Whole-space and interior Sobolev embeddings #
Let E be a finite-dimensional real inner product space of dimension n, with an additive Haar
measure, let 1 ≤ p < n and let p⋆ = np/(n - p) be the Sobolev conjugate, so that
1/p⋆ + 1/n = 1/p. The embedding W^{1,p}_0(Ω) ↪ L^{p⋆}(Ω) of
TauCeti.W1p0.sobolevEmbeddingL needs vanishing boundary values. This file removes that
hypothesis in the two situations where no boundary regularity is needed.
- The whole space. Test functions are dense in
W^{1,p}(E)forp < ∞(TauCeti.W1p.mem_w1p0Submodule_top), soW^{1,p}(E) = W^{1,p}_0(E)and the embedding applies to everyu ∈ W^{1,p}(E), with the bound‖u‖_{p⋆} ≤ C ‖∇u‖_pby the gradient alone. - The interior of a domain. If
Vhas compact closure inside an open setΩ, a smooth cutoffχequal to1nearclosure Vand compactly supported inΩturnsu ∈ W^{1,p}(Ω)intoχu ∈ W^{1,p}(E)(TauCeti.W1p.localizeL), which agrees withuonV. The whole-space embedding then boundsuinL^{p⋆}(V)byC ‖u‖_{W^{1,p}(Ω)}. Here the full norm is needed on the right, since the cutoff produces the termu ∇χ.
As V has finite measure, Hölder's inequality also gives the bound in L^q(V) for every
q ≤ p⋆. No boundedness or regularity of Ω itself is assumed.
For arbitrary finite source exponents, the interior inclusion into finite L^s is also bounded
whenever the source exponent q satisfies q ≤ s and 1/q - 1/s < 1/n.
This includes q = n and q > n. The proof localizes
into a bounded zero-boundary domain and uses the finite-measure Sobolev inequality. For q > n,
the native Morrey embedding also gives a bounded interior inclusion into L^∞.
Main declarations #
TauCeti.W1p.sobolevEmbeddingTopL: the embeddingW^{1,p}(E) →L[ℝ] L^{p⋆}(E), withTauCeti.W1p.coeFn_sobolevEmbeddingTopL(it does not change the function) andTauCeti.W1p.norm_sobolevEmbeddingTopL_le(the Gagliardo--Nirenberg--Sobolev bound).TauCeti.W1p.exists_eLpNorm_value_le_of_ae_eq: a bounded linear map intoL^{p⋆}(V)representing the values givesL^q(V)bounds forq ≤ p⋆whenVhas finite measure.TauCeti.W1p.exists_lpEmbeddingL_of_eLpNorm_value_le: conversely, a uniformL^r(V)bound on the values gives a bounded linear map intoL^r(V)representing them.TauCeti.W1p.exists_sobolevEmbeddingL_of_isCompact_closure: the interior embeddingW^{1,p}(Ω) →L[ℝ] L^{p⋆}(V).TauCeti.W1p.exists_eLpNorm_value_le_of_isCompact_closureandTauCeti.W1p.memLp_value_of_isCompact_closure: the interior estimate‖u‖_{L^q(V)} ≤ C ‖u‖_{W^{1,p}(Ω)}and integrability forq ≤ p⋆.TauCeti.W1p.exists_sobolevEmbeddingL_of_subcritical_gain: the bounded interior inclusion for finite source exponentqand target exponentswithq ≤ sand1/q - 1/s < 1/n.TauCeti.W1p.exists_lpInftyEmbeddingL_of_isCompact_closure: the supercritical interior inclusion intoL^∞.
References #
- L. C. Evans, Partial Differential Equations, 2nd ed., §5.6.1, Theorems 1 and 2.
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Theorem 9.9 and Corollary 9.10.
The whole space #
The Sobolev embedding W^{1,p}(E) ↪ L^{p⋆}(E) on the whole space. For 1 ≤ p and
1/p⋆ + 1/n = 1/p with p⋆ < ∞, inclusion is a continuous linear map. Unlike on a domain, no
boundary condition is needed: every u ∈ W^{1,p}(E) is a limit of test functions.
TauCeti.W1p.coeFn_sobolevEmbeddingTopL says the image of u is u itself, and
TauCeti.W1p.norm_sobolevEmbeddingTopL_le bounds it by the gradient alone.
Instances For
The whole-space Sobolev embedding does not change the function.
The Gagliardo--Nirenberg--Sobolev inequality on W^{1,p}(E). The image of u under the
whole-space Sobolev embedding is bounded in L^{p⋆} by the Lᵖ norm of the gradient of u
alone, with Mathlib's constant MeasureTheory.SNormLESNormFDerivOfEqConst.
The whole-space Sobolev embedding is injective.
Lower exponents on a set of finite measure #
Lower exponents on a set of finite measure. If a bounded linear map
S : W^{1,p}(Ω) → L^{p⋆}(V) represents the values, S u = u almost everywhere on V, and V
has finite measure, then ‖u‖_{L^q(V)} ≤ C ‖u‖_{W^{1,p}(Ω)} for every q ≤ p⋆, by Hölder's
inequality on V.
Inclusions from value bounds. A uniform estimate ‖u‖_{L^r(V)} ≤ C ‖u‖_{W^{1,p}(Ω)} on a
subset V ⊆ Ω makes restricting the values of u ∈ W^{1,p}(Ω) to V a continuous linear map
W^{1,p}(Ω) → L^r(V).
The interior of a domain #
The interior Sobolev embedding. If V has compact closure contained in the open set Ω,
1 ≤ p and 1/p⋆ + 1/n = 1/p with p⋆ < ∞, then restricting u ∈ W^{1,p}(Ω) to V is a
continuous linear map W^{1,p}(Ω) → L^{p⋆}(V). Neither boundedness nor boundary regularity of
Ω is assumed.
The interior Sobolev inequality. If V has compact closure contained in the open set
Ω, 1 ≤ p and 1/p⋆ + 1/n = 1/p with p⋆ < ∞, then for every q ≤ p⋆ there is a constant
C with ‖u‖_{L^q(V)} ≤ C ‖u‖_{W^{1,p}(Ω)} for all u ∈ W^{1,p}(Ω). The full Sobolev norm is
needed on the right: on a bounded Ω, a nonzero constant has vanishing gradient.
Interior integrability. If V has compact closure contained in the open set Ω, every
u ∈ W^{1,p}(Ω) lies in L^q(V) for every q ≤ p⋆, where 1/p⋆ + 1/n = 1/p and p⋆ < ∞.
Interior Sobolev embedding with any finite subcritical gain, including when the source exponent equals or exceeds the dimension.
A bounded linear interior Sobolev inclusion for every finite subcritical gain.
The supercritical interior inclusion into L^∞ is bounded and preserves the Sobolev value.