PRINCIPAL CURVATURE ESTIMATES FOR THE CONVEX LEVEL SETS OF SEMILINEAR ELLIPTIC EQUATIONS. Sun-Yung Alice Chang, Xi-Nan Ma, Paul Yang

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1 DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 28, Number 3, November 2010 doi: /dcds xx pp. 1 XX PRINCIPAL CURVATURE ESTIMATES FOR THE CONVEX LEVEL SETS OF SEMILINEAR ELLIPTIC EQUATIONS Su-Yug Alice Chag, Xi-Na Ma, Paul Yag Departmet of Mathematics, Priceto Uiversity Priceto NJ 08544, USA Departmet of Mathematics, Uiversity of Sciece ad Techology of Chia Hefei, , Ahui Provice, Chia Departmet of Mathematics, Priceto Uiversity Priceto NJ 08544, USA Dedicated to Professor Nireberg o his 85th birthday Abstract. We give a positive lower boud for the pricipal curvature of the strict covex level sets of harmoic fuctios i terms of the pricipal curvature of the domai boudary ad the orm of the boudary gradiet. We also exted this result to a class of semi-liear elliptic partial differetial equatios uder certai structure coditio. 1. Itroductio. The covexity of the level sets of the solutios of elliptic partial differetial equatios is a classical subject. For istace, Ahlfors [1] cotais the well-kow result that level curves of Gree fuctio o simply coected covex domai i the plae are the covex Jorda curves. I 1931, Gerge [8] proved the star-shapeess of the level sets of Gree fuctio o 3-dimesioal star-shaped domai. I 1956, Shiffma [20] studied the miimal aulus i R 3 whose boudary cosists of two closed covex curves i parallel plaes P 1, P 2. He proved that the itersectio of the surface with ay parallel plae P, betwee P 1 ad P 2, is a covex Jorda curve. I 1957, Gabriel [7] proved that the level sets of the Gree fuctio o a 3-dimesioal bouded covex domai are strictly covex, see also the book by Hormader [9]. Lewis [13] exteded Gabriel s result to p-harmoic fuctios i higher dimesios. Caffarelli-Spruck [6] geeralized the results [13] to a class of semiliear elliptic partial differetial equatios. Usig the idea of Caffarelli- Friedma [4], Korevaar [12] gave a ew proof o the results of [13, 6] by applyig the costat rak theorem of the secod fudametal form of the covex level sets of p-harmoic fuctio. A survey of this subject is give by Kawohl [11]. For more 2000 Mathematics Subject Classificatio. 35J05, 53J67. Key words ad phrases. Curvature estimate, level sets, semiliear elliptic equatio. Research of the first amed author ad third amed author are supported i part by DMS The first amed author also gratefully ackowledges partial support from the Mierva Research Foudatio ad the Charles Simoyi Edowmet fud durig the academic year while visitig the Istitute for Advaced Study. Research of the secod amed author was supported by NSFC No ad BaiRe Program i Chiese Academy of Scieces. Part of this work was doe while the secod amed author was visitig the Istitute for Advaced Study i , where he was supported by Natioal Sciece Foudatio uder No. DMS ad Zürich Fiacial Services. 1

2 2 SUN-YUNG ALICE CHANG, XI-NAN MA AND PAUL YANG recet related extesios, please see the papers by Biachii-Logietti-Salai [3] ad Bia-Gua-Ma-Xu [2]. Now we tur to the curvature estimates of the level sets of the solutios of elliptic partial differetial equatios. For 2-dimesioal harmoic fuctio ad miimal surface with covex level curves, Ortel-Scheider [19], Logietti [14] ad [15] proved that the curvature of the level curves attais its miimum o the boudary. Jost-Ma-Ou [10] ad Ma-Ye-Ye [18] proved that the Gaussia curvature ad the pricipal curvature of the covex level sets of 3-dimesioal harmoic fuctio attais its miimum o the boudary. I this paper, usig the strog maximum priciple, we obtai a pricipal curvature estimates for the strictly covex level set of higher dimesioal harmoic fuctio ad a class of semiliear elliptic partial differetial equatios. Our curvature estimate is i terms of the pricipal curvature of the boudary ad the boudary gradiet of the solutio of elliptic partial differetial equatios. Now we state our result. Theorem 1.1. Let Ω be a bouded domai i R, 2, a u b ad u C 4 (Ω) C 2 ( Ω) be a solutio for u = f(x, u) 0 i Ω. (1.1) Assume u = 0 i Ω. If the level sets of u are strictly covex with respect to ormal Du, ad let k 1 be the least pricipal curvature of the level sets. Cosider the followig two assertios (A 1 ) ad (A 2 ): (A 1 ) The fuctio Du k 1 attais it miimum o the boudary; (A 2 ) The fuctio Du 2 k 1 attais it miimum o the boudary. The we have the followig: Case 1: Suppose f = 0, the (A 1 ) is valid. Case 2: Suppose f = f(u). If f u 0, the (A 1 ) is valid; if f u 0, the (A 2 ) is valid. Case 3: Suppose f = f(x). If F (t, x) := t 3 f(x) is a covex fuctio for (t, x) (0, + ) Ω (or for f > 0 ad f 1 2 is cocave), the (A 1 ) is valid. Case 4: Suppose f = f(x, u). If f u 0 ad F u (t, x) := t 3 f(x, u) is a covex fuctio for (t, x) (0, + ) Ω for every choice of u (a, b), the (A 1 ) is valid. If the level sets of the solutio u i the above Theorem 1.1 are strictly covex with respect to ormal Du, the it is proved i [16, 17] that the orm of gradiet u attais its maximum ad miimum o the boudary. Combiig this fact with Gabriel ad Lewis theorem [7, 13], we have the followig cosequece. Corollary 1.2. Let u C ( Ω) satisfy u = 0 i Ω = Ω 0 \ Ω 1, u = 0 o Ω 0, u = 1 o Ω 1, (1.2) where Ω 0 ad Ω 1 are bouded covex smooth domais i R, 2, Ω 1 Ω 0. Let k 1 be the least pricipal curvature of the level sets of u i Ω, the we have the followig estimates mi k 1 mi Ω Ω k 1 mi Ω0 Du max Ω1 Du.

3 PRINCIPAL CURVATURE ESTIMATES FOR THE LEVEL SETS 3 We give a applicatio to the followig semiliear elliptic boudary value problem, its strict covexity of the level sets had bee obtaied i Caffarelli-Spruck [6] ad Korevaar [12]. Corollary 1.3. Let u C 4 (Ω) C 2 ( Ω) satisfy u = f(u) i Ω = Ω 0 \ Ω 1, u = 0 o Ω 0, u = 1 o Ω 1, (1.3) where Ω 0 ad Ω 1 are bouded covex smooth domais i R, 2, Ω 1 Ω 0. We assume f is C 2 icreasig fuctio ad f(0) = 0, ad let k 1 be the least pricipal curvature of the level sets of u i Ω, the we have the followig estimates. mi k 1 mi k 1( mi Ω 0 Du Ω Ω max Ω1 Du )2. Assumig u = 0, Biachii-Logietti-Salai [3] proved the covexity of the level sets of solutio u for some semiliear elliptic equatio i covex rig with Dirichlet boudary coditios as i (1.3). It follows from the costat rak theorem of the secod fudametal form of the covex level sets i [12], that the level sets are strictly covex. For the Poisso equatio, our structure coditio is the same as theirs. Now we outlie the proof of the Theorem 1.1. Let {a ij } be the symmetry curvature matrix o the strict covex level sets defied i (2.4), ad let {a ij } be its iverse matrix. We cosider the auxiliary fuctio ϕ(x, ξ) := Du θ a ij ξ i ξ j, where ξ = (ξ 1,...ξ 1 ) R 1, ξ = 1. For suitable choice θ, we shall derive the followig elliptic iequality ϕ 0 mod ϕ i Ω, (1.4) here we have suppressed the terms cotaiig the gradiet of ϕ with locally bouded coefficiets, the we apply the strog maximum priciple to obtai the results. I sectio 2, we first give brief defiitio o the covexity of the level sets, the obtai the curvature matrix a ij of the level sets of a fuctio, which appeared i [2, 5]. I sectio 3, we treat the semiliear elliptic partial differetial equatio ad complete the proof of Theorem 1.1. The mai techique i the proof of theorems cosists i rearragig the third derivatives terms usig the equatio ad the first derivatives coditio for ϕ. 2. The curvature matrix of level sets. I this sectio, we shall give the brief defiitio o the covexity of the level sets, the itroduce the curvature matrix (a ij ) of the level sets of a fuctio, which appeared i [2]. Firstly, we recall some fudametal otatios i classical surface theory. Assume a surface Σ R is give by the graph of a fuctio v i a domai i R 1 : x = v(x ), x = (x 1, x 2,, x 1 ) R 1. Defiitio 2.1. We defie the graph of fuctio x = v(x ) is covex with respect to the upward ormal ν = 1 W ( v 1, v 2,, v 1, 1) if the secod fudametal form b ij = v ij W of the graph x = v(x ) is oegative defiite, where W = 1 + v 2.

4 4 SUN-YUNG ALICE CHANG, XI-NAN MA AND PAUL YANG The pricipal curvature κ = (κ 1,, κ 1 ) of the graph of v, beig the eigevalues of the secod fudametal form relative to the first fudametal form. We have the followig well-kow formula. Lemma 2.2. ([5]) The pricipal curvature of the graph x = v(x ) with respect to the upward ormal ν are the eigevalues of the symmetric curvature matrix { } v il, (2.1) a il = 1 W v iv j v jl W (1 + W ) v lv k v ki W (1 + W ) + v iv l v j v k v jk W 2 (1 + W ) 2 where the summatio covetio over repeated idices is employed. Now we give the defiitio of the covex level sets of the fuctio u. Let Ω be a domai i R ad u C 2 (Ω), its level sets ca be usually defied i the followig sese. Defiitio 2.3. Assume u = 0 i Ω, we defie the level set of u passig through the poit x o Ω as Σ u(xo) = {x Ω u(x) = u(x o )}. Now we shall work ear the poit x o where u(x o ) = 0. By the implicit fuctio theorem, locally the level set Σ u(xo) ca be represeted as a graph x = v(x ), x = (x 1, x 2,, x 1 ) R 1, ad v(x ) satisfies the followig equatio u(x 1, x 2,, x 1, v(x 1, x 2,, x 1 )) = u(x o ). The the first fudametal form of the level set is g ij (1 + v 2 ) 1 2 = u u. The upward ormal directio of the level set is = δ ij + uiuj u, ad W = 2 ν = u u u (u 1, u 2,, u 1, u ). (2.2) Let h ij = u 2 u ij + u u i u j u u j u i u u i u j, (2.3) the the secod fudametal form of the level set of fuctio u is b ij = vij W = u hij u u. 3 Defiitio 2.4. For the fuctio u C 2 (Ω) we assume u 0 i Ω. Without loss of geerality we ca let u (x o ) 0 for x o Ω. We defie locally the level set Σ u(xo) = {x Ω u(x) = u(x o )} is covex with respect to the upward ormal directio ν if the secod fudametal form b ij is oegative defiite. Remark 2.5. If we let u be the upward ormal of the level set Σ u(xo) at x o, the u (x o ) > 0 by (2.2). Acccordig to the defiitio 2.4, if the level set Σ u(xo) is covex with respect to the ormal directio u, the the matrix (h ij (x o )) is opositive defiite. Now we obtai the represetatio of the curvature matrix (a ij ) of the level sets of the fuctio u with the derivative of the fuctio u, { 1 u i u l h jl u j u l h il a ij = u u 2 h ij + W (1 + W )u 2 + W (1 + W )u 2 u } iu j u k u l h kl W 2 (1 + W ) 2 u 4. (2.4) From ow o we deote u i u l h jl u j u l h il B ij = W (1 + W )u 2 + W (1 + W )u 2, C ij = u iu j u k u l h kl W 2 (1 + W ) 2 u 4, (2.5)

5 PRINCIPAL CURVATURE ESTIMATES FOR THE LEVEL SETS 5 ad A ij = h ij + B ij C ij, (2.6) the the symmetric curvature matrix of the level sets of u ca be represeted as 1 [ ] 1 a ij = hij u u 2 + B ij C ij = u u 2 A ij. (2.7) We ed this sectio with the followig Codazzi coditio which will be used i the ext sectios. Propositio 2.6. (see [2]) Deote a ij,k = aij x k for 1 i, j, k 1, the at the poit where u = u > 0, u i = 0, a ij,k is commutative i i, j, k, i.e. Proof. Direct calculatio shows a ij,k = a ik,j. a ij,k = u 1 u ijk + u 2 (u ij u k + u ik u j + u jk u i ). (2.8) The right had side of (2.8) is obviously commutative i i, j, k. 3. Pricipal curvature estimates of level set of Poisso equatio. I this sectio, we prove the Theorem 1.1. We study the followig equatio u = f(x, u) 0 i Ω. (3.1) Proof of Theorem 1.1: Sice the level sets of u are strictly covex with respect to ormal Du, the curvature matrix a ij of the level sets is positive defiite i Ω. Let a ij be the iverse matrix of a ij. We cosider the auxiliary fuctio ϕ(x, ξ) := Du θ a ij (x)ξ i ξ j, where ξ = (ξ 1,...ξ 1 ) R 1, ξ = 1. For suitable choice θ, we shall derive the followig elliptic iequality ϕ 0 mod ϕ i Ω, (3.2) where we modify the terms of the gradiet of ϕ with locally bouded coefficiets. The by the stadard strog maximum priciple, we get the result immediately. I order to prove (3.2) at a arbitrary poit x o Ω, as i Caffarelli-Friedma [4], we choose the ormal coordiate at x o. We have metioed i remark 2.5, sice the level sets of u are strictly covex with respect to ormal Du, by rotatig the coordiate system suitably by T xo, we may assume that u i (x o ) = 0, 1 i 1 ad u (x o ) = u > 0. Ad we ca further assume ξ = e 1, the matrix {u ij }(x o ) (1 i < j 1) is diagoal ad u ii (x o ) < 0. Cosequetly we ca choose T xo to vary smoothly with x o. If we ca establish (3.2) at x o uder the above assumptio, the go back to the origial coordiates we fid that (3.2) remai valid with ew locally bouded coefficiets o ϕ i (3.2), depedig smoothly o the idepedet variable. Thus it remais to establish (3.2) uder the above assumptio. Now we write ϕ(x) := Du θ a 11. From ow o, all the calculatios will be doe at the fixed poit x 0. Step1: we first compute the formula (3.19) Takig first derivative of ϕ, we get ϕ α = θ 2 Du θ 2 Du 2 αa 11 + Du θ a 11 α, (3.3)

6 6 SUN-YUNG ALICE CHANG, XI-NAN MA AND PAUL YANG sice it follows that a 11 α = 1 k,l=1 a 1k a 1l a kl,α, (3.4) a 11,α = θ u α u a 11 u θ a 2 11ϕ α. (3.5) From ow o, we follow the covetio: the Greek idices 1 α, β, γ, the Lati idices 1 i, j, k, l 1. Sice a 11 αα = 1 k,l,r,s=1 [a 1r a ks a 1l a kl,α a rs,α + a 1k a 1r a ls a kl,α a rs,α ] 1 = 2(a 11 ) 2 a kk a 2 1k,α (a 11 ) 2 a 11,αα, k=1 1 k,l=1 a 1k a 1l a kl,αα Takig derivative of equatio (3.3) oce more, ad usig (3.5), it follows that 1 a 2 11ϕ αα = Du θ a 11,αα + 2 Du θ a kk a 2 1k,α + θ Du θ 1 a 11 u αα From the equatio (3.1), k=1 θ(θ + 2) Du θ 2 u 2 αa 11 + θ Du θ 2 a 11 γ=1 u 2 αγ + 2θa 2 11u 1 u α ϕ α. u 2 θ a 2 11 ϕ = u 2 +[θ a 11,αα + 2u 2 α,γ=1 1 k=1 u 2 αγ + θu D f θ(θ + 2) a kk a 2 1k,α + 2θa 2 11u 1 θ u α ϕ α u 2 α]a 11. (3.6) Now we use (3.5) ad the Codazzi idetity (2.6) to treat the followig term i (3.6). 1 k=1 a kk a 2 1k,α = a 11 = 1,α k 1 a 2 11,α + +a 3 11u 2θ 1 a kk a 2 kk,1 +,α k 1 a kk a 2 1k,α + a kk a 2 kk,1 + θ 2 u 2 ϕ 2 α 2θa 2 11u 1 θ a kk a 2 1k,α a 11 u 2 α u α ϕ α. (3.7)

7 PRINCIPAL CURVATURE ESTIMATES FOR THE LEVEL SETS 7 From (3.6) ad (3.7), it follows that u 2 θ a 2 11 ϕ = u 2 +[θ 1 a 11,αα + 2u 2 a kk a 2 kk,1 + 2u 2 α,γ=1 +2a 3 11u 2 2θ Now we calculate the term u 2 αγ + θu D f + θ(θ 2) ϕ 2 α 2θa 2 11u 1 θ u 2 Let D = u u 2, the at x o, we get It follows that D α = 3u 2 u α, a 11,αα. 1,k α u 2 α]a 11 a kk a 2 1k,α u α ϕ α. (3.8) D αα = 5u u 2 α + 3u 2 u αα + u D αα = 5u By (2.4), we have u 2 α + 3u 2 D f + u Takig derivative of equatio (3.10), it follows that By (2.6) By (2.5), at x o we have We get u 2 γ=1 α,γ=1 u 2 αγ. u 2 αγ. (3.9) A 11 = u u 2 a 11. (3.10) A 11,α = a 11 D α + Da 11,α, A 11,αα = Da 11,αα + 2a 11,α D α + a 11 D αα. (3.11) A 11,αα = h 11,αα + B 11,αα C 11,αα. (3.12) C 11,αα = 0. (3.13) a 11,αα = u 1 +u 1 h 11,αα u 1 B 11,αα [a 11 D αα + 2D α a 11,α ]. (3.14) Takig first ad secod derivatives of equatio (2.5) o B ij, we have 1 B 11,αα = 4 l=1 u 1α u lα h 1l W (1 + W )u 2. (3.15)

8 8 SUN-YUNG ALICE CHANG, XI-NAN MA AND PAUL YANG Hece usig u jj = u a jj ad W (x o ) = 1, we get u 1 1 B 11,αα = 4 = 2u 11 u By (3.5), it follows that D α a 11,α = 6θa 11 2u 1 l=1 u 2 1αh 11 W (1 + W )u 3 u 2 1α = 2a 3 11u 2 + 2a 11 u 2 1. (3.16) u 2 α 6a 2 11u 1 θ u α ϕ α. (3.17) Combiig (3.14) with (3.9) ad (3.16)-(3.17), we get u 2 a 11,αα = u 1 h 11,αα + [(5 + 6θ) u 2 α + 3u D f + From (3.8) ad (3.18), it follows that u 2 θ a 2 11 ϕ = u 1 +2a 3 11u 2 + 2a 11 u 2 1 6a 2 11u 1 θ h 11,αα + 2u 2 +[(θ + 1) α,γ=1 +(θ 2 + 4θ + 5) 1,k α α,γ=1 u 2 αγ]a 11 u α ϕ α. (3.18) u 2 αγ + (3 + θ)u D f u 2 α + 2u 2 1]a 11 +2a 3 11u 2 + 2a 3 11u 2 2θ (6 + 2θ)a 2 11u 1 θ ϕ 2 α 1 a kk a 2 1k,α + 2u 2 a kk a 2 kk,1 u α ϕ α. (3.19) STEP 2: I this step we calculate the followig term i (3.19) h 11,αα, u 1 i order to derive the formula (3.32). By (2.3), we have ad h 11,α = 2u u α u 11 + u 2 u 11α + u α u u u 1 u 1α 2u α u 1 u 1 2u u 1α u 1 2u u 1 u 1α, (3.20) h 11,αα = 2u u αα u u u α u 11α 2u u 1αα u 1 + u 2 u 11αα +2u 11 u 2 α + 2u u 2 1α 4u α u 1α u 1 4u u 1α u 1α. (3.21)

9 PRINCIPAL CURVATURE ESTIMATES FOR THE LEVEL SETS 9 From the equatio (3.1), we fid Sice u 1 h 11,αα = u u 11αα + 2u u α u 11α 4 +2u 1 u u αα 2u 1 u 1α u 1α 2a 11 u 2 1α 4u 1 u 1 = u D 11 f + 2u 11 D f 2u 1 D 1 f u 1α u 1α 2a 11 4u 1 u 1 u 1α u α u αα1 u 2 α u α u 11α u 2 α + 2u 1 u u 2 1α u 1α u α. (3.22) A jj = Da jj where D := u u 2. (3.23) Takig derivative of equatio (3.23), we fid that Similar usig (3.20), at x o, A jj,α = a jj D α + Da jj,α, (3.24) A jj,α = h jj,α = 2u u jα u j 2u u α u jj u 2 u jjα. (3.25) From (3.24)- (3.25) ad (3.9), for 1 j 1, we have The for 2 j 1, we get u jjα = u a jj,α + 2u 1 u j u jα a jj u α. (3.26) From (3.5) ad (3.26), it follows that so Usig (3.28), we have 4 u jj1 = u a jj,1 u 1 a jj. (3.27) u 11α = u a 11,α + 2u 1 u 1 u 1α u α a 11 = (1 + θ)u α a u 1 u 111 = (3 + θ)u 1 a 11 + u 1 θ a 2 11ϕ 1, u 11 = (1 + θ)u a u 1 u α u 11α = 4(1 + θ)a 11 +8u 1 u 1 u 1 u 1α + u 1 θ a 2 11ϕ α, (3.28) u 2 α u u 1 θ a 2 11ϕ. (3.29) u 1α u α + 4a 2 11u 1 θ u α ϕ α. (3.30)

10 10 SUN-YUNG ALICE CHANG, XI-NAN MA AND PAUL YANG By (3.27), (3.29) ad the equatio (3.1), we treat the other third derivative term, 4 u 1α u 1α = 4u 11 u 11 4u 1 u 1 1 = 4u 11 u 11 4u 1 D 1 f + 4u 1 j=1 u jj1 1 = 4u 1 u 111 4u 11 u 11 4u 1 D 1 f + 4u 1 j=2 u jj1 1 = 4u u 1 a kk,1 4u 1 D 1 f 4(1 + θ)a 11 u 2 1 Combiig (3.22), (3.30)-(3.31), we have u 1 4(1 + θ)u a 2 11u 4u a ii i=2 +4a 2 11u 1 θ u 1 ϕ 1 + 4a 3 11u 2 θ ϕ. (3.31) 1 h 11,αα = 4u u 1 a kk,1 6u 1 D 1 f + u D 11 f 2u a 11 D f (6 + 4θ)a 11 (2 + 4θ)u u a u a 2 11u 1 θ u 2 α (4 + 4θ)a 11 u u 1 u u 2 1 u 1 ϕ 1 + 4a 2 11u 1 θ 1 a ii + 4a 3 11u 2 θ ϕ i=1 u α ϕ α. (3.32) STEP 3: The coclusio of the proof. Now we combie the (3.19) ad (3.32), it follows that u 2 θ a 2 11 ϕ = 2u 2 1,k α 1 1 a kk a 2 1k,α + 2u 2 a kk a 2 kk,1 4u u 1 a kk,1 6u 1 D 1 f + u D 11 f + (1 + θ)u a 11 D f 1 +(θ 2 1)a 11 u 2 α (2 + 4θ)a 11 u 2 1 4u 2 1 a ii + 2a 3 11u 2 +6u 1 u u 2 1 (2 + 4θ)u u a (θ + 1)a 11 +4a 2 11u 1 θ +2a 3 11u 2 2θ u 1 ϕ 1 + 4a 3 11u 2 θ ϕ ϕ 2 α (2 + 2θ)a 2 11u 1 θ i=1 α,γ=1 u 2 αγ u α ϕ α. (3.33)

11 PRINCIPAL CURVATURE ESTIMATES FOR THE LEVEL SETS 11 Sice 1 1 a kk a 2 kk,1 4u u 1 2u 2 It follows that u 2 θ 1 a kk,1 2u 2 1(a 11 i=1 a ii ). (3.34) a 2 11 ϕ u D 11 f 6u 1 D 1 f + (1 + θ)u a 11 D f + 6u 1 u u (θ 2 1)a 11 u 2 α 4θa 11 u 2 1 6u 2 1 a ii + 2a 3 11u 2 (2 + 4θ)u u a (θ + 1)a 11 α,γ=1 i=1 u 2 αγ mod ϕ, (3.35) where we modify the terms of the gradiet of ϕ with locally bouded coefficiets, from ow o we omit ϕ. From the equatio, at x o, we have 1 1 u = f u ii = f + u a ii. (3.36) i=1 Now we let λ i = a ii > 0, 1 i 1, use the followig abbreviatio σ k (λ) = λ i1 λ ik, 1 i 1< <i k 1 ad let σ k (λ i) deote the summatio i which the terms ivolvig λ i are deleted. Combiig (3.35)- (3.36), it follows that u 2 θ a 2 11 ϕ u D 11 f + (1 + θ)u λ 1 D f 6u 1 D 1 f + 6u 1 u 2 1f +(θ 2 + θ)λ 1 f 2 + 2(θ 2 + θ)u λ 1 σ 1 (λ 1)f + 2(θ 2 θ 1)u λ 2 1 f i=1 +λ 1 u 2 [(θ 1) 2 λ (θ + 1) 2 σ 1 2 (λ 1) +2(θ 2 θ 1)λ 1 σ 1 (λ 1) 2(1 + θ)σ 2 (λ 1)] 1 +(θ 1) 2 λ 1 u (θ + 1) 2 λ 1 u 2 j. (3.37) Now we make the followig choice of θ to complete the proof. Case 1: if f = 0, for = 3, we let θ = 0 the for ϕ := a 11, we get j=2 u 2 a 2 11 ϕ λ 1 (u u 2 23) + λ 1 u 2 3(λ 1 λ 2 ) 2 0. (3.38) If 3, we ca choose θ = 1. The from (3.37), for ϕ := Du 1 a 11 satisfies u 3 a 2 11 ϕ 4λ 1 u λ 2 1u 2 [λ 1 + σ 1 (λ)] 0. (3.39) Case 2: if f = f(x) 0, we ca choose θ = 1. The from (3.37), for ϕ := Du 1 a 11 satisfies u 3 a 2 11 ϕ u f 11 6u 1 f 1 + 6u 1 u 2 1f +2u λ 2 1 f + 4λ 1 u λ 2 1u 2 [λ 1 + σ 1 (λ)]. (3.40) If the fuctio (t, x) t 3 f(x) is a covex fuctio for x Ω ad t (0, + ) (or for f > 0 ad f 1 2 is a cocave fuctio). So the matrix {2ff ij 3f i f j } is oegative defiite. The u f 11 6u 1 f 1 + 6u 1 u 2 1f 0. (3.41)

12 12 SUN-YUNG ALICE CHANG, XI-NAN MA AND PAUL YANG It follows that u 3 a 2 11 ϕ 2u λ 1 2 f + 4λ 1 u λ 2 1u 2 [λ 1 + σ 1 (λ)] 0. (3.42) Case 3: for f = f(u) 0, so i this case we have D 11 f = f u u 11 = u λ 1 f u, u D 11 f + (1 + θ)u λ 1 D f 6u 1 D 1 f = θu 2 λ 1 f u. (3.43) Usig (3.37) ad (3.43), we ca make the followig choice. Whe f u 0, we let θ = 1. The from (3.37), for ϕ := Du 1 a 11 satisfies u 3 a 2 11 ϕ u 2 λ 1 f u + 6u 1 u 2 1f + 2u λ 2 1 f + 4λ 1 u λ 2 1u 2 [λ 1 + σ 1 (λ)] 0. (3.44) Whe f u 0, we let θ = 2. From (3.37), for ϕ := Du 2 a 11 satisfies a 2 11 ϕ 2u 2 λ 1 f u + 6u 1 u 2 1f + 6λ 1 f u λ 1 σ 1 (λ 1)f + 2u λ 2 1 f + λ 1 u λ 1 j=2 +λ 1 u 2 [λ σ 1 2 (λ 1) + 2λ 1 σ 1 (λ 1) 6σ 2 (λ 1)] λ 1 u 2 [λ σ 1 2 (λ 1) + 2λ 1 σ 1 (λ 1) 6σ 2 (λ 1)]. (3.45) For = 2 or = 3, σ 2 (λ 1) = 0. iequalities, i.e. for 4, we have The we have By (3.45)-(3.46), it follows that Case 4: for f = f(x, u) 0. Sice so for θ = 1, we get u 2 j For 4 we use the followig Maclauri [ σ 2(λ 1) C 2 2 ] 1 σ 1 (λ 1) 2 2, σ 2 (λ 1) 3 2( 2) σ2 1(λ 1) 9σ 1 2 (λ 1) 6σ 2 (λ 1) 0. (3.46) a 2 11 ϕ 0. D 1 f = f 1, D 11 f = f 11 u λ 1 f u, (3.47) u D 11 f + (1 + θ)u λ 1 D f 6u 1 D 1 f = u f 11 6u 1 f 1 u 2 λ 1 f u. If f u 0 ad for ay u (a, b) the fuctio (t, x) t 3 f(x, u) is a covex fuctio for x Ω ad t (0, + ) (or for f > 0 ad f 1 2 is a cocave fuctio for

13 PRINCIPAL CURVATURE ESTIMATES FOR THE LEVEL SETS 13 x), the we also have (3.41). Now let θ = 1. From (3.37), for ϕ := Du 1 a 11, it follows that u 3 a 2 11 ϕ u f 11 6u 1 f 1 + 6u 1 u 2 1f u 2 λ 1 f u + 2u λ 2 1 f + 4λ 1 u λ 2 1u 2 [λ 1 + σ 1 (λ)] The we complete the proof of the Theorem u λ 1 2 f + 4λ 1 u λ 2 1u 2 [λ 1 + σ 1 (λ)] 0. (3.48) Remark 3.7. Recetly Ma-Ou-Zhag [17] obtaied the Gaussia curvature lower boud estimate for the covex level sets of harmoic fuctio o covex rig. Ackowledgmets. The secod amed author would like to thak Prof. P.Gua for useful discussios o this subject. REFERENCES [1] L. V. Ahlfors, Coformal ivariats: Topics i geometric fuctio theory, McGraw-Hill Series i Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johaesburg, 1973(pp 5 6). [2] B. J. Bia, P. Gua, X. N. Ma ad L. Xu, A microscopic covexity priciple for the level sets of solutio for oliear elliptic partial differetial equatios, to appear i Idiaa Uiv. Math. J.. [3] C. Biachii, M. Logietti ad P. Salai, Quasicocave solutios to elliptic problems i covex rigs, Idiaa Uiv. Math. J., 58 (2009), [4] L. Caffarelli ad A. Friedma, Covexity of solutios of some semiliear elliptic equatios, Duke Math. J., 52 (1985), [5] L. A. Caffarelli, L. Nireberg ad J. Spruck, Noliear secod order elliptic equatios IV: Starshaped compact Weigarte hypersurfaces, Curret topics i partial differetial equatios, Y.Ohya, K. Kasahara ad N. Shimakura (eds), Kiokuize, Tokyo, 1985, [6] L. Caffarelli ad J. Spruck, Covexity properties of solutios to some classical variatioal problems, Comm. Partial Differ. Equatios, 7 (1982), [7] R. Gabriel, A result cocerig covex level surfaces of 3-dimesioal harmoic fuctios, J. Lodo Math.Soc., 32 (1957), [8] J. J. Gerge, Note o the Gree fuctio of a star-shaped three dimesioal regio, Amer. J. Math., 53 (1931), [9] L. Hormader, Notios of Covexity, reprit of the 1994 editio. Moder Birkhauser Classics, Birkhauser Bosto, Ic., Bosto, MA, [10] J. Jost, X. N. Ma ad Q. Z. Ou, Curvature estimates i dimesios 2 ad 3 for the level sets of p-harmoic fuctios i covex rigs, preprit. [11] B. Kawohl, Rearragemets ad Covexity of Level Sets i PDE, Lectures Notes i Math., 1150, Spriger-Verlag, Berli, [12] N. J. Korevaar, Covexity of level sets for solutios to elliptic rig problems, Comm. Partial Differ. Equatios, 15, 1990, [13] J. L. Lewis, Capacitary fuctios i covex rigs, Arch. Ratioal Mech. Aal., 66 (1977), [14] M. Logietti, Covexity of the level lies of harmoic fuctios, (Italia) Boll. U. Mat. Ital. A 6 (1983), [15] M. Logietti, O miimal surfaces bouded by two covex curves i parallel plaes, J. Diff. Equatios, 67 (1987), [16] M. Logietti ad P. Salai, O the Hessio matrix ad Mikowski additio of quasicovex fuctios, J. Math. Pures Appl., 88 (2007), [17] X. N Ma., Q. Z. Ou ad W. Zhag, Gaussia Curvature estimates for the covex level sets of p-harmoic fuctios, Comm. Pure Appl. Math., 63 (2008), o.7, [18] X. N. Ma, J. Ye ad Y. H. Ye, Pricipal curvature estimates for the level sets of harmoic fuctios ad miimal graph i R 3, to appear i Commu. Pure Appl. Aal..

14 14 SUN-YUNG ALICE CHANG, XI-NAN MA AND PAUL YANG [19] M. Ortel ad W. Scheider, Curvature of level curves of harmoic fuctios, Caad. Math. Bull., 26 (1983), [20] M. Shiffma, O surfaces of statioary area bouded by two circles, or covex curves, i parallel plaes, Aals of Math., 63 (1956), Received March 2010; revised April address: chag@math.priceto.edu address: xia@ustc.edu.c address: yag@math.priceto.edu