27.02.2018

Geodesic flows: notes 5

Geodesian flows, notes 5: symplectic isoperimetric inequalities

If ΩRnΩ⊂Rn is a bounded open set with smooth boundary, the isoperimetric inequality states that

Voln(Ω)n1n1nω1/nnVoln1(Ω)Voln(Ω)n−1n≤1nωn1/nVoln−1(∂Ω)

with equality iff ΩΩ is a ball (here ωn=Voln(B1)ωn=Voln(B1)). This inequality is equivalent with the (Gagliardo-Nirenberg-)Sobolev inequality W1,1(Rn)Lnn1(Rn)W1,1(Rn)⊂Lnn−1(Rn) with sharp constant,

uLnn11nω1/nnuL1.‖u‖Lnn−1≤1nωn1/n‖∇u‖L1.

(One direction follows by applying the Sobolev inequality to a smoothed version of χΩχΩ, and the other direction involves the coarea formula and writing the Lnn1Lnn−1 norm of a function in terms of super-level sets, see e.g. this link.)

Isoperimetric inequalities have been studied in many different geometries besides RnRn (sphere, Gaussian measure, Riemannian, ...). We intend to discuss a rather different looking variant, called a symplectic isoperimetric inequality, based on Viterbo (the notions that appear will not be explained at this time):

Theorem. One has

d(L)n/2cnVoln(L)d(L)n/2≤cnVoln(L)

where LL is a (nn-dimensional) Lagrangian manifold in R2nR2n and d(L)d(L) is the so called displacement energy of LL.

(The displacement energy dd is a "special capacity" on open subsets of a symplectic manifold and it satisfies d(A)>0d(A)>0 for nonempty open AR2nA⊂R2nd(A)d(B)d(A)≤d(B) if ABA⊂B, and d(F(A))=d(A)d(F(A))=d(A) if FF is a symplectic diffeomorphism. See Hofer-Zehnder, Symplectic invariants and Hamiltonian dynamics.)

Why is such an inequality called isoperimetric? This is related to celebrated work of Gromov on symplectic geometry in the 1980s. In particular, Gromov proved that if LL is any closed Lagrangian manifold in R2n=CnR2n=Cn, there is a holomorphic map f:D¯¯¯¯Cnf:D¯→Cn (where DD is the unit disc in CC) which maps D∂D into LL. A subsequent result of Chekanov shows (presumably, I did not have access to the paper) that such a map can be chosen so that

Area(f(D¯¯¯¯))dnVoln(L)2/n.Area(f(D¯))≤dnVoln(L)2/n.

If n=1n=1, any 11-manifold LL in R2R2 is Lagrangian, and if LL is a smooth curve bounding a simply connected domain, presumably this is a version of the standard isoperimetric inequality in R2R2. This is related to the (vague) idea of symplectization introduced by Arnold, which means in particular that for a statement related to submanifolds of a smooth manifold, there should be an analogous symplectic statement phrased in terms of Lagrangian manifolds.

Let us work out a consequence of the symplectic isoperimetric inequality, which implies a version of the Aleksandrov-Bakelman-Pucci maximum principle. This is an important tool in the Krylov-Safonov estimates (De Giorgi-Nash-Moser for nondivergence form equations), and these are fundamental for the regularity theory of fully nonlinear elliptic PDE.

Theorem. (ABP) If ΩRnΩ⊂Rn is a bounded smooth domain, and if uC2(Ω)C(Ω¯¯¯¯)u∈C2(Ω)∩C(Ω¯), then

supΩusupΩu+C[Ω|det(D2u)|dx]1/nsupΩu≤sup∂Ωu+C[∫Ω|det(D2u)|dx]1/n

where D2u=(jku)nj,k=1D2u=(∂jku)j,k=1n is the Hessian. (It is important for applications that the integral can actually be taken over the subset of ΩΩ where uutouches its concave envelope, but we will not worry about this.)

We derive this from the following variant of the symplectic isoperimetric inequality, also proved by Viterbo:

Theorem. If KRnK⊂Rn is compact, and if LL is a Lagrangian manifold contained in TKT∗K, then

γ(L)ncn,KVoln(L)γ(L)n≤cn,KVoln(L)

where γ(L)γ(L) is another symplectic invariant.

Let uC2c(Ω)u∈Cc2(Ω), and define

L={(x,u(x));xΩ}.L={(x,∇u(x));x∈Ω}.

This turns out to be a Lagrangian manifold in R2nR2n. To compute its (usual Riemannian) volume, we note that the tangent vectors of LL are of the form vj=(ej,ju)vj=(ej,∂j∇u) and the metric on LL (induced by the Euclidean metric on R2nR2n is given by

gjk=vjvk=δjk+juku=δjk+l=1n(D2u)jl(D2u)lk.gjk=vj⋅vk=δjk+∂j∇u⋅∂k∇u=δjk+∑l=1n(D2u)jl(D2u)lk.

Thus g=I+(D2u)2g=I+(D2u)2, and

Voln(L)=ΩdVL(x)=Ω|det(gjk(x))|1/2dx=Ωdet(I+(D2u)2)−−−−−−−−−−−−−√dx.Voln(L)=∫ΩdVL(x)=∫Ω|det(gjk(x))|1/2dx=∫Ωdet(I+(D2u)2)dx.

It also turns out that for this LL, one has γ(L)=maxΩuminΩuγ(L)=maxΩu−minΩu. Thus the symplectic isoperimetric inequality gives

(maxΩuminΩu)ncn,ΩΩdet(I+(D2u)2)−−−−−−−−−−−−−√dx.(maxΩu−minΩu)n≤cn,Ω∫Ωdet(I+(D2u)2)dx.

Since u|Ω=0u|∂Ω=0 we have maxu0maxu≥0 and minu0minu≤0, and thus replacing uu by λuλu and letting λλ→∞ gives that

uLCn,Ωdet(D2u)1/nL1,uC2c(Ω).‖u‖L∞≤Cn,Ω‖det(D2u)‖L11/n,u∈Cc2(Ω).

(A related paper on symplectic Brunn-Minkowski. Some references for ABP: Caffarelli-Cabre, Taylor PDE 3 (section 14.13), Gilbarg-Trudinger (section 9.1), slides of Silvestre.)

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