Volume decay and concentration of high-dimensional Euclidean balls -- a PDE and variational perspective
aa r X i v : . [ m a t h . HO ] M a y VOLUME DECAY AND CONCENTRATION OF HIGH-DIMENSIONALEUCLIDEAN BALLS — A PDE AND VARIATIONAL PERSPECTIVE
SIRAN LI
Abstract.
It is a well-known fact — which can be shown by elementary calculus — that thevolume of the unit ball in R n decays to zero and simultaneously gets concentrated on the thinshell near the boundary sphere as n ր ∞ . Many rigorous proofs and heuristic arguments areprovided for this fact from different viewpoints, including Euclidean geometry, convex geometry,Banach space theory, combinatorics, probability, discrete geometry, etc. In this note we give yetanother two proofs via the regularity theory of elliptic partial differential equations and calculusof variations. The problem
A well-known fact in high-dimensional Euclidean geometry, with which we may be familiarsince the very first calculus class, can be stated as follows:
Theorem 1.1.
Let B n = { x ∈ R n : | x | < } be the Euclidean unit ball in R n . The volume of B n gets concentrated near the boundary sphere ∂ B n = { x ∈ R n : | x | = 1 } and tends to as n ր ∞ . Under the MathOverflow question “
What’s a nice argument that shows the volume of theunit ball in R n approaches ? ” posted about years ago ([1]), nearly a dozen elegant andsurprising answers are provided. Contributors to the solutions and discussions include manyrenowned mathematicians: Greg Kuperberg, Timothy Gowers, Ian Agol, Bill Johnson, Gil Kalai,Pete L. Clark, Anton Petrunin... The answers employ techniques from radically different fieldsof mathematics, ranging from combinatorics to the geometry of Banach spaces.The aim of this note is give yet another two proofs of Theorem 1.1 using the knowledgeabout harmonic functions and/or harmonic maps. More generally, we show that Theorem 1.2.
For each n = 1 , , , . . . let u n be a harmonic function in B n . Assume thatthe Dirichlet energies of u n in B n are uniformly bounded. Then the energies decay to andincreasingly concentrate on ∂ B n as n ր ∞ . Remark 1.3.
In view of Theorem 1.2, our proof of Theorem 1.1 shall assume a priori that
Vol( B n ) are uniformly bounded in n . In fact, Vol( B n ) are known to be maximised at n = 5 . Onthe other hand, we shall prove Theorem 1.2 for weakly harmonic functions, i.e. functions thatsatisfy the Laplace equation in the distributional sense. Date : May 26, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
High-dimensional geometry; volume of balls. . The strategy
Consider a harmonic function u = ( u , u , . . . , u n ) : B n → R n , ∆ u i = n X j =1 ∂ u i ∂x j = 0 (2.1)for each i ∈ { , , . . . , n } . A fundamental property of harmonic function is the energy decay phenomenon: For each R ∈ ]0 , , the Dirichlet energy E ( R ) := Z B R |∇ u ( x ) | d x satisfies E ( R/ ≤ θ E ( R ) (2.2)for some number θ strictly less than . Throughout B nR ≡ B R := { x ∈ R n : | x | < R } ; we dropthe superscript n when there is no confusion about the dimension. By a standard argument inthe regularity theory of elliptic PDEs, we may strengthen Eq. (2.2) to the following form: forany < r < R ≤ there holds E ( r ) ≤ C (cid:16) rR (cid:17) β E ( R ) , (2.3)where C is a universal constant (namely, independent of any parameters) and β ≃ n . As n getslarge, the factor ( r/R ) β decays exponentially. It means that, given two arbitrary concentric balls B R and B r , the Dirichlet energy E R is always concentrated in the shell B R ∼ B r . One can nowconclude 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 har-monic functions (between Euclidean domains) are Dirichlet energy minimisers : Id B n = arg min ( E [ v ] = Z B n |∇ v | d x : v ∈ W , ( B n ) and v ( ω ) = ω for all ω ∈ ∂ B n ) . (2.4)Here W , ( B n ) denotes the Sobolev space of finite-energy maps: W , ( B n ) := ( w : B n → R n : Z B n (cid:16) |∇ w | + | w | (cid:17) d x ) . (2.5)Eq. (2.4) is tantamount to the stationariness of u = Id B n with respect to both inner and outer variations , i.e. , the one-parameter families of smooth variations which deform and thedomain and the range of u , respectively. These together imply that E (1) ≤ c Z ∂B n |∇ tan u | dΣ , (2.6)where ∇ tan and dΣ are respectively the gradient and surface measure on the unit sphere; c ∼O (1 /n ) . From here, a rescaling and iteration argument as before will lead us to Eq. (2.3).In the following two sections we make the above discussions rigorous, thus giving two moreproofs of Theorem 1.1. Our arguments can be found, in one form or another, in any standardtextbook on elliptic PDEs and calculus of variations. We refer the readers to [2] by Qing Han andFang-Hua Lin, and [4] by Leon Simon, among many other references. For background materialson mollifiers and elementary inequalities, see [3] by Elliott Lieb and Michael Loss. . The PDE proof
Gradient estimate.
Let us take u ∈ W , ( B n ) to be any weak ( i.e. , distributional) solutionfor the Laplace equation (2.1). We show for all large p that the L p -norm of ∇ u over B / canbe controlled by the L -norm of u over B = B n . Lemma 3.1.
For each p ∈ ]2 , ∞ [ , there is a constant C depending only on p such that k∇ u k L p ( B / ) ≤ C k u k L ( B ) . Proof.
It is well-known that harmonic functions satisfy the mean-value property. So, for a sym-metric mollifier J on R n , pointwise we have u = J δ ⋆ u for each δ ∈ ]0 , / , where J δ ( x ) := δ − n J ( x/δ ) and ⋆ is the convolution. Thus, by Young’s convolution inequality we can bound k∇ u k L p ( B / ) ≤ k∇ J δ k L q ( B / ) k u k L ( B ) , where q is determined by /q = 1 /p + 1 / . A simple scaling argument gives us k∇ J δ k L q ( B / ) ≤ δ − k∇ J k L q ( B ) . Now one may complete the proof by fixing δ and J . (cid:3) Energy decay.
Next let us deduce that
Lemma 3.2.
For any ≤ r < R ≤ there holds E ( r ) ≤ C (cid:16) rR (cid:17) β E ( R ) . (3.1) Here C is a universal constant, and β ∈ ] n/ , n [ is a dimensional constant. In fact, β can bechosen as close to n as we want.Proof. By considering u R ( x ) := u ( x/R ) it suffices to prove for R = 1 and r ∈ ]0 , / . We applythe Hölder inequality, Lemma 3.1, and the scaling Vol( B r ) / Vol( B n ) = r n to obtain E ( r ) := Z B r |∇ u | d x ≤ ® Z B / |∇ u | p d x ´ p î Vol( B r ) ó p − p ≤ ( C ) k u k L ( B ) î Vol( B r ) ó p − p ≡ ( C ) r n ( p − p î Vol( B n ) ó p − p E (1) . Here p is an arbitrary number in ]2 , ∞ [ . We select β := n ( p − p and note that β ր n as p ր ∞ .In addition, the volume of the unit ball is uniformly bounded in n ; hence, there is a universalconstant C which bounds ( C ) î Vol( B n ) ó p − p from the above. The proof is now complete. (cid:3) Conclusion.
Now we are at the stage of presenting
Proof of Theorem 1.1 and Theorem 1.2.
By Lemma 3.2, for each r ∈ [0 , one has E ( r ) ≤ C r β E (1) . Since β ր ∞ and C is universal, we have E ( r ) ց as n ր ∞ . Since r ∈ [0 , isarbitrary, we can conclude that E (1) ց . Energy concentration follows directly from Eq. (3.1).Hence Theorem 1.2 is proved. On the other hand, clearly u = Id B n is a harmonic function. ItsDirichlet energy is given by E ( r ) = Z B r |∇ x | d x = n Vol( B r ) . ending r ր , we find that Vol( B n ) decays no slower than O (1 /n ) . This yields Theorem 1.1. (cid:3) The variational proof
Inner and outer variations.
It is well-known that a harmonic function u : B n → R n isa Dirichlet-minimiser ; that is, u minimises E (1) among all the finite-energy maps attaining thesame values on ∂ B n (see Eq. (2.4)). In particular, consider the following two types of variations: • ( Inner variation ). Consider φ ∈ C ∞ ( B n , R n ) and ψ in t ( x ) := x + tφ ( x ) . • ( Outer variation ). Consider e φ ∈ C ∞ ( B n × R n ; R n ) such that e φ ( x, u ) = 0 near ∂ B n × R n , |∇ u e φ ( x, u ) | ≤ C and | e φ ( x, u ) | + |∇ x e φ ( x, u ) | ≤ C (1 + | u | ) for universal constants C and C . Then we set ψ out t ( x, u ) := u ( x ) + t e φ ( x, u ) .Here, ψ in t and ψ out t are one-parameter families of boundary-preserving diffeomorphismsobtained by deforming the domain and the range of u , respectively. The minimality of u yields ddt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t =0 ( Z B n (cid:12)(cid:12)(cid:12)(cid:12) ∇ Ä u ◦ ψ in t ( x ) ä (cid:12)(cid:12)(cid:12)(cid:12) d x ) = 0 , (4.1) ddt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t =0 ( Z B n (cid:12)(cid:12)(cid:12)(cid:12) ∇ ψ out t ( x, u ) (cid:12)(cid:12)(cid:12)(cid:12) d x ) = 0 . (4.2)As is standard in calculus of variation, we take φ ( x ) = η ( | x | ) x and e φ ( x, u ) = η ( | x | ) u . If,furthermore, the test function η is chosen to tend to the indicator function [0 ,r [ , then a directcomputation from Eqs. (4.1) and (4.2) gives us ( n − Z B r |∇ u | d x = r Z ∂ B r |∇ u | dΣ − r Z ∂ B r | ∂ ν u | dΣ , (4.3) Z B r |∇ u | d x = n X i =1 Z ∂ B r u i ( ∂ ν u ) i dΣ . (4.4)In the above, r ∈ [0 , is arbitrary, ν is the outward unit normal vectorfield, and dΣ is the(Riemannian) surface measure as before.4.2. Proof of Theorems 1.2 and 1.1.
In this subsection we show
Lemma 4.1. If u : B n → R n is a non-constant Dirichlet minimiser for n ≥ , then E (1) < n − H (1) . (4.5) Definition 4.2. H denotes the surface-Dirichlet energy: H ( r ) := Z ∂ B r |∇ tan u | dΣ . ∇ tan is the tangential gradient on ∂ B r , i.e., the gradient associated to the Levi-Civita connectionon the round sphere ∂ B r . 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) ց as n ր ∞ . The identity map u ( x ) = x is a Dirichlet minimiser with E ( r ) = n Vol( B r ) , so we have Vol( B n ) ց . In fact, it follows that Vol( B n ) decays no slowerthan O (1 /n ) . (cid:3) hat’s left now is to present a proof of Lemma 4.1. It follows fairly straightforwardly fromthe formulae of inner and outer variations, Eqs. (4.3) and (4.4). Proof of Lemma 4.1.
Assume for contradiction that E (1) ≥ n − H (1) . We may compute H (1) by subtracting the angular derivatives from the total derivatives: H (1) = Z ∂ B n (cid:16) |∇ u | − | ∂ ν u | (cid:17) dΣ . By Eq. (4.3) for inner variations we get H (1) = Z ∂ B n |∇ u | dΣ + n − Z B n |∇ u | d x − Z ∂ B n |∇ u | dΣ= 12 Z ∂ B n |∇ u | dΣ + n − Z B n |∇ u | d x, which is no greater than ( n − E (1) / by assumption. It implies that Z ∂ B n |∇ u | dΣ = 0 . But this forces ∂ ν u to vanish in the L -norm on ∂ B n , which in turn implies that E (1) = 0 bythe outer variation Eq. (4.4). Thus u is a constant on B n . (cid:3) References [1] MathOverflow question, “What’s a nice argument that shows the volume of the unit ball in R n approaches ?”
Analysis: second edition , Graduate Studies in Mathematics, vol. 14,American Mathematical Society, Providence, RI, 2001[4] Leon Simon,
Theorems on regularity and singularity of energy minimizing maps , Based on lecture notes byNorbert Hungerbühler. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996
Siran Li: Department of Mathematics, Rice University, MS 136 P.O. Box 1892, Houston,Texas, 77251, USA.