Abstract
It is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in ℝn decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n↗∞.
Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc.
In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.
1 The problem
A well-known fact in high-dimensional Euclidean geometry, with which we may be familiar since the very first calculus class, can be stated as follows.
Theorem 1.1.
Let Bn={x∈Rn:|x|<1} be the Euclidean unit ball in Rn.
The volume of Bn gets concentrated near the boundary sphere ∂Bn={x∈Rn:|x|=1} and tends to 0 as n↗∞.
Under the MathOverflow question “What’s a nice argument that shows the volume of the unit ball in ℝn approaches 0?” posted about 10 years ago [4], nearly a dozen elegant and surprising answers are provided.
Contributors to the solutions and discussions include Greg Kuperberg, Timothy Gowers, Ian Agol, Bill Johnson, Gil Kalai, Pete L. Clark, Anton Petrunin, etc. (see [4]).
The answers employ techniques from radically different fields of mathematics, ranging from combinatorics to the geometry of Banach spaces.
The aim of this note is to give yet another two proofs of Theorem 1.1 using the knowledge about harmonic functions and/or harmonic maps.
More generally, we show the following.
Theorem 1.2.
For each n=1,2,3,…, let un be a harmonic function in Bn.
Assume that the Dirichlet energies of un in Bn are uniformly bounded.
Then the energies decay to 0 and increasingly concentrate on ∂Bn as n↗∞.
Remark 1.3.
In view of Theorem 1.2, our proof of Theorem 1.1 shall assume a priori that Vol(𝐁n) are uniformly bounded in n.
In fact, Vol(𝐁n) are known to be maximised at n=5.
On the other hand, we shall prove Theorem 1.2 for weakly harmonic functions, i.e. functions that satisfy the Laplace equation in the distributional sense.
2 The strategy
Consider a harmonic function u=(u1,u2,…,un):𝐁n→ℝn,
(2.1)Δui=∑j=1n∂2ui∂xj2=0
for each i∈{1,2,…,n}.
A fundamental property of harmonic function is the energy decay phenomenon: for each R∈]0,1[, the Dirichlet energy
𝐄(R):=∫𝐁R|∇u(x)|2dx
satisfies
(2.2)𝐄(R2)≤θ𝐄(R)
for some number θ strictly less than 1.
Throughout, 𝐁Rn≡𝐁R:={x∈ℝn:|x|<R}; we drop the superscript n when there is no confusion about the dimension.
By a standard argument in the regularity theory of elliptic PDEs, we may strengthen equation (2.2) to the following form: for any 0<r<R≤1, there holds
(2.3)𝐄(r)≤C(rR)β𝐄(R),
where C is a universal constant (namely, independent of any parameters) and β≃n.
As n gets large, the factor (rR)β decays exponentially.
It means that, given two arbitrary concentric balls 𝐁R and 𝐁r, the Dirichlet energy 𝐄R is always concentrated in the shell 𝐁R∼𝐁r.
One can now conclude by taking the identity harmonic map u(x)=x.
We may also deduce Theorem 1.1 from calculus of variations.
It is well known that harmonic functions (between Euclidean domains) are Dirichlet energy minimisers,
(2.4)𝐈𝐝𝐁n=argmin{𝐄[v]=∫𝐁n|∇v|2dx:v∈W1,2(𝐁n)andv(ω)=ωfor allω∈∂𝐁n}.
Here, W1,2(𝐁n) denotes the Sobolev space of finite-energy maps,
W1,2(𝐁n):={w:𝐁n→ℝn:∫𝐁n(|∇w|2+|w|2)dx}.
Equation (2.4) is tantamount to the stationariness of u=𝐈𝐝𝐁n with respect to both inner and outer variations, i.e., the one-parameter families of smooth variations which deform and the domain and the range of u, respectively.
These together imply that
𝐄(1)≤c1∫∂Bn|∇tanu|2dΣ,
where ∇tan and dΣ are respectively the gradient and surface measure on the unit sphere; c1∼𝒪(1n).
From here, a rescaling and iteration argument as before will lead us to equation (2.3).
In the following two sections, we make the above discussions rigorous, thus giving two more proofs of Theorem 1.1.
Our arguments can be found, in one form or another, in any standard textbook on elliptic PDEs and calculus of variations.
We refer the readers to [1] by Qing Han and Fang-Hua Lin, and [3] by Leon Simon, among many other references.
For background materials on mollifiers and elementary inequalities, see [2] by Elliott Lieb and Michael Loss.
3 The PDE proof
3.1 Gradient estimate
Let us take u∈W1,2(𝐁n) to be any weak (i.e., distributional) solution for the Laplace equation (2.1).
We show for all large p that the Lp-norm of ∇u over 𝐁12 can be controlled by the L2-norm of u over 𝐁1=𝐁n.
Lemma 3.1.
For each p∈]2,∞[, there is a constant C2 depending only on p such that
∥∇u∥Lp(𝐁12)≤C2∥u∥L2(𝐁1).
Proof.
It is well known that harmonic functions satisfy the mean-value property.
So, for a symmetric mollifier J on ℝn, pointwise, we have u=Jδ⋆u for each δ∈]0,12], where Jδ(x):=δ-nJ(xδ) and ⋆ is the convolution.
Thus, by Young’s convolution inequality, we can bound
∥∇u∥Lp(𝐁12)≤∥∇Jδ∥Lq(𝐁12)∥u∥L2(𝐁1),
where q is determined by 1q=1p+12.
A simple scaling argument gives us ∥∇Jδ∥Lq(𝐁12)≤δ-1∥∇J∥Lq(𝐁1).
Now one may complete the proof by fixing δ and J. ∎
3.2 Energy decay
Next let us deduce the following.
Lemma 3.2.
For any 0≤r<R≤1, there holds
(3.1)𝐄(r)≤C3(rR)β𝐄(R).
Here, C3 is a universal constant, and β∈]n2,n[ is a dimensional constant.
In fact, β can be chosen as close to n as we want.
Proof.
By considering uR(x):=u(xR), it suffices to prove for R=1 and r∈]0,12].
We apply the Hölder inequality, Lemma 3.1 and the scaling Vol(𝐁r)/Vol(𝐁1n)=rn to obtain
𝐄(r):=∫𝐁r|∇u|2dx≤{∫𝐁12|∇u|2pdx}1p[Vol(𝐁r)]p-1p≤(C2)2∥u∥L2(𝐁1)2[Vol(𝐁r)]p-1p≡(C2)2rn(p-1)p[Vol(𝐁1n)]p-1p𝐄(1).
Here, p is an arbitrary number in ]2,∞[.
We select β:=n(p-1)p and note that β↗n as p↗∞.
In addition, the volume of the unit ball is uniformly bounded in n; hence, there is a universal constant C3 which bounds (C2)2[Vol(𝐁1n)]p-1p from the above.
The proof is now complete.
∎
3.3 Conclusion
Now we are at the stage of presenting the following proof.
Proof of Theorem 1.1 and Theorem 1.2.
By Lemma 3.2, for each r∈[0,1[, one has 𝐄(r)≤C3rβ𝐄(1).
Since β↗∞ and C3 is universal, we have 𝐄(r)↘0 as n↗∞.
Since r∈[0,1[ is arbitrary, we can conclude that 𝐄(1)↘0.
Energy concentration follows directly from equation (3.1).
Hence Theorem 1.2 is proved.
On the other hand, clearly u=𝐈𝐝𝐁n is a harmonic function.
Its Dirichlet energy is given by
𝐄(r)=∫𝐁r|∇x|2dx=nVol(𝐁r).
Sending r↗1, we find that Vol(𝐁n) decays no slower than 𝒪(1n).
This yields Theorem 1.1.
∎
4 The variational proof
4.1 Inner and outer variations
It is well known that a harmonic function u:𝐁n→ℝn is a Dirichlet-minimiser; that is, u minimises 𝐄(1) among all the finite-energy maps attaining the same values on ∂𝐁n (see equation (2.4)).
In particular, consider the following two types of variations.
(Inner variation).
Consider ϕ∈C0∞(𝐁n,ℝn) and ψint(x):=x+tϕ(x).
(Outer variation).
Consider ϕ~∈C∞(𝐁n×ℝn;ℝn) such that ϕ~(x,u)=0 near ∂𝐁n×ℝn, |∇uϕ~(x,u)|≤C4 and |ϕ~(x,u)|+|∇xϕ~(x,u)|≤C5(1+|u|) for universal constants C4 and C5.
Then we set
ψoutt(x,u):=u(x)+tϕ~(x,u).
Here, ψint and ψoutt are one-parameter families of boundary-preserving diffeomorphisms obtained by deforming the domain and the range of u, respectively.
The minimality of u yields
(4.1)ddt|t=0{∫𝐁n|∇(u∘ψint(x))|2dx}=0,
(4.2)ddt|t=0{∫𝐁n|∇ψoutt(x,u)|2dx}=0.
As is standard in calculus of variation, we take ϕ(x)=η(|x|)x and ϕ~(x,u)=η(|x|)u.
If, furthermore, the test function η is chosen to tend to the indicator function 𝟙[0,r[, then a direct computation from equations (4.1) and (4.2) gives us
(4.3)(n-2)∫𝐁r|∇u|2dx=r∫∂𝐁r|∇u|2dΣ-2r∫∂𝐁r|∂νu|2dΣ,
(4.4)∫𝐁r|∇u|2dx=∑i=1n∫∂𝐁rui(∂νu)idΣ.
In the above, r∈[0,1] is arbitrary, ν is the outward unit normal vector field, and dΣ is the (Riemannian) surface measure as before.
4.2 Proof of Theorems 1.2 and 1.1
In this subsection, we show the following lemma.
Lemma 4.1.
If u:Bn→Rn is a non-constant Dirichlet minimiser for n≥3, then
𝐄(1)<2n-2𝐇(1).
Definition 4.2.
𝐇 denotes the surface-Dirichlet energy,
𝐇(r):=∫∂𝐁r|∇tanu|2dΣ.
∇tan is the tangential gradient on ∂𝐁r, i.e., the gradient associated to the Levi–Civita connection on the round sphere ∂𝐁r.
Assuming Lemma 4.1, we may immediately deduce Theorems 1.2 and 1.1.
Proof of Theorem 1.1 and Theorem 1.2.
By assumption, 𝐇(1) is bounded independent of n.
Hence 𝐄(1)↘0 as n↗∞.
The identity map u(x)=x is a Dirichlet minimiser with 𝐄(r)=nVol(𝐁r), so we have Vol(𝐁n)↘0.
In fact, it follows that Vol(𝐁n) decays no slower than 𝒪(1n2).
∎
What is left now is to present a proof of Lemma 4.1.
It follows fairly straightforwardly from the formulae of inner and outer variations, equations (4.3) and (4.4).
Proof of Lemma 4.1.
Assume for contradiction that 𝐄(1)≥2n-2𝐇(1).
We may compute 𝐇(1) by subtracting the angular derivatives from the total derivatives,
𝐇(1)=∫∂𝐁n(|∇u|2-|∂νu|2)dΣ.
By equation (4.3) for inner variations, we get
𝐇(1)=∫∂𝐁n|∇u|2dΣ+n-22∫𝐁n|∇u|2dx-12∫∂𝐁n|∇u|2dΣ=12∫∂𝐁n|∇u|2dΣ+n-22∫𝐁n|∇u|2dx,
which is no greater than (n-2)𝐄(1)2 by assumption.
It implies that
∫∂𝐁n|∇u|2dΣ=0.
But this forces ∂νu to vanish in the L2-norm on ∂𝐁n, which in turn implies that 𝐄(1)=0 by the outer variation equation (4.4).
Thus u is a constant on 𝐁n.
∎
