De Giorgi's isoperimetric inequality #
Let Ω be a bounded convex open subset of a finite-dimensional real inner product space E of
dimension n, and let μ be an additive Haar measure. For u ∈ W^{1,p}(Ω), 1 ≤ p ≤ ∞, and
levels k < l, this file proves De Giorgi's isoperimetric inequality
(l - k) · |{u ≥ l}| · |{u ≤ k}| ≤ μ(B(0, 1)) · (diam Ω) ^ (n + 1) · ∫_{k < u < l} |∇u|,
all sets being taken inside Ω. On a ball of radius R the constant is
μ(B(0, 1)) · (2R) ^ (n + 1); its value depends on the normalization of μ.
The inequality quantifies how a Sobolev function passes from low to high values: if both
{u ≤ k} and {u ≥ l} occupy a fixed proportion of Ω, then ∇u carries a definite amount of
mass on the strip {k < u < l} in between. This is the ingredient of De Giorgi's proof of
Hölder continuity for weak solutions of divergence-form equations with bounded measurable
coefficients that turns a lower bound on the measure of {u ≤ k} into a decay of the measure of
{u ≥ l} along a sequence of levels. Only the L¹ norm of ∇u on the strip enters, which is
why the inequality is stated for every exponent p and proved at p = 1.
Main declarations #
TauCeti.W1p.sub_mul_measureReal_mul_measureReal_le_of_convex: the inequality on a bounded convex domain.TauCeti.W1p.sub_mul_measureReal_mul_measureReal_le_of_eq_ball: the inequality on a ball of radiusR, with constantμ(B(0, 1)) (2R) ^ (n + 1).TauCeti.W1p.sub_mul_measureReal_mul_measureReal_le_of_ball_subset: the same inequality for a ball contained in a larger Sobolev domain.TauCeti.W1p.sq_sub_mul_measureReal_mul_measureReal_le_of_ball_subset: its squared form foru ∈ W^{1,2}(Ω), with theL²energy of the truncation(u - k)⁺on the ball.
References #
- E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino (1957).
- Q. Han, F. Lin, Elliptic Partial Differential Equations, Chapter 4.
- L. Caffarelli, A. Vasseur, The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics, Discrete Contin. Dyn. Syst. Ser. S (2010).
De Giorgi's isoperimetric inequality. Let Ω be a bounded convex open set in a
finite-dimensional real inner product space of dimension n, let u ∈ W^{1,p}(Ω) with
1 ≤ p ≤ ∞, and let k < l. Then, all sets being taken inside Ω,
(l - k) · |{u ≥ l}| · |{u ≤ k}| ≤ μ(B(0, 1)) · (diam Ω) ^ (n + 1) · ∫_{k < u < l} |∇u|.
A function that is both ≤ k and ≥ l on sets of positive measure must have a weak gradient
of definite L¹ mass on the strip {k < u < l}.
De Giorgi's isoperimetric inequality on a ball. For u ∈ W^{1,p}(B(c, R)),
1 ≤ p ≤ ∞, and levels k < l, all sets being taken inside the ball,
(l - k) · |{u ≥ l}| · |{u ≤ k}| ≤ μ(B(0, 1)) · (2R) ^ (n + 1) · ∫_{k < u < l} |∇u|.
The power R ^ (n + 1) is the one forced by scaling.
De Giorgi's isoperimetric inequality on a ball contained in the Sobolev domain.
For u ∈ W^{1,p}(Ω), a ball B(c, R) ⊆ Ω, and levels k < l, all level sets and the gradient
integral being restricted to the ball,
(l - k) · |{u ≥ l}| · |{u ≤ k}| ≤ μ(B(0, 1)) · (2R) ^ (n + 1) · ∫_{k < u < l} |∇u|.
This is the ball inequality applied to the Sobolev restriction of u.
De Giorgi's isoperimetric inequality, squared form. For u ∈ W^{1,2}(Ω), a ball
B(c, R) ⊆ Ω, and levels k < l with w = (u - k)⁺ ∈ L²(Ω), all level sets being restricted
to the ball,
((l - k) · |{u ≥ l}| · |{u ≤ k}|)² ≤ (μ(B(0, 1)) · (2R) ^ (n + 1))² · |{k < u < l}| · ∫_B |∇w|²,
where B = B(c, R).
This is the isoperimetric inequality on the ball followed by the Cauchy–Schwarz inequality on the
strip {k < u < l}, where ∇u = ∇w.