ - s/d the theorem names, with one constant, depending on the domain, the dimension, the
base exponent and the rung count alone, bounding its seminorm by a uniform L^{p₀} bound on the
family.
Landing exponents of clause (ii) #
Clause (ii) #
Clause (ii) of the Sobolev embedding at n/p ∉ ℕ. The rung count is
⌊n/p⌋, which the two hypotheses p₀ s < d < p₀ (s + 1) pin down, and the Hölder exponent is
⌊n/p⌋ - n/p + 1. One constant bounds the Hölder seminorm on the closure of the domain by a
uniform L^{p₀} bound on the family, and the members it applies to are those of depth at most
m - 1 - s, which is k - 1 - ⌊n/p⌋. The supremum is bounded by the same constant, so the
estimate is on the whole C^{0,γ} norm.
Clause (ii) of the Sobolev embedding at n/p ∈ ℕ. The rung count n/p
sends the landing reciprocal to zero, so the ladder reaches every finite exponent and the Hölder
exponent may be any value in (0,1).
Clause (ii) with classical derivatives #
Clause (ii) at n/p ∉ ℕ as a C^{k-1-⌊n/p⌋,γ} statement. One family of representatives
serves every index at once: bounded and γ-Hölder on the closure of the domain under the constant
the clause names, differentiable on the domain with the next members as partial derivatives, and
n times continuously differentiable there whenever the supply leaves n orders above it.
Clause (ii) at n/p ∈ ℕ as a C^{k-1-⌊n/p⌋,γ} statement. The same family of
representatives, at any Hölder exponent in (0,1).