Research Article Uniqueness of Solutions to a Nonlinear Elliptic Hessian Equation

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1 Applied Mathematics Volme 2016, Article ID , 5 pages Research Article Uniqeness of Soltions to a Nonlinear Elliptic Hessian Eqation Siyan Li School of Mathematics and Applied Statistics, Faclty of Engineering and Information Sciences, University of Wollongong, Wollongong, NSW 2522, Astralia Correspondence shold be addressed to Siyan Li; sl296@owmail.ed.a Received 29 Jne 2016; Accepted 6 November 2016 Academic Editor: Carlos Conca Copyright 2016 Siyan Li. This is an open access article distribted nder the Creative Commons Attribtion License, which permits nrestricted se, distribtion, and reprodction in any medim, provided the original work is properly cited. Throgh an Alexandrov-Fenchel ineqality, we establish the general Brnn-Minkowski ineqality. Then we obtain the niqeness of soltions to a nonlinear elliptic Hessian eqation on S n. 1. Introdction According to a general Brnn-Minkowski ineqality, we obtain a proof of the niqeness of soltions to the following flly nonlinear elliptic Hessian eqation: σ k ( ij +δ ij )=f p 1 on S n, (1) where is the spport fnction of convex bodies, ij are the second-order covariant derivations of with respect to any orthonormal frame {e 1,e 2,...,e n } on S n, δ ij is the standard Kronecker symbol, S n is the nit sphere of n-dimension, f is a positive fnction defined on S n, k {1,2,...,n}, p>1,and σ k is the kth elementary symmetric fnction defined as follows: for λ=(λ 1,λ 2,...,λ n ) R n, σ k (λ) = λ i1 λ i2 λ ik. 1 i 1 <i 2 < <i k n The definition can be extended to any symmetric matrix W R n n by σ k (W) = σ k (λ(w)), whereλ(w) = (λ 1 (W), λ 2 (W),...,λ n (W)) is the eigenvale vector of W. Eqation (1) arrives from the geometry of convex bodies. A compact convex sbset of Eclidean (n+1)-space R n+1 with nonempty interiors is called a convex body. An important concept related to a convex body Q is its spport fnction. Definition 1. Let M (the bondary of a convex body Q) bea smooth, closed, niformly convex hypersrface enclosing the (2) origin in R n+1. Assme that M is parameterized by its inverse Gass map X:S n M R n+1 ;thespport fnction of M (or Q) is defined by (x) = x, X (x), x S n, (3) where, denotes the standard inner prodct in R n+1. is convex after being extended as a fnction of homogeneos degree 1 in R n+1. Conversely, any continos convex fnction of homogeneos degree 1 determines a convex body as follows: Q={y R n+1 :y x (x), x S n }. (4) From some basic concepts to spport fnction, Minkowski sm[seedefinition4],andmixedvolmes[seedefinition5], Minkowski developed a set of theories related to convex bodies. If k=nand p=1,(1)is the Monge-Ampère eqation corresponding to the classical Minkowski problem det ( ij +δ ij )=f on S n, (5) which has been solved by Nirenberg [1], Pogorelov [2, 3], Cheng and Ya [4], and many others. When p=1,(1)isthe classical Christoffel-Minkowski problem: σ k ( ij +δ ij )=f on S n. (6)

2 2 Applied Mathematics Anecessarycondition[3]for(6)tohaveasoltionis S n x i f (x) ds=0, i=1,2,...,n+1, (7) where ds is the standard area form on S n. Gan et al. [5] obtained that (7) is sfficient for (6) to have an admissible soltion [see Definition 6]. Firey [6] generalized the Minkowski sm to p-sm [see Definition 4] from p = 1 to p 1 in1962.later,ltwak [7] extended the classical srface area measre to the psmcases.alsoin[7],ltwakfirstintrodcedthegeneral Minkowski problem, which is called L p -Minkowski problem thereafter. In the smooth category, L p -Minkowski problem is eqivalent to considering the following Monge-Ampère eqation: det ( ij +δ ij )=f p 1 on S n. (8) The niqeness of L p -Minkowski problem for p>1and p = n+1(the niqeness holds p to a dilation if p = n + 1) has been solved in [7]. However, the niqeness for p<1is difficlt and still open. In [8], Jian et al. obtained that, for any n 1<p<0, there exists a positive fnction f C (S n ) to garantee that (8) has two different soltions, which means that we need more conditions to consider the niqeness. When considering cases 1 k<n,attentionispaidtothe generalized Christoffel-Minkowski problem. In the smooth category, we need to stdy the k-hessian eqation (1). For (1), H et al. [9] got the existence and niqeness of soltions to (1) when 1 k < n and p > k + 1 nder appropriate conditions. However, the niqeness of (1) when p<1has not been solved well. In this paper, we stdy the niqeness of (1) for p>1. Or main reslt is the following. Theorem 2. Sppose is a positive admissible soltion of σ k ( ij +δ ij )=f p 0 on S n, (9) where 1 k<n, k Z, p 0 R + \{k},andf is a positive fnction defined on the nit sphere S n and the niqeness holds. If p 0 =k, the niqeness holds p to a dilation, which means that if solves (9), {a : a R + } are the whole soltionsof(9). Remark 3. Here, we rewrite (1) by (9), where p 0 =p 1. The organization of this paper is as follows. In Section 2, we show some basic concepts and lemmas which have been obtained by Gan et al. in [10]. In Section 3, we prove two sefl propositions according to the methods in [11]. In the last section, we prove the main theorem. 2. Preliminaries Definition 4. Given two convex bodies Q 1 and Q 2 in R n+1 with respective spport fnctions 1, 2,andλ, μ 0 (λ+μ > 0), the Minkowski sm λq 1 +μq 2 R n+1 is defined by the convex body whose spport fnction is λ 1 +μ 2. For p 1,letQ 1 and Q 2 be two convex bodies containing the origin in R n+1 in their interiors, and λ, μ 0 (λ+μ>0). The convex body λ Q 1 + p μ Q 2,whosespportfnctionis given by (λ p 1 +μp 2 )1/p, is called Firey s p-sm of Q 1 and Q 2, where + p meansthep-smmation and meansfirey s mltiplication. Definition 5. Let Q 1,Q 2,...,Q r be convex bodies in R n+1 and the volme of their Minkowski sm Q=λ 1 Q 1 +λ 2 Q 2 + +λ r Q r, λ i 0, (10) is an (n + 1)th degree homogeneos polynomial of the family λ 1,λ 2,...,λ r.specially,thevolmeofq is Vol (Q) = Vol (λ 1 Q 1 +λ 2 Q 2 + +λ r Q r ) = r λ i1 λ i2 λ in+1 V(Q i1,q i2,...,q in+1 ), i 1,i 2,...,i n+1 =1 (11) where the fnctions V are symmetric. Then V(Q 1,Q 2,..., Q n+1 ) is called the Minkowski mixed volme of Q 1,Q 2,..., Q n+1. Definition 6. For k {1,2,...,n},letΓ k be the convex cone in R n which is determined by Γ k = {λ R n :σ 1 (λ) >0,σ 2 (λ) >0,...,σ k (λ) >0}. (12) Afnction C 2 (S n ) is called k-convex if W (x) ={ ij (x) +(x) δ ij } Γ k, x S n, (13) and is called an admissible soltion to (1) if is k-convex and satisfies (1). Definition 7. Let A 1,A 2,...,A m be symmetric real k k matrices, λ 1,λ 2,...,λ m R; the determinant of λ 1 A λ m A m is a homogeneos polynomial of degree k in λ 1,λ 2,...,λ m.namely, det (λ 1 A 1 + +λ m A m ) = m λ i1 λ ik D k (A i1,...,a ik ). i 1,...,i k =1 (14) In fact, the coefficient λ i1 λ ik depends only on A i1,...,a ik ; they are niqely determined. D k (A 1,...,A k ) is called the mixed discriminant of A 1,...,A k. For later applications, we collect some reslts here which have been proved in [10]. Lemma 8. Let 1, 2,..., n+1 be the spport fnction of convex bodies Q 1,Q 2,...,Q n+1, respectively. Denoting Minkowski mixed volme V(Q 1,Q 2,...,Q n+1 ) by V( 1, 2,..., n+1 ) and W m ={( m ) ij + m δ ij }, m=1,2,...,n+1, (15)

3 Applied Mathematics 3 V( 1, 2,..., n+1 ) = S n 1 D n (W 2,W 3,...,W n+1 )ds, (16) where D n (W 2,W 3,...,W n+1 ) is the mixed discriminant [see Definition 7] of W 2,W 3,...,W n+1. Remark 9. For all 1 k n,setting k+2 = = n+1 =1, V( 1,...,,1,...,1)fl V ( 1, 2,..., ) = S n 1 D k (W 2,W 3,...,W )ds, (17) where D k (W 2,W 3,...,W ) is the mixed discriminant of W 2,W 3,...,W. Frthermore, if 1 = 2 = = n+1 =, denote V( 1, 2,..., n+1 ) fl V() and V ( 1, 2,..., ) fl V (); with eqality if and only if 0 constants a, a 1,...,a n+1. Proof. We only need to prove that = a 1 + n+1 i=1 a ix i for some F (t) =V 1/() ((1 t) 0 +t 1 ) (22) is concave on [0, 1].Setting t =(1 t) 0 +t 1, t [0,1],we have F (t) =V 1/() ( t, t,..., t ). (23) By the symmetric mltilinear property of V,it is obvios that F (t) =V 1/() 1 ( + 1, t,..., t ), k t,..., t )V ( 0 (24) V () = S n det ( ij +δ ij )ds, V () = S n σ k ( ij +δ ij )ds. (18) Lemma 10. V is a symmetric mltilinear form on (C 2 (S n )) n+1. Lemma 11. For any fnction C 2 (S n ), W={ ij +δ ij }, 1 k<n,wehavetheminkowskitypeintegralformla, S n σ k (W) ds= S n σ (W) ds, (19) F (t) =kv 1/() 2 ( [V ( t,..., t ) t,..., t ) k 1 V ( 0 + 1, 0 + 1, t,..., t ) V 2 ( 0 + 1, t,..., t )] 0, k (25) where ds is the standard area element on S n. The following is a form of Alexandrov-Fenchel ineqality for positive k-convex fnctions which comes from [10]. Lemma 12 (Alexandrov-Fenchel ineqality). If 1, 2,..., k are k-convex, 1 is positive, and there exists l {2,3,...,k} sch that l 0on S n,,foranyv C 2 (S n ), V 2 (V, 1, 2,..., k ) V ( 1, 1, 2,..., k )V (V, V, 2,..., k ), (20) with eqality if and only if V = a 1 + n+1 i=1 a ix i for some constants a, a 1,...,a n Two Important Propositions Now we prove two important propositions. The methods we se are from [11]. Proposition 13. Sppose 0, 1 >0are k-convex; V 1/() ((1 t) 0 +t 1 ) (1 t) V 1/() ( 0 )+tv 1/() ( 1 ), t [0, 1], (21) where the last ineqality ses (20); ths F is a concave fnction on [0, 1]. The eqality condition is checked easily. Proposition 14 (general Brnn-Minkowski ineqality). Spposing 0, 1 >0are k-convex, S n 1 σ k (( 0 ) ij with eqality if and only if 0 constants a, a 1,...,a n+1. Proof. Setting V 1/() ( 1 )V 1 1/() ( 0 ), F (t) =V 1/() ((1 t) 0 +t 1 ) (26) = a 1 + n+1 i=1 a ix i for some (1 t) V 1/() ( 0 ) tv 1/() ( 1 ), (27) F(0) = F(1) = 0. By(21),F(t) 0; thsf (0) 0; namely, ( 0 )V ( 0 + 1, 0,..., 0 ) V 1/() 1 +V 1/() ( 0 ) V 1/() ( 1 ) 0. k (28)

4 4 Applied Mathematics Then V 1/() 1 ( 0 ) S n ( )σ k (( 0 ) ij By (19), and +V 1/() ( 0 ) V 1/() ( 1 ). V 1/() 1 ( 0 ) S n 1 σ k (( 0 ) ij V 1/() ( 1 ), S n 1 σ k (( 0 ) ij V 1/() ( 1 )V 1 1/() ( 0 ). (29) (30) (31) 1. (36) V Similarly, we have V/ 1.Ths V. Case 2 (0 <p 0 <k). We have p 0 σ k ( ij +δ ij )=V p 0 σ k (V ij + Vδ ij ); (37) V () = S n σ k ( ij +δ ij )ds = S n ( V )p 0+1 Vσk (V ij + Vδ ij )ds p 0 +1 σ k (V ij + Vδ ij )ds] [ S n (38) 4. Proof of Theorem 2 Now we prove Theorem 2. The main methods are from [7, 12]. Proof. Assming that (9) has two soltions and V, we consider the eqation in the following three cases. Case 1 (p 0 >k). Spposing x 0 is the maximm vale point of G=/V,atx 0,wehave 0= ln G= V V, 0 2 ln G=( 2 that is, Hence = 2 therefore 2 V V ; ( )2 2 ) ( 2 V V ( V)2 V 2 ) 2 (32) 2 V V. (33) f p 0 (x 0 )= k (x 0 )σ k ( ij +δ ij)(x 0 ) k (x 0 )σ k ( V ij V +δ ij)(x 0 ) = k (x 0 ) V k (x 0 ) fvp 0 (x 0 ); (34) p0 k (x 0 ) V p0 k (x 0 ) G(x 0 )= (x 0) 1; (35) V (x 0 ) p 0 Vσ k (V ij + Vδ ij )ds] [ S n V (kp 0+k)/() (V) V p 0 V 1 (p 0+1)/() (V), (p V 0 +1)/() () (V) =V(p 0+1)/() () wherewehavesedhölder ineqality in the first ineqality and sed (26) in the second one. Hence V () = V (V), which forces both the eqalities to hold. By the eqality condition, there exists a constant a R sch that V =a. By (9), we know a=1. Therefore, V. Case 3 (p 0 =k). According to Case 2, when p 0 =k,wehave V () = S n σ k ( ij +δ ij )ds= S n ( V ) Vσ k (V ij + Vδ ij σ k (V ij + Vδ ij )ds] )ds [ S n k Vσ k (V ij + Vδ ij )ds] V () V [ S k n (V) V k (V) =V () ; (39) all the eqalities hold. Ths there exists a R,schthat V =a. Therefore {a : a R + } are the whole soltions of (9). NowwecompletetheproofofTheorem2. Competing Interests The athor declares no competing interests. References [1] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Commnications on Pre and Applied Mathematics,vol.6,no.3,pp ,1953.

5 Applied Mathematics 5 [2] A. V. Pogorelov, On existence of a convex srface with a given sm principal radii of crvatre, Uspekhi Matematicheskikh Nak,vol.8,no.3,pp ,1953(Rssian). [3]A.V.Pogorelov,The Mltidimensional Minkowski Problem, Winston, Washington, DC, USA, [4]S.Y.ChengandS.T.Ya, Onthereglarityofthesoltion of the n-dimensionalminkowski problem, Commnications on Pre and Applied Mathematics,vol.29,no.5,pp ,1976. [5] P. Gan, X.-N. Ma, and F. Zho, The Christofel-Minkowski problem. III. Existence and convexity of admissible soltions, Commnications on Pre and Applied Mathematics,vol.59,no. 9, pp , [6] W. J. Firey, p-means of convex bodies, Mathematica Scandinavica,vol.10,pp.17 24,1962. [7] E. Ltwak, The Brnn-Minkowski-Firey theory I: mixed volmes and the Minkowski problem, Differential Geometry,vol.38,no.1,pp ,1993. [8] H. Jian, J. L, and X.-J. Wang, Nonniqeness of soltions to the L p -Minkowski problem, Advances in Mathematics,vol.281, pp ,2015. [9] C. Q. H, X.-N. Ma, and C. Shen, On the Christoffel- Minkowski problem of Firey s p-sm, Calcls of Variations and Partial Differential Eqations, vol.21,no.2,pp , [10] P. F. Gan, X.-N. Ma, N. Trdinger, and X. Zh, A form of Alexandrov-Fenchel ineqality, Pre and Applied Mathematics Qarterly,vol.6,no.4,pp ,2010. [11] R. Schneider, Convex Bodies: the Brnn-Minkowski Theory, Cambridge University Press, Cambridge, UK, [12] K.-S. Cho and X.-J. Wang, The L p -Minkowski problem and the Minkowski problem in centroaffine geometry, Advances in Mathematics,vol.205,no.1,pp.33 83,2006.

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On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R^3

On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R^3 University of Wollongong Research Online Faclty of Engineering and Information Sciences - Papers: Part A Faclty of Engineering and Information Sciences 2015 On the niqeness of Lp-Minkowski problems: The