Sobolev inequalities on sets of finite measure #
Let E be a real normed space of dimension n ≥ 1 with an additive Haar measure μ, and let
ω = μ(B(0, 1)). On a set Ω of finite measure, W^{1,p}_0(Ω) embeds in L^q(Ω) whenever
1 ≤ p ≤ q < ∞ and δ = 1/p - 1/q < 1/n, with the explicit bound
‖u‖_{L^q} ≤ n⁻¹ ω ^ (-1/n) ((1 - δ) / (1/n - δ)) ^ (1 - δ) μ(Ω) ^ (1/n - δ) ‖Du‖_{Lᵖ}.
Unlike the Gagliardo–Nirenberg–Sobolev inequality TauCeti.W1p.eLpNorm_value_le_mul_enorm_gradient,
which holds on every Ω with a constant independent of Ω but only for p < n and at the
critical exponent 1/q = 1/p - 1/n, this bound covers every dimension and every p, at the price
of the factor μ(Ω) ^ (1/n - δ). That factor is what makes it scale correctly: on a ball of
radius R it is a multiple of R ^ (1 - n δ), which is exactly how ‖u‖_{L^q} / ‖Du‖_{Lᵖ}
scales under dilation. In particular W^{1,2}_0 of a ball embeds in some L^q with q > 2 in
every dimension, including n = 1 and n = 2, where no critical exponent is available. The
borderline case p = n is TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient.
A C¹ function u with compact support in Ω is bounded pointwise by the Riesz potential of
‖Du‖ of order one, ‖u x‖ ≤ (n ω)⁻¹ ∫_Ω ‖Du y‖ ‖x - y‖ ^ (1 - n) dy
(TauCeti.enorm_le_lintegral_enorm_fderiv_mul_enorm_sub_rpow), and that potential maps
Lᵖ(Ω) to L^q(Ω) with the stated constant (TauCeti.eLpNorm_setLIntegral_enorm_sub_rpow_mul_le
with κ = 1/n). The estimate then passes from test functions to their closure W^{1,p}_0(Ω)
(TauCeti.W1p.eLpNorm_value_le_of_forall_testFunction).
Main declarations #
TauCeti.eLpNorm_le_mul_measure_rpow_mul_eLpNorm_fderiv_of_lt: the bound for compactly supportedC¹functions.TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient_of_lt: the bound onW^{1,p}_0(Ω).TauCeti.W1p.eLpNorm_value_le_mul_rpow_mul_enorm_gradient_of_eq_ball: the scale-invariant form on a ballB(x₀, R), with constant a multiple ofR ^ (1 - n δ).TauCeti.W1p.exists_forall_eLpNorm_value_le_mul_enorm_gradient_of_measure_ne_top: some exponentq > pand constantSwith‖u‖_{L^q} ≤ S ‖∇u‖_{Lᵖ}onW^{1,p}_0(Ω).TauCeti.W1p0.exists_strongDual_apply_eq_setIntegral_of_memLp: forr > max 1 (2n/(n+2)), anLʳ(Ω)function acts onW^{1,2}_0(Ω)as a continuous linear functionalv ↦ ∫_Ω f v.
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Lemmas 7.12 and 7.14, and Theorem 7.10.
The Sobolev inequality on sets of finite measure, for compactly supported functions. Let
n be the dimension and ω = μ(B(0, 1)), and let 1 ≤ p ≤ q < ∞ with δ = 1/p - 1/q < 1/n.
If u is C¹ with compact support inside a measurable set Ω, then
‖u‖_{L^q} ≤ n⁻¹ ω ^ (-1/n) ((1 - δ) / (1/n - δ)) ^ (1 - δ) μ(Ω) ^ (1/n - δ) ‖Du‖_{Lᵖ}.
The Sobolev inequality on W^{1,p}_0(Ω) for Ω of finite measure. Let n ≥ 1 be the
dimension and ω = μ(B(0, 1)), and let p ≤ q < ∞ with δ = 1/p - 1/q < 1/n. If Ω has finite
measure, then every u ∈ W^{1,p}_0(Ω) lies in L^q(Ω), with
‖u‖_{L^q} ≤ n⁻¹ ω ^ (-1/n) ((1 - δ) / (1/n - δ)) ^ (1 - δ) μ(Ω) ^ (1/n - δ) ‖∇u‖_{Lᵖ}.
No regularity of Ω is assumed.
The Sobolev inequality on W^{1,p}_0 of a ball. Let n ≥ 1 be the dimension and
ω = μ(B(0, 1)), and let p ≤ q < ∞ with δ = 1/p - 1/q < 1/n. If Ω = B(x₀, R), then every
u ∈ W^{1,p}_0(Ω) satisfies
‖u‖_{L^q} ≤ n⁻¹ ω ^ (-δ) ((1 - δ) / (1/n - δ)) ^ (1 - δ) R ^ (1 - n δ) ‖∇u‖_{Lᵖ}.
The power R ^ (1 - n δ) is the one dictated by scaling, so the remaining constant depends only
on n, δ and the normalization of μ.
A Sobolev inequality on W^{1,p}_0(Ω) with a gain of integrability. If Ω has finite
measure, n ≥ 1 and p < ∞, there are an exponent q > p and a constant S such that
‖u‖_{L^q} ≤ S ‖∇u‖_{Lᵖ} for every u ∈ W^{1,p}_0(Ω). This holds in every dimension; see
TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient_of_lt for explicit q and
S.
An Lʳ function acts continuously on W^{1,2}_0(Ω). Let n ≥ 1 be the dimension, let
Ω have finite measure, and let r > 1 with n (2 - r) < 2 r, that is r > 2n/(n + 2) (which
is automatic for n ≤ 2). For f ∈ Lʳ(Ω), the pairing v ↦ ∫_Ω f v is a continuous linear
functional on W^{1,2}_0(Ω). This is the right-hand side of the weak
formulation of an elliptic equation with Lʳ data.