Morrey's inequality for C¹ functions #
Let E be a finite-dimensional real normed space of dimension n, let μ be an additive Haar
measure on E, and let p > n. This file proves Morrey's inequality: a C¹ function whose
derivative lies in Lᵖ is Hölder continuous of exponent 1 - n / p, with
‖u x - u y‖ ≤ C(n, p, μ) * ‖x - y‖ ^ (1 - n / p) * ‖Du‖_{Lᵖ}.
The constant is explicit. Writing ω = μ(B(0, 1)) and K = n ω (p - 1) / (p - n), it is
C = 2 ^ (n + 1) / (n ω) * K ^ (1 - 1 / p) * 2 ^ (1 - n / p). For n ≥ 2 it blows up as
p ↓ n, as it must, since the embedding fails in the borderline case p = n. For n = 1 one has
K = ω independently of p, and no blow-up occurs.
The proof starts from the pointwise potential estimate behind the Poincaré–Wirtinger inequality
(TauCeti.enorm_sub_setAverage_le_of_starConvex), which bounds the deviation of u x from a mean
of u by the Riesz potential ∫ ‖Du y‖ ‖x - y‖ ^ (1 - n) dy. For p > n, Hölder's inequality
bounds that potential by ‖Du‖_{Lᵖ}, because the conjugate power ‖x - y‖ ^ ((1 - n) p') of
the kernel is integrable near the pole exactly when p > n. Comparing u x and u y with the
mean of u over a ball containing both points gives the Hölder estimate.
Main declarations #
TauCeti.setLIntegral_mul_enorm_sub_rpow_one_sub_finrank_le: forp > n, the Riesz potential of order one ofg, over a set of radiusDabout the pole, is at mostK ^ (1 - 1 / p) * D ^ (1 - n / p) * ‖g‖_{Lᵖ}.TauCeti.enorm_sub_setAverage_le_of_starConvex_of_finrank_lt: the deviation ofu xfrom the mean ofuover a subset of a star-convex domain, bounded by‖Du‖_{Lᵖ}.TauCeti.enorm_sub_setAverage_le_of_convex_of_finrank_lt: the same on a bounded convex domain.TauCeti.enorm_sub_le_of_mem_ball_of_finrank_lt: Morrey's inequality on a ball.TauCeti.enorm_sub_le_of_contDiff_of_finrank_lt: Morrey's inequality on the whole space.TauCeti.holderWith_of_contDiff_of_finrank_lt: aC¹function with derivative inLᵖ,p > n, is Hölder continuous of exponent1 - n / p.
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 7.17.
- L. C. Evans, Partial Differential Equations, §5.6.2, Theorem 4.
The Riesz potential of order one is bounded on Lᵖ for p > n. If Ω lies in
closedBall x D and p exceeds the dimension n of the space, then the Riesz potential at x
of a function g on Ω is at most K ^ (1 - 1 / p) * D ^ (1 - n / p) times the Lᵖ(Ω) norm of
g, where K = n μ(B(0, 1)) (p - 1) / (p - n).
The Morrey potential estimate for the mean. If u is C¹ on an open set Ω which is
star-convex about x and contained in closedBall x D, and p exceeds the dimension n of the
space, then for every S ⊆ Ω of positive measure the deviation of u x from the mean of u
over S is at most D ^ n / (n μ(S)) * K ^ (1 - 1 / p) * D ^ (1 - n / p) times the Lᵖ(Ω)
norm of the derivative of u, where K = n μ(B(0, 1)) (p - 1) / (p - n).
The Morrey potential estimate on a convex domain. If u is C¹ on a bounded convex
open set Ω, x ∈ Ω, and p exceeds the dimension n of the space, then for every S ⊆ Ω
of positive measure the deviation of u x from the mean of u over S is at most
d ^ n / (n μ(S)) * K ^ (1 - 1 / p) * d ^ (1 - n / p) times the Lᵖ(Ω) norm of the derivative
of u, where d = diam Ω and K = n μ(B(0, 1)) (p - 1) / (p - n).
Morrey's inequality on a ball. If u is C¹ on ball z r and p exceeds the dimension
n of the space, then for all x, y in the ball, ‖u x - u y‖ is at most
2 ^ (n + 1) / (n ω) * K ^ (1 - 1 / p) * (2 r) ^ (1 - n / p) times the Lᵖ norm of the
derivative of u on the ball, where ω = μ(B(0, 1)) and K = n ω (p - 1) / (p - n).
Morrey's inequality. If u is C¹ on the whole space and p exceeds the dimension n
of the space, then ‖u x - u y‖ is at most
2 ^ (n + 1) / (n ω) * K ^ (1 - 1 / p) * (2 ‖x - y‖) ^ (1 - n / p) times the Lᵖ norm of the
derivative of u, where ω = μ(B(0, 1)) and K = n ω (p - 1) / (p - n).
Morrey's inequality, Hölder form. If u is C¹ on the whole space, p exceeds the
dimension n of the space, and the derivative of u lies in Lᵖ, then u is Hölder continuous
of exponent 1 - n / p, with constant
2 ^ (n + 1) / (n ω) * K ^ (1 - 1 / p) * 2 ^ (1 - n / p) * ‖Du‖_{Lᵖ}, where ω = μ(B(0, 1))
and K = n ω (p - 1) / (p - n).