EGULAITY OF NONLOCAL MINIMAL CONES IN DIMENSION 2 OVIDIU SAVIN AND ENICO VALDINOCI Abstrct. We show tht the only nonlocl s-miniml cones in 2 re the trivil ones for ll s 0, 1). As consequence we obtin tht the singulr set of nonlocl miniml surfce hs t most n 3 Husdorff dimension. 1. Introduction Nonlocl miniml surfces were introduced in [2] s boundries of mesurble sets E whose chrcteristic function χ E minimizes certin H s/2 norm. More precisely, for ny s 0, 1), the nonlocl s-perimeter functionl Per s E, Ω) of mesurble set E in n open set Ω n is defined s the Ω-contribution of χ E in χ E H s/2, tht is 1) Per s E, Ω) := LE Ω, n \ E) + LE \ Ω, Ω \ E), where LA, B) denotes the double integrl dx dy LA, B) :=, A,B mesurble sets. x y n+s A set E is s-miniml in Ω if Per s E, Ω) is finite nd A B Per s E, Ω) Per s F, Ω) for ny mesurble set F for which E \ Ω = F \ Ω. We sy tht E is s-miniml in n if it is s-miniml in ny bll B for ny > 0. The boundry of s-miniml sets re referred to s nonlocl s-miniml surfces. The theory of nonlocl miniml surfces developed in [2] is similr to the theory of stndrd miniml surfces. In fct s s 1, the s-miniml surfces converge to the clssicl miniml surfces nd the functionl in 1) fter multipliction by fctor of the order of 1 s)) Gmm-converges to the clssicl perimeter functionl see [3, 1]). In [2] it ws shown tht nonlocl s-miniml surfces re C 1,α outside singulr set of Husdorff dimension n 2. The precise dimension of the singulr set is determined by the problem of existence in low dimensions of nontrivil globl s-miniml cones i.e. n s-miniml set E such tht te = E for ny t > 0). In the cse of clssicl miniml surfces Simons theorem sttes tht the only globl miniml cones in dimension n 7 must be hlf-plnes, which implies tht the Husdorff dimension of the singulr set of miniml surfce in n is n 8. In [4], the uthors used these results to show tht if s is sufficiently close to 1 the sme holds for s-miniml surfces i.e. globl s-miniml cones must be hlf-plnes if n 7 nd the Husdorff dimension of the singulr set is n 8. Given the nonlocl chrcter of the functionl in 1), it seems more difficult to nlyze globl s- miniml cones for generl vlues of s 0, 1). The purpose of this short pper is to show tht tht there re no nontrivil s-miniml cones in the plne. Our theorem is the following. Theorem 1. If E is n s-miniml cone in 2, then E is hlf-plne. 1
2 OVIDIU SAVIN AND ENICO VALDINOCI From Theorem 1 bove nd Theorem 9.4 of [2], we obtin tht s-miniml sets in the plne re loclly C 1,α. Corollry 1. If E is s-miniml set in Ω 2, then E) Ω is C 1,α -curve. In higher dimensions, the result of Theorem 1 nd the dimensionl reduction performed in [2] imply tht ny nonlocl s-miniml surfce in n is loclly C 1,α outside singulr set of Husdorff dimension n 3. Corollry 2. Let E be nonlocl s-miniml surfce in Ω n nd let Σ E E Ω denote its singulr set. Then H d Σ E ) = 0 for ny d > n 3. The ide of the proof of Theorem 1 is the following. If E 2 is s-miniml cone then we construct set Ẽ s trnsltion of E in B /2 which coincides with E outside B. Then the difference between the energies of the extension) of Ẽ nd E tends to 0 s. This implies tht lso the energy of E Ẽ is rbitrrily close to the energy of E. On the other hnd if E is not hlf-plne the set Ẽ E cn be modified loclly to decrese its energy by fixed smll mount nd we rech contrdiction. In the next section we introduce some nottion nd obtin the perturbtive estimtes tht re needed for the proof of Theorem 1 in Section 3. 2. Perturbtive estimtes We strt by introducing some nottion. Nottion. We denote points in n by lower cse letters, such s x = x 1,..., x n ) n nd points in n+1 + := n 0, + ) by upper cse letters, such s X = x, x n+1 ) = x 1,..., x n+1 ) n+1 +. The open bll in n+1 of rdius nd center 0 is denoted by B. Also we denote by B + := B n+1 + the open hlf-bll in n+1 nd by S n + := S n n+1 + the unit hlf-sphere. The frctionl prmeter s 0, 1) will be fixed throughout this pper; we lso set := 1 s 0, 1). The stndrd Eucliden bse of n+1 is denoted by {e 1,..., e n+1 }. Whenever there is no possibility of confusion we identify n with the hyperplne n {0} n+1. The trnspose of squre mtrix A will be denoted by A T, nd the trnspose of row vector V is the column vector denoted by V T. We denote by I the identity mtrix in n+1. We introduce the functionl 2) E u) := B + ux) 2 x n+1 dx. which is relted to the s-miniml sets by n extension problem, s shown in Section 7 of [2]. More precisely, given set E n with loclly finite s-perimeter, we cn ssocite to it uniquely its extension function u : + n+1 whose trce on n {0} is given by χ E χ n \E nd which minimizes the energy functionl in 2) for ny > 0. We recll see Proposition 7.3 of [2]) tht E is s-miniml in n if nd only if its extension u is miniml for the energy in 2) under compct perturbtions whose trce in n {0} tkes the vlues ±1. More precisely, for ny > 0, 3) E u) E v) for ny v tht grees with u on B + {x n+1 > 0} nd whose trce on n {0} is given by χ F χ n \F for ny mesurble set F which is compct perturbtion of E in B.
EGULAITY OF NONLOCAL MINIMAL CONES 3 Next we estimte the vrition of the functionl in 2) with respect to horizontl domin perturbtions. For this we introduce stndrd cutoff function Given > 0, we let ϕ C 0 n+1 ), with ϕx) = 1 if X 1/2 nd ϕx) = 0 if X 3/4. 4) Y := X + ϕx/)e 1. Then we hve tht X Y = Y X) is diffeomorphism of n+1 + s long s is sufficiently lrge possibly in dependence of ϕ). Given mesurble function u : n+1 +, we define 5) u + Y ) := ux). Similrly, by switching e 1 with e 1 or ϕ with ϕ in 4)), we cn define u Y ). In the next lemm we estimte discrete second vrition for the energy E u). Lemm 1. Suppose tht u is homogeneous of degree zero nd E u) < +. Then 6) E u + ) + E u ) 2E u) C n 3+, for suitble C 0, depending on ϕ nd u. Proof. We strt with the following observtion. Let = 1,..., n+1 ) n+1 nd 1...... n+1 0...... 0 A :=... 0...... 0 with 1 + 1 0. Then direct computtion shows tht 7) I + A) 1 = I 1 A A = I 1 + 1 deti + A). Now, we define nd Notice tht 1 if /2 X, χ X) := 0 otherwise 1 ϕx/)...... n+1 ϕx/) MX) := 1 0...... 0.... 0...... 0 8) M = O1/) χ. Let now X) := det D X Y X) = deti + MX)) = 1 + 1ϕX/) = 1 + tr MX).
4 OVIDIU SAVIN AND ENICO VALDINOCI By 7), we see tht 9) Also, 1/ = 1 + O1/), therefore, by 8), DX Y ) 1 = I + M ) 1 = I M. M M T 10) = O1/ 2 )χ. Now, we perform some chin rule differentition of the domin perturbtion. For this, we tke X to be function of Y nd the functions u, Y, χ, M nd will be evluted t X, while u + will be evluted t Y e.g., the row vector X u is short nottion for X ux), while Y u + stnds for Y u + Y )). We use 5) nd 9) to obtin Y u + = Xu D Y X = X u D X Y ) 1 = X u I M ). Also, by chnging vribles, dy = det D X Y dx = dx. Accordingly Y u + 2 yn+1 dy = X u I M ) I M ) T X n+1 dx ) = X u I M M T + MMT X n+1 dx ) 1 ) = X u + tr M I M M T + MMT X n+1 dx. Hence, from 10), Y u + 2 yn+1 dy = X u 1 + tr M ) I M M T + O1/ 2 )χ ) X n+1 dx. The similr term for Y u my be computed by switching ϕ to ϕ which mkes M switch to M): thus we obtin Y u 2 yn+1 dy = X u 1 tr M ) I + M + M T + O1/ 2 )χ ) X n+1 dx. By summing up the lst two expressions, fter simplifiction we conclude tht Y 11) u + 2 + Y u 2) ) yn+1 dy = 2 1 + O1/ 2 )χ X u 2 x n+1 dx. On the other hnd, the function gx) := X ux) 2 x n+1 is homogeneous of degree 2, hence χ X u [ ] 2 x n+1 dx = g dx = gϑϱ) dϑ ϱ n dϱ B + = /2 ϱ n+ 2 [ S n + B + \B+ /2 gϑ) dϑ ] /2 dϱ = C n+ 1, S n +
EGULAITY OF NONLOCAL MINIMAL CONES 5 for suitble C depending on u. This nd 11) give tht Y u + 2 + Y u 2) B + yn+1 dy 2 = O1/ 2 ) χ X u 2 x B + n+1 dx = O1/ 2 ) C n+ 1, which completes the lemm. B + X u 2 x n+1 dx Lemm 1 turns out to be prticulrly useful when n = 2. In this cse 6) implies tht 12) E u + ) + E u ) 2E u) C s, nd the right hnd side becomes rbitrrily smll for lrge. As consequence, we lso obtin the following corollry. Corollry 3. Suppose tht E is n s-miniml cone in 2 nd tht u is the extension of χ E χ 2 \E. Then 13) E u + ) E u) + C s. Proof. Since E is cone, we know tht u is homogeneous of degree zero see Corollry 8.2 in [2]): thus, the ssumptions of Lemm 1 re fulfilled nd so 12) holds true. From the minimlity of u see 3)), we infer tht which together with 12) gives the desired clim. E u) E u ), 3. Proof of Theorem 1 We rgue by contrdiction, by supposing tht E 2 is n s-miniml cone different thn hlfplne. By Theorem 10.3 in [2], E is the disjoint union of finite number of closed sectors. Then, up to rottion, we my suppose tht sector of E hs ngle less thn π nd is bisected by e 2. Thus, there exist M 1 nd p E B M on the e 2 -xis) such tht p ± e 1 2 \ E. Let > 4M be sufficiently lrge. Using the nottion of Lemm 1 we hve 14) u + Y ) = uy e 1), for ll Y B + 2M, nd u + Y ) = uy ) for ll Y n+1 + \ B+, where u is the extension of χ E χ 2 \E. We define v X) := min{ux), u + X)} nd w X) := mx{ux), u + X)}. Denote P := p, 0) 3. We clim tht 15) Indeed, by 14) while u + < w = u in neighborhood of P, nd u < w = u + in neighborhood of P + e 1. u + P ) = up e 1) = χ E χ 2 \E)p e 1 ) = 1 up ) = χ E χ 2 \E)p) = 1.
6 OVIDIU SAVIN AND ENICO VALDINOCI Similrly, u + P + e 1) = up ) = 1 while up + e 1 ) = 1. This nd the continuity of the functions u nd u + t P, respectively P + e 1, give 15). We point out tht E u) E v ), thnks to 14) nd the minimlity of u. This nd the identity imply tht 16) E v ) + E w ) = E u) + E u + ), E w ) E u + ). Now we observe tht w is not minimizer for E 2M with respect to compct perturbtions in B + 2M. Indeed, if w were minimizer we use u w nd the first fct in 15) to conclude u = w in B + 2M from the strong mximum principle. However this contrdicts the second inequlity in 15). Therefore, we cn modify w inside compct set of B 2M nd obtin competitor u such tht E 2M u ) + δ E 2M w ), for some δ > 0, independent of since w restricted to B + 2M is independent of, by 14)). The inequlity bove implies 17) E u ) + δ E w ), since u nd w gree outside B + 2M. Thus, we use 16), 13) nd 17) to conclude tht E u ) + δ E w ) E u + ) E u) + C s. Accordingly, if is lrge enough we hve tht E u ) < E u), which contrdicts the minimlity of u. This completes the proof of Theorem 1. Acknowledgments OS hs been supported by NSF grnt 0701037. EV hs been supported by MIU project Nonliner Elliptic problems in the study of vortices nd relted topics, EC project ε: Elliptic Pde s nd Symmetry of Interfces nd Lyers for Odd Nonlinerities nd FIB project A&B: Anlysis nd Beyond. Prt of this work ws crried out while EV ws visiting Columbi University. eferences [1] L. Ambrosio, G. de Philippis, L. Mrtinzzi, Gmm-convergence of nonlocl perimeter functionls, Mnuscript Mth. 134 2011), no. 3 4, 377 403. [2] L. A. Cffrelli, J.-M. oquejoffre nd O. Svin, Nonlocl miniml surfces, Comm. Pure Appl. Mth. 63 2010), no. 9, 1111 1144. [3] L. A. Cffrelli nd E. Vldinoci, Uniform estimtes nd limiting rguments for nonlocl miniml surfces, Clc. Vr. Prtil Differentil Equtions 41 2011), no. 1 2, 203 240. [4] L. A. Cffrelli nd E. Vldinoci, egulrity properties of nonlocl miniml surfces vi limiting rguments, Preprint, http://www.m.utexs.edu/mp rc/c/11/11-69.pdf Ovidiu Svin Mthemtics Deprtment, Columbi University, 2990 Brodwy, New York, NY 10027, USA. Emil: svin@mth.columbi.edu Enrico Vldinoci
EGULAITY OF NONLOCAL MINIMAL CONES 7 Diprtimento di Mtemtic, Università degli Studi di Milno, Vi Cesre Sldini 50, 20133 Milno, Itly. Emil: enrico.vldinoci@unimi.it