Concave An

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An Optimal Relative Isoperimetric Inequality in Concave Cylindrical Domains in IR

INHOKIM*

Departmentof Mathematics,SeoulNational University,San56-1, Shinlim-Dong, Kwanack-gu, Seoul, 151-742,Korea

(Received2 February 1999; Revised 25 March1999)

Weproveanoptimalrelativeisoperimetric inequalityinconcavecylindricaldomains in 1

n,

whichgeneralizes the well-knowntwo-dimensional relativeisoperimetric inequality L >27rAin aplanarsectorwithangle greater thanorequalto

Keywords." Isoperimetric inequality; Sobolev inequality;Geodesic metricspace 1991 MathematicsSubjectClassification: ^Primary51M16; Secondary 53C21

1.

INTRODUCTION

The aim ofthispaperis topointout anoptimal relativeisoperimetric inequality inconcavecylindrical domains in I1ⁿwhichplays animpor- tant role in the Sobolev inequalities and the mixed boundary value problemsofpartialdifferential equations.

Let Sbea sectorin

I12

with thesectorangle0

<

27r.Itiswell-known that adomainf

c

Ssatisfiestheoptimalrelativeisoperimetric inequality

Length(0Ft- 0S) > 20Area(Ft) Length(0f- 0S) > 2zrArea(Ft)

if0

<

rr,

(1)

if 0

>_

7r.

(2)

*E-mail:[email protected].

97

In

[3]

Lionsand Pacella have obtainedageneralizationof

(1)

^tosub- domains ofconvexcones in

Rn,

n

>

2

(see

^also

[4]).

^{In the samepaper} theyhave also pointedoutsomeapplicationstosymmetrizationprob- lems and Sobolev inequalities.

In

thispaperwewill giveahigherdimensional generalization of the inequality

(2).

^{To state the resultwe first fixnotations. Let}

f:

^{1t 11}

be a Lipschitz continuous convex function i.e. for any

A

E

[0, 1]

^and

anyt,t2^E

f[(1 A)tl + At2] < (1 A)f(tl)+ f(t2).

Let

W

be theclosed cylindricalconcave

(the

^{complementofa}

convex)

domain in

n,

n

>

2,definedby

W-- {(Xl,X2,...,Xn)Ix2 _<f(x)}.

For n 2,

W

^isjust the set

V

of points on and under the graph of

f

in Ii²and,forn

>

2,

W

^istheproductof

V

and

n-2.

With this notation we state

MAIN THEOREM Let f be a compact domain in

W

^with

rectifiable

boundaryOf. Thenwehavethe relative isoperimetricinequality

Voln-1 (Of O]/Y) ^>_

ⁿ

Woln() (n-1)/n,

where

wn

^isthevolume

of

^{theunitballin}

n.

The equalitycase occurs

if

^and

only

if

^{glis aEuclidean}

half

^{ballwithOf OVa;anEuclidean}

half

^sphere.

Remark For n--2, we recover the well-known relativeisoperimetric inequalityi.e.for^g/c

V

Length(0f- 0V)

²

27rArea(ft).

By

the well-known arguments^{we can} deduce optimal Sobolev type inequality.

COROLLARY Suppose theconvex

function f

^iscontinuously

differentia-

ble.Let beasmooth

function

with compactsupportin 14;

(note

^that may notvanish on

OW).

Thenwehave

JW ^{ix7 _>}

ⁿ

^I1

We nowsketch a briefoutline of theproofof the MainTheorem.

Precisestatements willbe giveninthenexttwo sections.Weconsiderthe abstract geodesic metricspace (in^thesenseof Alexandrov and

Gromov)

A4 definedbythedisjointunionof

W

^withitselfunderthe identifica- tionalong

OW.

WewillshowA//isnonpositivelycurved in thesenseof

Gromov,

i.e. CAT(0)-inequalityholds forany geodesic trianglein3//.

Since A4 is piecewise-linear we may apply the optimal isoperimetric inequality in

[2]

recently proved by Caoand Escobar. Thentherequired relative isoperimetric inequality in

W

follows from the existence of canonical isometric reflection in i.e. the reflection with respect to

OW.

2.

PRELIMINARIES

Welist someaspects of thetheoryof geodesicmetricspaceinthesense ofAlexandrovandGromovwhich willbe usedintheproof. Fordetails werefer the readerto

[1]

and

[2].

Let

(A’, d)

^bealocally-compact, geodesic^metricspace.Thenanytwo pointsp and q ofA’canbe connectedby^ageodesic(i.e.distance-realizing

curve).

Wedenotethisgeodesicbyff-. Let^A

A(a0,

al,

a2)

^beageodesic triangle with vertices _{a0, al} and a2. The corresponding comparison triangleA’ A’

(a, a, a)

^{is a}trianglein^]12withthesamesidelengths as

A.

A is said tosatisfy CAT(0)-inequality (comparison inequality of Alexandrov andToponogov)if, foranypE and the corresponding pointp’ E

aa’

^{2such that}

^d(al,p) dR2(dl,p’),

^wehave the comparison inequality

d(ao,p) < dR2(ao,p’).

Ais said tosatify CAT*(0)-inequality if, for any pE and q aoa2 and thecorrespondingpoints

p’

_aoaland

q’ aoa

^2,wehave

d(p, q) <_ d:(p’, q’).

Wenowrecall threeLemmasfrom

[1].

LEMMA1

[1,

^Lemma

3]

^CAT(O)-inequality andCAT* (O)-inequalityare equivalentonanyconvexsubset

of

^X.

LEMMA

2

[1,

^Lemma

4]

^Let

A(ao,

b,

c)

^and

A(al,

b,

c)

be geodesic triangles satisfyingCAT*(O)-inequality.Suppose

aob

^tA

bal

^{is a}(minimal) geodesic between

ao

^andal. Then the geodesic triangle

A(ao,

al,

c)

^also

satisfies

^theCAT*(O)-inequality.

LEMMA 3

[1,

^Corollary

5]

^Let

"1

^and,’9(2be geodesic spaces satisfying CAT*(O)-inequality.

Suppose A1

^{C 2(1}

andA2

^{C2(2areclosedandconvex}

subsets such that thereisan isometry

of A1

^onto

A2.

Then the glueing2(

of

2(1 and2(2under the

identification of

^{A and}

A2

^{viatheisometry isalsoa}

geodesic spacesatisfying CAT*(O)-inequalitywiththecanonicalglueing metricgivenby

f ^dxj(p,q)

^{ifp,qE 2(j (j=}

^1,2)

dx(p,q) inf[dxl(p,r) + dx:(q,(r))] if

^{p 2(1 and q 2(2.}

rEAl

A

geodesicspace²⁽is said tobe nonpositively curvedinthesenseof GromovifCAT(0)-inequality is satisfied,locally. Itwasrecently proved by Cao and Esocobar that the Euclidean isoperimetric inequality remains true in a simply-connected, complete, piecewise-linear space which isnonpositivelycurvedinthesenseofGromov.

THEOREM

[2]

^Let

.M

ⁿ be a simply-connected, complete,piecewise- linear

manifold

^{which is} nonpositively curved in the sense

of

^Gromov.

Then,

for

^{anycompactdomain9t}

^c

^iV[with

rectifiable

^{boundaryOf,the}

Euclidean isoperimetricinequality

Voln-1 (O) nwln/n ^Vol,(ff)

^(n-1)/n

holds, and the equality occurs

if

^andonly

if

^{f is} isometric to a Eucli- dean ball.

3.

PROOF OF THE MAIN THEOREM

First notethat the

convex

functionfdefiningthe cylindrical domain14 may be assumedtobepiecewise-linear bysimple approximation argu- ments.

IV

^isclearlyageodesicspacewithrespecttothe intrinsicmetric i.e. forany p, q

W,

dw(p, q)

^inf

Length(c),

where the infimumistakenoverallcontinuouspaths^clying in

W

and connecting p to q. With respect tothis metric the boundary 0"14; is a closed andconvexsubsetof

W.

Notealso that

OW

is isometricto

IR n-l,

because

cOW

^istheproductof thegraph

off:

^IR

^IR

^andIR

n-2. Now

we considerthe piecewise-linear geodesicspace

Mn

obtained from glueing

W

^withitselfunder the identification along

OW.

We claimthat

M

is nonpositively curved in thesenseofGromov. InviewofLemmas and 3 itsufficestocheck the validity ofCAT*(0)-inequalityin14;.

Let^A

A(a,

b,

c)

^{be a}geodesic triangle in

W.

^IfA is anEuclidean triangle, there isnothingtoprove. If A lies entirely ⁱⁿ

OW, CAT*(0)-

inequalityisalsotrivialbecause

OW

isisometrictoIR

-1.

The remaining case canbe handledwiththeaidofLemma2.In fact,sincethe

functionf

ispiecewise-linear and the closeddomain

W

is concave(i.e. ^{the com-} plementofa convexdomain in

n),

^{we can}decomposeA into theunion of finite number ofEuclideantriangles

Aj,

^{j 1,2,...,l.}

Hence,

apply- ingLemma2 successively,weconclude that A satisfies CAT*(0)-inequal- ity andthereby^wehaveprovedthe claim.

Let

f be any compactdomain in

W

with rectifiableboundary^0[2.

Note that the space 3d has the canonical isometric reflection (with respectto

OW)

and thus thereis acompactdomain

2 c

2M,theunion of f with itself under the identification along 0f 0W, satisfying the following volumerelation

Voln(fi) 2Voln()),

Vol_

(Off)

^2Vol_l

(0f2 cOW).

Since

AA

^{is a} simply-connected, complete,piecewise-linear andnon- positively curved space wemay apply the isoperimetric inequality by Caoand Escobar andthen,substituting the above volume relation,we obtainthe required^relativeisoperimetricinequality. The equalitycase follows easily from the equalitycaseof

[2]

^{and the}proofiscompleted.

Acknowledgments

This work was partially supported by

GARC,

BSRI-94-1416. The author wouldlike tothank ProfessorJ.Choe for bringing the paper

[2]

tohis attentionand for fruitful conversations.

References

[1] W.Ballmann, Singular spaces of nonpositive curvature, Surlesgroupeshyperboliques d’apres MikhaelGromov, E.Ghys andP.de laHarpe (Eds.)Birkhauser(1990).

[2] J.CaoandJ.Escobar,Anisoperimetric comparison theorem forPL-manifoldsof non-positivecurvature, preprint.

[3] P.L.LionsandF.Pacella, Isoperimetric inequalities forconvexcones,Proc.A.M.S.109 477-485(1990).

[4] P.L.Lions,F. Pacella andM.Tricarico,Bestconstants inSobolev inequalities for functionsvanishing onsomepart of theboundary and related questions,Indiana Univ.

Math.J. 37 301-324(1988).