Morrey's embedding for W^{1,p}(Ω) in the interior of a domain #
Let Ω be an open subset of a finite-dimensional real inner product space of dimension n, with
an additive Haar measure μ, and let n < p ≤ ∞. This file proves that every u ∈ W^{1,p}(Ω) is
locally Hölder continuous in Ω with exponent α = 1 - n / p, in the form of the local Morrey
inequality
|u*(x) - u*(y)| ≤ C ‖∇u‖_{Lᵖ(B(x, 2|x - y|))} |x - y|^α whenever closedBall x (2|x - y|) ⊆ Ω,
where u* is the precise representative of u (the limit of its averages over shrinking balls,
which agrees with u almost everywhere on Ω by TauCeti.W1p.ae_eq_preciseRepresentative) and
C depends only on n, p and μ. No regularity of ∂Ω is needed, and the endpoint p = ∞
is included: there α = 1, so a function of W^{1,∞}(Ω) is locally Lipschitz in Ω.
Morrey's embedding on the whole space, TauCeti.W1p.exists_holderWith_ae_eq_value, is proved by
approximation with test functions, which are dense in W^{1,p}(ℝⁿ) but not in W^{1,p}(Ω). Here
the estimate is instead a consequence of Morrey's Dirichlet growth theorem
(TauCeti.W1p.dist_preciseRepresentative_le_of_setIntegral_norm_gradient_le): by Hölder's
inequality on a ball B(z, r),
∫_{B(z, r)} ‖∇u‖ ≤ μ(B(z, r))^(1 - 1/p) ‖∇u‖_{Lᵖ(B(z, r))} = ω^(1 - 1/p) r^(n - 1 + α) ‖∇u‖_{Lᵖ},
with ω = μ(B(0, 1)), which is exactly the growth condition of that theorem.
Main declarations #
TauCeti.W1p.setIntegral_ball_norm_gradient_le: Hölder's inequality on a ball, which gives the growth condition.TauCeti.W1p.exists_dist_preciseRepresentative_le: the local Morrey inequality.TauCeti.W1p.exists_holderOnWith_preciseRepresentative: the precise representative isα-Hölder continuous, with constantC ‖∇u‖_{Lᵖ(Ω)}, on every set of diameter at mostRwhose closed2R-thickening lies inΩ.TauCeti.W1p.continuousOn_preciseRepresentative: the precise representative is continuous onΩ.TauCeti.W1p.exists_holderOnWith_preciseRepresentative_of_isCompact: it isα-Hölder continuous on every compact subset ofΩ.TauCeti.W1p.exists_lipschitzOnWith_preciseRepresentative_of_isCompact: a function ofW^{1,∞}(Ω)is Lipschitz on every compact subset ofΩ.TauCeti.W1p.preciseRepresentative_eq_morreyRepresentative: on the whole space, for finitep > n, the precise representative is the canonical representativeTauCeti.W1p.morreyRepresentativeof the whole-space embedding.
References #
- C. B. Morrey, Multiple Integrals in the Calculus of Variations, Theorem 3.5.2.
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorems 7.17 and 7.19.
- L. C. Evans, Partial Differential Equations, §5.6.2, Theorem 4, and §5.8.2, Theorem 4.
The gradient of a W^{1,p} function grows like a power of the radius. If B(z, r) ⊆ Ω,
then by Hölder's inequality
∫_{B(z, r)} ‖∇u‖ ≤ ω^(1 - 1/p) r^(n (1 - 1/p)) ‖∇u‖_{Lᵖ(B(z, r))},
where n is the dimension and ω is the measure of the unit ball. For p > n the exponent
n (1 - 1/p) is n - 1 + α with α = 1 - n / p, which is the growth condition of Morrey's
Dirichlet growth theorem.
The local Morrey inequality. Let n < p ≤ ∞, where n is the dimension, and let
α = 1 - n / p. There is a constant C > 0, depending only on n, p and the Haar measure μ,
such that the following holds. Let u ∈ W^{1,p}(Ω) and let x, y be points with
closedBall x (2 |x - y|) ⊆ Ω. Then the precise representative u* of u satisfies
|u*(x) - u*(y)| ≤ C ‖∇u‖_{Lᵖ(B(x, 2 |x - y|))} |x - y|^α.
Morrey's embedding in the interior of a domain. Let n < p ≤ ∞, where n is the
dimension, and let α = 1 - n / p. There is a constant C > 0, depending only on n, p and the
Haar measure μ, such that the following holds. Let u ∈ W^{1,p}(Ω), and let K be a set of
diameter at most R whose closed 2R-thickening lies in Ω. Then the precise representative of
u is α-Hölder continuous on K with constant C ‖∇u‖_{Lᵖ(Ω)}.
Continuity of W^{1,p} functions for p above the dimension. If n < p ≤ ∞, where n
is the dimension, the precise representative of any u ∈ W^{1,p}(Ω), which agrees with u almost
everywhere on Ω, is continuous on Ω.
The continuous precise representative of a W^{1,∞}(Ω) function is bounded
pointwise on Ω by the essential supremum of its value.
The endpoint local Morrey estimate. A W^{1,∞}(Ω) precise representative satisfies
an interior Lipschitz estimate with constant C ‖∇u‖, where C > 0 depends only on the
space and Haar measure. The estimate applies whenever the closed ball of radius twice the
distance of the two points lies in Ω.
Morrey's embedding on compact subsets. If n < p ≤ ∞, where n is the dimension, the
precise representative of any u ∈ W^{1,p}(Ω) is Hölder continuous with exponent 1 - n / p on
every compact subset of Ω.
W^{1,∞} functions are locally Lipschitz. The precise representative of any
u ∈ W^{1,∞}(Ω) is Lipschitz on every compact subset of Ω.
On the whole space, for finite p above the dimension, the precise representative of
u ∈ W^{1,p}(ℝⁿ) is the canonical Morrey representative TauCeti.W1p.morreyRepresentative: both
are continuous and agree with u almost everywhere.