Some remarks on uniqueness and regularity of Cheeger sets

Some Remarks on Uniqueness and Regularity of Cheeger Sets

V. CASELLES(*) - A. CHAMBOLLE(**) - M. NOVAGA(***)

ABSTRACT- We show that generically the subsets ofR^Nwith finitevolumehavea unique Cheeger set, in the sense that there always exists a nearby set which has a unique Cheeger set. We also prove that Cheeger sets areC^1;1, when the am- bient set isC^1;1.

1. Introduction.

Given a nonempty set VR^N with finite volume, we call Cheeger constant ofVthequantity

hV :ˆ min

FV

P(F) jFj ; …1†

wherejFjdenotes deN-dimensional volume ofF,P(F) denotes the peri- meter ofF[5], and the minimum is taken over all nonempty sets of finite perimeter contained in V. A Cheeger set of V is any set GV which minimizes (1).

For any setFof finite perimeter inR^N, let us define l_F :ˆP(F)

jFj :

(*) Indirizzo dell'A.: Departament de Tecnologia, Universitat Pompeu-Fabra, Barcelona, Spain.

E-mail: [email protected]

(**) Indirizzo dell'A.: CMAP, CNRS UMR 7641, Ecole Polytechnique, 91128 Palaiseau Cedex, France.

E-mail: [email protected]

(***) Indirizzo dell'A.: Dipartimento di Matematica Pura ed Applicata, UniversitaÁ di Padova, Via Trieste 63, 35121 Padova, Italy.

E-mail: [email protected]

Notice that for any Cheeger setGofVit holdslGˆhV, as a consequence Gis a Cheeger set ofVif and only ifGsolves the minimum problem (whose valueis zero):

minFVP(F) h_VjFj:

…2†

Finding the Cheeger sets of a given set V is, in general, a difficult task. This task is simplifie d ifV is a convex set andNˆ2. In that case, there is a unique Cheeger set and is given by V^RB_R where V^R:ˆ fx2V:dist(x; @V)>RgandR>0 is such thatjV^Rj ˆpR²[2, 23]

(wedenoteby XY theset fx‡y:x2X;y2Yg). In particular, we observe that the Cheeger set ofV is convex. Both features, uniqueness and convexity of the Cheeger set are due to the convexity ofV(a simple counterexample is given in [23] when V is not convex).

The uniqueness of the Cheeger set inside bounded convex subsets of R^N was proved in [13] when the convex body is uniformly convex and of classC², and in [1] in the general case. In the convex case, the C^1;1 reg- ularity of Cheeger sets is a consequence of the results in [18, 19, 28].

Moreover, a Cheeger set can be characterized in terms of the mean cur- vature of its boundary: the sum of the principal curvatures being bounded by the Cheeger constant (see [17, 6, 23, 2] for Nˆ2 and [3, 1] for the general case).

Let us comment on the role played by the Cheeger constant in other contexts. Given an open bounded setVR^Nwith Lipschitz boundary and p2(1;1), the Cheeger constant ofVpermits to give a lower bound on the first eigenvalue of the p-Laplacian on V with Dirichlet boundary condi- tions. Indeed, if we define

l_p(V):ˆ min

06ˆv2W^1;p₀ (V)

R

V jrvj^pdx R

V jvj^pdx ; …3†

then

l_p(V) h_V p

p: …4†

This result was proved in [15] whenpˆ2 and extended to anyp2(1;1) in [21]. Whenpˆ1 the first eigenvalue of the 1-Laplacian is defined by

l1(V):ˆ min

06ˆv2BV(V)

R

VjDvj ‡ R

@VjvjdH^{N 1} R

Vjvjdx ; …5†

whereBV(V) denotes the space of functions of bounded variation inV. Then l1(V)ˆhV and both problems are equivalent in the following sense: a functionu2BV(V) is a minimum of (5) if and only if almost every level set is a Cheeger set (see [22]). These results have been extended in several di- rections, in particular, using weighted volume and perimeter [11, 7] and for anisotropic versions of the perimeter [24]. Let us also recall that Cheeger sets are related to the global behavior of solutions of the time-dependent constant-mean-curvature equation under vanishing initial condition and Dirichlet boundary data [26]. Finally, we mention an interesting inter- pretation of the Cheeger constant in terms of the max flow min cut theorem [27, 20].

Theplan of thepaper is thefollowing: in Section 2 weshow theex- istence of the maximal and the minimal Cheeger sets, inside any set V R^Nof finite volume. In Section 3 we prove that there exists a unique Cheeger set, up to arbitrarily small perturbations ofV. Finally, in Section 4 we show that Cheeger sets are always of classC^1;1, out of a singular set of dimension at mostN 8, whenV is also of classC^1;1. In Remark 4.2, we point out that the uniqueness and regularity results can be extended to minimizers of (2), withh_V replaced by a genericl>h_V.

2. Maximal and minimal Cheeger sets.

DEFINITION2.1. LetVbe a measurable set inR^Nof finite volume. We say that a Cheeger set X V is a maximal Cheeger set if Y X for all Cheeger sets Y V. We say that X is a minimal Cheeger set if either YX o r Y\Xˆ ;for all Cheeger sets Y V.

LEMMA2.2. Let X;Y be twoCheeger sets inV. Then X[Y and X\Y (if non-empty) are also Cheeger sets inV.

PROOF. SinceX;Y are Cheeger sets, we have

P(X[Y)‡P(X\Y)P(X)‡P(Y)ˆh_V(jXj‡jYj)ˆh_V(jX[Yj‡jX\Yj):

Now, using that

P(X\Y) jX\Yj h_V wehavethat

P(X[Y)hVjX[Yj:

As a consequenceX[Yis Cheeger, henceP(X[Y)ˆhVjX[Yj. Then, we deduce thatP(X\Y)ˆhVjX\Yj, that isX\Y is also a Cheeger set. p

As a consequence of Lemma 2.2, we obtain:

LEMMA 2.3. There exists a maximal Cheeger set CmaxV. Moreover C_maxis a bounded set.

The second assertion easily follows from standard density estimates for solutions of (2): there exists r₀>0 and d>0 such that if r5r₀, either jB_r(x)\C_maxj>d, or there existsr⁰5rwithjB_r⁰(x)\C_maxj ˆ0, see [4]. In particular, it shows that the set of points whereC_max(or any other Cheeger set ofV) has Lebesgue density zero is an open set. This is not true for the points of density one, at least ifVis not open, as shown by the example of a setVwith empty interior.

LEMMA 2.4. Let X;Y be twoCheeger sets in V. Assume that X is minimal, that is, it contains no other Cheeger set inside. Then either XY o r X\Yˆ ;. In particular, twodifferent minimal Cheeger sets are dis- joint.

PROOF. IfX\Yis nonempty, then it is also a Cheeger set contained in X. SinceXis minimal, wehaveX\Y ˆX, that isXY. p Recall that, by the isoperimetric inequality, there exists a constant aˆa(V)>0 such that any Cheeger set inVhas volume greater or equal toa.

LEMMA2.5. There are minimal Cheeger sets inVand they are finite in number. In particular, Cheeger sets of minimal volume are minimal Cheeger sets, and any Cheeger set contains a minimal Cheeger set.

PROOF. Consider the problem minfjXj:X is a Cheeger set of {V}.

Then any minimizing sequence has a subsequence converging to a set, say X, such thatXis a Cheeger set of minimal volume. By Lemma 2.2, the set Xdoes not intersect any other Cheeger set, therefore is minimal. Since any of such sets has a volumea, there are only finitely many of them. To prove the last assertion, we just take a minimal volume Cheeger set between the ones contained in the given Cheeger set. p

REMARK2.6. IfVis an open set andCis a minimizer of (2), by classical regularity results [25] we know that (@CnS)\V is analytic, whereSis a closed singular set of dimension at mostN 8. Moreover, ifV is of class C^1;1, then C is a minimizer of a prescribed curvature problem with cur- vatureinL¹ [8], hence@CnSis of classW^2;pfor allp51(see also [29] for thecaseNˆ3).

REMARK2.7. By a result of Giusti [17], an open setXVis a minimal Cheeger set iff X has finite perimeter and there is a solution of the ca- pillary problem inX(with vertical contact angle), i.e. there exists a vector fieldz:X!R^Nsuch thatjzj51 and divzˆhV.

REMARK2.8. The computation of the maximal Cheeger set has been the object of recent interest [12]. By adapting the proof of Proposition 4 in [3] one can prove the following result. LetV be a bounded subset of R^N with Lipschitz continuous boundary, and let u2BV(V)\L²(V) bethe solution of thevariational problem

(Q)_l: min

u2BV(V)\L²(V)

( Z

V

jDuj ‡l 2 Z

V

(u 1)²dx‡ Z

@V

jujdH^{N 1} )

: …6†

Then 0u1. Let Es:ˆ fusg, s2(0;1]. Then for any s2(0;1] we have

P(E_s) l(1 s)jE_sj P(F) l(1 s)jFj …7†

for anyFV. Ifl>0 is big enough, indeed greater than 1=kx_Vkwhere kx_Vk :ˆmax

( Z

R^N

ux_Vdx: u2BV(R^N);

Z

V

jDuj 1 )

;

then the level setfuˆ kuk₁gis the maximal Cheeger set ofV. In parti- cular, the maximal Cheeger set can be computed by solving (6), and for that wecan usethealgorithm in [14].

3. Uniqueness of Cheeger sets up to small perturbations.

We prove that the Cheeger set is unique, up to arbitrarily small per- turbations of the ambient setV.

THEOREM1. LetVR^Nbe an open set with finite volume. Then, for any compact set KVthere exists a bounded open setVK Vsuch that KV_KandV_Khas a unique Cheeger set.

PROOF. By Lemma 2.5, we know that V has a finitenumber of dis- joint minimal Cheeger sets. LetCbe a minimal Cheeger set ofV, le tVe be any open set such thatKVeV, and le tVK:ˆC[V. Notice thate C is also a (minimal) Cheeger set inVK, and wewant to show that it is the only one. Indeed, let Dbe a Cheeger set inV_K, then by Lemma 2.4 either DC orD\Cˆ ;. The latter cannot happen, since in this case wewould haveDV_KnCVe V, but thedistanceof D from the boundary of V cannot be positive, otherwise we could decrease the quotientP(D)=jDjby rescaling Dwith a factor larger than one. It then follows DC. By Remark 2.6 there exist singular sets S_C@C and

S_D@D, of dimension at mostN 8, such thatA_C:ˆ(@CnS_C)\Vand

A_D:ˆ(@DnS_D)\V areboth analytic solutions of thegeometric equa- tion (N 1)Hˆh_V, whe re H denotes the mean curvature. As a con- sequence, since H^{N 1}(AC\AD) H^{N 1}((@CnV)e \V)>0, by analytic continuation weget ADˆA_C. Moreprecisely, assumeby contradiction that wecan find x2A_C\A_D such that A_C\B_r(x)6ˆA_D\B_r(x) for all r>0. LettingTbethetangent hyperplaneto@Datx, wecan write@D and@Cas thegraph of two smooth functionsvandv, re spe ctive ly, ove r T\Br(x) forr>0 small enough. Identifying T\Br(x) withBrR^{N 1}, wehavethatv;v :Br!Rboth solvetheequation

div Dv 1‡ jDvj²

q ˆ hV:

…8†

Moreover, it holdsvv,v(0)ˆv(0) andv(~y)>v(~y) for somey~2B_r. LetBbean open ball such thatBB_r,v>vonBandv(y)ˆv(y) for somey2@B. Noticethat, sincebothvandvbelong toC¹(B)\C¹(B), the fact thatv(y)ˆv(y) also implies thatDv(y)ˆDv(y). InB, both func- tions solve(8). Letting now wˆv v, wehavethat w(y)ˆ0 and Dw(y)ˆ0, whilew>0 insideB. For anyx2Bwehave

0ˆdiv…DC…Dv…x†† DC…Dv…x†††

ˆdiv Z¹

0

D²C Dv…x† ‡t…Dv…x† Dv…x††

dt

! Dw…x†

!

; whereC(p)ˆ 

1‡ jpj² q

, so thatwsolves a linear, uniformly elliptic equa-

tion with smooth coefficients. Then Hopf's lemma [16] implies that Dw(y)nB(y)50, a contradiction. HenceA_C ˆAD, which is equivalent to

CˆD. p

REMARK3.1. Notice that, given any open setVwith finitevolume, for alle>0, wecan find a setVeVsuch thatjVnVej5eandVehas a unique Cheeger set. Indeed, considering as above a minimal Cheeger setCV, wecan define

V_e:ˆVn [

q2Q^N\VnC

B_r(q);

wherer(q)>0 is such thatBr(q)VnCand X

q2Q^N\VnC

r(q)5e:

Let Dbe a Cheeger set inVe different fromC, thenjDnCj>0. By the regularity result in Remark 2.6, it follows thatDnChas nonempty interior, which is impossibleby theconstruction ofVe.

Wecan requirethat alsoV_e is open but the construction is a bit more complicated. First, we need to remove from V a small closed ball inside each minimal Cheeger set different fromC. This ensures that any Cheeger setC⁰in the new set must containC. Then, we remove fromVa (possibly countable) union of closed balls contained in VnC, each one touching a connected component of@C\V.

REMARK3.2. For a general open setV, onemay also consider a dif- ferent notion of Cheeger set, based on the following definition of peri- meter:

P_V(E):ˆsup(Z

E

divfdx:f2C¹(V;R^N);jfj 1;divf2L¹(V) )

;

which coincides with the lower semicontinuous relaxation of the usual perimeter restricted to the compact subsets of V. Noticethat such no- tion of Cheeger set gives a higher Cheeger constant of V, which still verifies (4), and it coincides with the classical notion if, for instance,Vis thesubgraph of a continuous function near each point of its boundary.

We observe that Theorem 1 remains true also with this definition of Cheeger set.

4. Regularity of Cheeger sets in regular domains.

We now show that each Cheeger set ofVis of classC^1;1, ifVis also of classC^1;1.

THEOREM2. LetVbe a bounded open set with boundary of class C^1;1. Then any Cheeger set C ofV has boundary of class C^1;1, out of a closed singular setS@C of dimension at most N 8.

PROOF. We know that any Cheeger set is a solution of the variational problem (2). LetCbe a Cheeger set ofV, and letx₀2(@CnS)\@V, where thesingular setSis as in Remark 2.6. We may assume that nearx0,@Vis thegraph of aC^1;1functionf :B2r!RwhereB2ris an (N 1)-dimensional ball centered atx₀of radius 2r. Wemay as well assumethat@Cis thegraph of u:B_2r!R. Weknow thatu2W^2;p(B_2r) for any p51, in particular u2C^1;a(B_2r) for anya51. Weobservethatuis a solution of

min (Z

Br



1‡jrvj² q

‡h_Vv

dx: v2BV(B_r);vf;vj_@Brˆuj_@Br )

: …9†

The result follows by adapting the proof of regularity for the obstacle problem in [9]. Indeed, since @V is of class C^1;1,rf has modulus of con- tinuitys(r)kr,k>0. LettingL(x):ˆf(x0)‡ rf(x0)(x x0), wehave

L(x) kr²f(x)u(x) x2Br: Weshall provethat

u(x)L(x)‡Cr² x2B^r₂; …10†

for someconstantC>0. Weshall denotebyCa positiveconstant that may vary from line to line. Considerwˆu (L kr²)0, and observe thatu satisfies the equation

div ru 1‡ jruj² q

0 B@

1

CA‡h_V 0 x2B_r; …11†

with equality inDˆ fx2Br : u(x)>f(x)g. Dueto theregularity ofu, (11) can bewritten

a_ij(x)@_xixju‡h_V 0 x2B_r; …12†

whereaij2C^a(Br) areuniformly positive. It follows that alsowsatisfies (12) (and, still, with an equality inD) . Let noww1bethesolution of

a_ij(x)@_xixjw₁‡h_V ˆ 0 x2B_r;

withw₁j_@Br ˆwj_@Br 0. Observe thatw₁w. Without loss of generality, weassumethatx₀ˆ0. Letgˆh_V= min

x2Br Tr(A(x)

, whereA(x)ˆ(a_ij(x)), and Q(x)ˆ(g=2)(jxj² r²). Then, Q is a subsolution of (12) in B_r, with Qj_@Br ˆ0, so thatQw1inBr. In particular, wehavethat

aij(x)@xixj(w1 Q) ˆ hV Tr(A(x)) minBr Tr(A) 1 0

@

1

A x2Br;

and the right-hand side of this equation is bounded byCr^a (since A(x) is HoÈlderian of exponenta). Wehave

w₁(x₀)w(x₀)ˆu(x₀) (L(x₀) kr²)ˆf(x₀) (L(x₀) kr²)ˆkr²; while, sincew1 Q0, it satisfies a Harnack inequality [16, Thms 9.20 and 9.22] inB_r=2:

w1(x) Q(x) Cinf

Br (w1 Q)‡Cr² Cw1(x0)‡Cg

2r²‡Cr²Cr²;

hence also w₁(x)Cr², for anyx2B_r=2 (for someconstantC>0 which does not blow-up asr!0).

Let now w2:ˆw w1. Thefunction w2 satisfies 0w2w Q, w₂j_@Br ˆ0, and

a_ij(x)@_xixjw₂0 x2B_r; …13†

again, with an equality ifx2D. Considerx2B_ra point wherew₂reaches its maximum: then, either w2(x)ˆ0, in which casew2ˆ0 insideBr, or w2(x)>0, in which casewemust havex62D, since(13) is satisfied with an equality inD(it could bethatw₂ is constant and maximal inD, in which casewemay always assumex2@D\B_r).

Thus, either w₂ˆ0 in B_r, or u(x)ˆf(x). In particular, in thelatter case, we find that for anyx2Br,

w2…x† w2…x† w…x† Q…x† ˆu…x† …L…x† kr²† ‡g

2…r² jxj²† f…x† f…0† rf…0† x‡Cr²Cr²; so thatw(x)ˆw₁(x)‡w₂(x) Cr²ifx2B_r=2, which shows (10). p

REMARK4.1. Since the Cheeger sets ofVaresolutions of (2), ifVis of classC^1;1andCis a Cheeger set ofV, wehave(N 1)H_C(x)hV for a.e.

x2@C.

REMARK 4.2. We point out that Theorems 1 and 2 extend also to minimizers of (2), withhV replaced by anyl>hV (see [3]).

Acknowledgments. The first author acknowledges partial support by PNPGC project, reference MTM2006-14836. The second author is partially supported by the ``Agence Nationale de la Recherche'', project ANR-06- BLAN-0082.

REFERENCES

[1] F. ALTER- V. CASELLES,Uniqueness of the Cheeger set of a convex body.To appear in Nonlinear Analysis, TMA.

[2] F. ALTER- V. CASELLES- A. CHAMBOLLE, Evolution of Convex Sets in the Plane by the Minimizing Total Variation Flow, Interfaces and Free Boundaries,7(2005), pp. 29-53.

[3] F. ALTER - V. CASELLES - A. CHAMBOLLE, A characterization of convex calibrable sets inR^N, Math. Ann.,332(2005), pp. 329-366.

[4] L. AMBROSIO,Corso introduttivo alla teoria geometrica della misura ed alle superfici minime, Scuola Normale Superiore, Pisa, 1997.

[5] L. AMBROSIO- N. FUSCO- D. PALLARA,Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, 2000.

[6] G. BELLETTINI- V. CASELLES- M. NOVAGA,The Total Variation Flow inR^N, J. Differential Equations,184(2002), pp. 475-525.

[7] G. BUTTAZZO- G. CARLIER- M. COMTE,On the selection of maximal Cheeger sets, Differential and Integral Equations,20(9) (2007), pp. 991-1004.

[8] E. BAROZZI- U. MASSARI,Regularity of minimal boundaries with obstacles, Rend. Sem. Mat. Univ. Padova,66(1982), pp. 129-135.

[9] L. A. CAFFARELLI, The obstacle problem revisited, TheJournal of Fourier Analysis and Applications,4(1998), pp. 383-402.

[10] L. A. CAFFARELLI- N. M. RIVIERE,On the rectifiability of domains with finite perimeter, Ann. Scuola NormaleSuperioredi Pisa,3(1976), pp. 177-186.

[11] G. CARLIER- M. COMTE, On a weighted total variation minimization problem, J. Funct. Anal.,250(2007), pp. 214-226.

[12] G. CARLIER- M. COMTE- G. PEYREÂ,Approximation of maximal Cheeger sets by projection, Preprint (2007).

[13] V. CASELLES- A. CHAMBOLLE- M. NOVAGA,Uniqueness of the Cheeger set of a convex body, Pacific Journal of Mathematics,232(1) (2007), pp. 77-90.

[14] A. CHAMBOLLE,An algorithm for total variation minimization and applica- tions, Journal of Mathematical Imaging and Vision,20(2004), pp. 89-97.

[15] J. CHEEGER, A lower bound for the smallest eigenvalue of the Laplacian,

Problems in Analysis, Princeton Univ. Press, Princeton (New Jersey, 1970), pp. 195-199.

[16] D. GILBARG - N. S. TRUDINGER, Elliptic partial Differential Equations of Second Order, Springer Verlag, 1998.

[17] E. GIUSTI, On the equation of surfaces of prescribed mean curvature.

Existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), pp. 111-137.

[18] E. H. A. GONZALEZ - U. MASSARI - I. TAMANINI, Minimal boundaries enclosing a given volume, Manuscripta Math.,34 (1981), pp. 381-395.

[19] E. GONZALEZ- U. MASSARI- I. TAMANINI,On the regularity of sets minimiz- ing perimeter with a volume constraint, Indiana Univ. Math. Journal, 32 (1983), pp. 25-37.

[20] D. GRIESER,The first eigenvalue of the Laplacian, isoperimetric constants, and the max-flow min-cut theorem, Arch. Math.,87(1) (2006), pp. 75-85.

[21] L. LEFTON- D. WEI,Numerical approximation of the first eigenpair of the p- laplacian using finite elements and the penalty method, Numer. Funct. Anal.

Optim.,18(3-4) (1997), pp. 389-399.

[22] B. KAWOHL- V. FRIDMAN,Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ.

Carolinae,44(2003), pp. 659-667.

[23] B. KAWOHL, T. LACHAND-ROBERT,Characterization of Cheeger sets for convex subsets of the plane, Pacific J. Math.,225(1) (2006), pp. 103-118.

[24] B. KAWOHL - M. NOVAGA, The p-Laplace eigenvalue problem as p!1and Cheeger sets in a Finsler metric, J. Convex Anal.,15(3) (2008), pp. 623-634.

[25] U. MASSARI, Esistenza e regolaritaÁ delle ipersuperfici di curvatura media assegnata inRⁿ, Arch. Rat. Mech. Anal.,55(1974), pp. 357-382.

[26] P. MARCELLINI- K. MILLER,Asymptotic growth for the parabolic equation of prescribed mean curvature, J. Differential Equations,51(3) (1984), pp. 326-358.

[27] G. STRANG, Maximal flow through a domain, Math. Programming, 26 (2) (1983), pp. 123-143.

[28] E. STREDULINSKY- W. P. ZIEMER,Area minimizing sets subject toa volume constraint in a convex set, J. Geom. Anal.,7(1997), pp. 653-677.

[29] J. TAYLOR,Boundary regularity for solutions to various capillarity and free boundary problems, Comm. in Partial Differential Equations,2(1977), 323-357.

Manoscritto pervenuto in redazione il 5 maggio 2009.