The borderline Sobolev embedding p = n and Trudinger's inequality #
Let E be a real normed space of dimension n ≥ 1 with an additive Haar measure μ, and let
ω = μ(B(0, 1)). In the borderline case p = n of the Sobolev embedding, W^{1,n}_0(Ω) does not
embed in L^∞(Ω) once n ≥ 2, but on a set Ω of finite measure it embeds in L^q(Ω) for every
finite q, with the explicit bound
‖u‖_{L^q} ≤ n⁻¹ ω ^ (-1/n) (q (1 - 1/n) + 1) ^ (1 - 1/n + 1/q) μ(Ω) ^ (1/q) ‖Du‖_{Lⁿ}
for every q ≥ n. For n = 1 the constant is bounded in q, in line with the embedding of
W^{1,1}_0 in L^∞. For n ≥ 2 it grows like q ^ (1 - 1/n) as q → ∞, and summing these
bounds in the exponential series gives Trudinger's inequality
∫_Ω exp ((|u| / (c ‖Du‖_{Lⁿ})) ^ (n / (n - 1))) ≤ e (nⁿ + 5) μ(Ω),
c = (2e) ^ (1 - 1/n) n⁻¹ ω ^ (-1/n).
This inequality in ℝ≥0∞ needs no finiteness assumption on μ(Ω), and is trivial when
μ(Ω) = ∞. When μ(Ω) < ∞ it makes |u| ^ (n / (n - 1)) exponentially integrable on Ω, so on
sets of finite measure the substitute for the failed embedding in L^∞ is an embedding in an
exponential Orlicz space.
The proof of the moment bound is that of Gilbarg–Trudinger, Theorem 7.15: the bound is the case
p = n, δ = 1/n - 1/q of the Sobolev inequality on sets of finite measure,
TauCeti.eLpNorm_le_mul_measure_rpow_mul_eLpNorm_fderiv_of_lt, which rests on the Riesz potential
of ‖Du‖ of order one, and on W^{1,n}_0(Ω) it is the same case of
TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient_of_lt.
For Trudinger's inequality, the moment bound at q = n k / (n - 1), k ≥ n - 1, reads
∫_Ω (|u| / (c ‖Du‖_{Lⁿ})) ^ (n k / (n - 1)) ≤ (k + 1) ^ (k + 1) (2e) ^ (-k) μ(Ω), and
(k + 1) ^ (k + 1) / k! ≤ (k + 1) e ^ (k + 1) makes the k-th term of the exponential series at
most e (k + 1) 2 ^ (-k) μ(Ω). The finitely many terms with k < n - 1 are bounded by the term
k = n - 1, using t ^ k ≤ 1 + t ^ (n - 1).
Main declarations #
TauCeti.eLpNorm_le_mul_measure_rpow_mul_eLpNorm_fderiv: the bound for compactly supportedC¹functions.TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient: the bound onW^{1,n}_0(Ω).TauCeti.setLIntegral_exp_norm_div_rpow_le: Trudinger's inequality for compactly supportedC¹functions, andTauCeti.setLIntegral_exp_norm_div_rpow_le_of_eLpNorm_fderiv_le, its form with an upper bound for‖Du‖_{Lⁿ}in place of the norm itself.TauCeti.W1p.setLIntegral_exp_abs_value_div_rpow_le: Trudinger's inequality onW^{1,n}_0(Ω).TauCeti.W1p.integrableOn_exp_mul_abs_value: its local form onW^{1,n}(Ω), without boundary condition:exp (c |u|)is integrable on every compact subset ofΩ, for everyc.
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 7.15 and its proof.
- N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483.
From the moment bounds to exponential integrability #
The borderline Sobolev inequality for compactly supported functions. Let n be the
dimension and ω = μ(B(0, 1)). If u is C¹ with compact support inside a set Ω,
then for every q ≥ n,
‖u‖_{L^q} ≤ n⁻¹ ω ^ (-1/n) (q (1 - 1/n) + 1) ^ (1 - 1/n + 1/q) μ(Ω) ^ (1/q) ‖Du‖_{Lⁿ}.
Trudinger's inequality for compactly supported functions. Let n ≥ 2 be the dimension
and ω = μ(B(0, 1)). If u is C¹ with compact support inside a set Ω, then
∫_Ω exp ((‖u‖ / (c ‖Du‖_{Lⁿ})) ^ (n / (n - 1))) ≤ e (nⁿ + 5) μ(Ω),
where c = (2e) ^ (1 - 1/n) n⁻¹ ω ^ (-1/n).
Trudinger's inequality with a bound on the derivative. Let n ≥ 2 be the dimension. If
u is C¹ with compact support inside a set Ω and ‖Du‖_{Lⁿ} ≤ A for some A > 0, then
∫_Ω exp ((‖u‖ / (c A)) ^ (n / (n - 1))) ≤ e (nⁿ + 5) μ(Ω),
with c as in TauCeti.setLIntegral_exp_norm_div_rpow_le. Since the bound A need not be the
norm of Du itself, a single exponent serves a whole family of functions.
The borderline Sobolev inequality on W^{1,n}_0(Ω). Let n be the dimension and
ω = μ(B(0, 1)). If Ω has finite measure, then every u ∈ W^{1,n}_0(Ω) lies in L^q(Ω) for
every finite q ≥ n, with
‖u‖_{L^q} ≤ n⁻¹ ω ^ (-1/n) (q (1 - 1/n) + 1) ^ (1 - 1/n + 1/q) μ(Ω) ^ (1/q) ‖∇u‖_{Lⁿ}.
No regularity of Ω is assumed.
Trudinger's inequality on W^{1,n}_0(Ω). Let n ≥ 2 be the dimension and
ω = μ(B(0, 1)). Every u ∈ W^{1,n}_0(Ω) satisfies
∫_Ω exp ((|u| / (c ‖∇u‖_{Lⁿ})) ^ (n / (n - 1))) ≤ e (nⁿ + 5) μ(Ω),
where c = (2e) ^ (1 - 1/n) n⁻¹ ω ^ (-1/n). So if μ(Ω) < ∞ and ∇u ≠ 0, then
exp (σ |u| ^ (n / (n - 1))) is integrable on Ω for σ = (c ‖∇u‖_{Lⁿ}) ^ (-n / (n - 1)),
although u need not be bounded. No regularity of Ω is assumed.
Local exponential integrability of W^{1,n}(Ω). Let n ≥ 2 be the dimension. For every
u ∈ W^{1,n}(Ω), every compact K ⊆ Ω and every real c, the function exp (c |u|) is
integrable on K. No boundary condition on u and no regularity of Ω is assumed.
This is the local form of Trudinger's inequality
TauCeti.W1p.setLIntegral_exp_abs_value_div_rpow_le: a cutoff of u lies in W^{1,n}_0(U) for
an open U ⊇ K of finite measure, and by Young's inequality exp (c |u|) is bounded by a constant
times exp ((|u| / A) ^ (n / (n - 1))) for every rate c, because n / (n - 1) > 1.