The Monge-Ampère Equation for Strictly(n?1)-convex Functions with Neumann Condition

Bin Deng

Department of Mathematics,University of Science and Technology of China,Hefei 230026,China.

Abstract.A C2 function on Rn is called strictly(n?1)-convex if the sum of any n?1 eigenvalues of its Hessian is positive.In this paper,we establish a global C2 estimates to the Monge-Ampère equation for strictly(n?1)-convex functions with Neumann condition. By the method of continuity,we prove an existence theorem for strictly(n?1)-convex solutions of the Neumann problems.

Key words:Neumann problem,(n?1)-convex,elliptic equation.

1 Introduction

Let Ω?Rnbe a bounded convex domain andν(x)be the outer unit normal atx ∈?Ω.Supposef ∈C2(Ω)is positive andIn this paper,we mainly consider the following equations of Monge-Ampère type with Neumann condition,

where the matrixform=n?1,with the elements as follows,

a linear combination ofuij,whereandis the generalized Kronecker symbol.All indexesi,j,αi,βi,···come from 1 ton.

For general 1≤m ≤n?1,the matrixcomes from the following operatoras in[2]and[10].First,note that(uij)n×ninduces an operatorUon Rnby

where{e1,e2,···,en}is the standard orthogonal basis of Rn.We further extendUto act on the real vector space∧mRnby

whereis the standard basis for∧mRn.ThenWis the matrix ofunder this standard basis.It is convenient to denote the multi-index byWe only consider the increasing multi-index,that is,1≤α1＜···＜αm ≤n.By the dictionary arrangement,we can arrange all increasing multi-indexes from 1 toand useNαdenote the order number of the multi-indexi.e.,Nα=1 forα=(12···m),···.We also usedenote the index set{α1,···,αm}without confusion.It is not hard to see that

if the index setequals to the index setbut;and also

if the index sets{α1,···,αm}and{β1,···,βm}have more than one different element.Specifically,forn=3,m=2,we have

It follows thatWis symmetrical and diagonal with(uij)n×ndiagonal.The eigenvalues ofWare the sums of eigenvalues of(uij)n×n.Denoted byμ(D2u)=(μ1,···,μn)the eigenvalues of the Hessian and bythe eigenvalues ofW.Generally,for anyk=1,2,···,we define thekthelementary symmetry function by

We also setS0=1.In particular,we have

Ifm=1,the equation(1.1)is known as Monge-Ampère equation.

Then we define the generalized G?arding's cone by,for

In the absence of ambiguity,we omit the subscriptnfor simplicity.Obviously,andNormally,we say aC2functionuis convex if any eigenvalue of the Hessian is nonnegative,equivalentlySimilarly,we give the following definition ofm-convexity.

For the Dirichlet problem in Rn,many results are known.For example,the Dirichlet problem of Laplace equation is studied in[8],Caffarelli-Nirenberg-Spruck[1],and Ivochkina[16]solved the Dirichlet problem of Monge-Ampère equation,and Caffarelli-Nirenberg-Spruck[2]solved the Dirichlet problem of general Hessian equations even including the case considered here.For the general Hessian quotient equation,the Dirichlet problem is solved by Trudinger in[31].Finally,Guan[9]treated the Dirichlet problem for general fully nonlinear elliptic equation on the Riemannian manifolds.

Also,the Neumann or oblique derivative problem of partial differential equations was widely studied.For a priori estimates and the existence theorem of Laplace equation with Neumann boundary condition,we refer to the book[8].Also,we can see the book written by Lieberman[17]for the Neumann or oblique derivative problem of linear and quasilinear elliptic equations.In 1987,Lions-Trudinger-Urbas solved the Neumann problem of Monge-Ampère equation in the celebrated paper[21].For the the Neumann problem ofk-Hessian equations,Trudinger[32]established the existence theorem when the domain is a ball,and he conjectured(in[32],page 305)that one can solve the problem in sufficiently smooth uniformly convex domains.Recently,Ma and Qiu[22]gave a positive answer to this problem and solved the the Neumann problem ofk-Hessian equations in uniformly convex domains.After their work,the research on the Neumann problem of other equatios has made progresses(see,e.g.,[3,4,23,33]).

Similarly tom-convexity for the Hessian(see Definition 1.1),we can formulate the notion ofm-convexity for curvature operator and second fundamental forms of hypersurfaces.There are large amount literature in differential geometry on this subject.For example,Sha[25]and Wu[34]introduced them-convexity of the sectional curvature of Riemannian manifolds and studied the topology for these manifolds.In a series interesting papers,Harvey and Lawson([11-13])introduce some generalized convexities on the solutions of the nonlinear elliptic Dirichlet problem,m-convexity is a special case.Han-Ma-Wu[10]obtained an existence theorem ofm-convex starshaped hypersurface with prescribed mean curvature. More recently,in the complex space Cncase,Tosatti and Weinkove([29,30])solved the Monge-Ampère equation for(n?1)-plurisubharmonic functions on a compact K?hler manifold,where the(n?1)-plurisubharmonicity means the sum of anyn?1 eigenvalues of the complex Hessian is nonnegative.

From the above geometry and analysis reasons,it is naturally to study the Neumann problem(1.1).In[6],the author considered the following Neumann problem for general fully nonlinear equations

Eq.(1.1)is a special case of(1.6)whenm=n?1,k=n.Parallel to Definition 1.1,we give

Forwe obtained an existence theorem of thek-admissible solution with less geometric restrictions to the boundary.Forandwe

Definition 1.2.We say u is k-admissible if.Particularly,if ,u is strictlym-convex.got an existence theorem if Ω is strictly(m,k0)-convex,i.e.,,whereκ=(κ1,···,κn?1)denote the principal curvatures of?Ω with respect to its inner normal?ν. We didn't prove the existence for strictlym-convex solution for the equation(1.6)in[6].Particularly,form=n?1(maybe the most interesting case except the casem=1),we got the existence of thek-admissible solution fork≤n?1 only except that of the(n?1)-convex solution fork=n.In this paper,given a strong geometric restriction to the boundary,we can prove the existence of strictly(n?1)-convex solution to the Neumann problem(1.1).

We always denoteκ=(κ1,···,κn?1)the principal curvature andthe mean curvature of the boundary.We now state the main result of this paper as follows.

Theorem 1.1.SupposeΩ?Rn(n≥3)is a bounded strictly convex domain with C4boundary.Denote κmax(x)(κmin(x))the maximum(minimum)principal curvature at x∈?Ωsuch that

Let f ∈C2(Ω)be a positive function and Then there exists a unique strictly(n?1)-convex solution of the Neumann problem(1.1).

We may point out that the curvature condition(1.7)is only used to obtain the upper bound for the double normal derivative in Lemma 4.3.When the dimensionnis large,it is easy to see that the domain Ω is almost a ball.As a special case,forn=3,H=κmax+κmin,we have

Corollary 1.1.SupposeΩ?R3is a bounded strictly convex domain with C4boundary.Denote κmax(x)(κmin(x))the maximum(minimum)principal curvature at x ∈?Ωsuch that κmax＜Let f ∈C2(Ω)be a positive function and .Then there exists a unique strictly2-convex solution of the Neumann problem(1.1).

The rest of this paper is arranged as follows.In Section 2,we give some basic properties of the elementary symmetric functions and some notations.In Section 3,we establish a prioriC0estimates and global gradient estimates.In Section 4,we show the proof of the global estimates of second order derivatives.Finally,we can prove the existence theorem by the method of continuity in Section 5.

2 Preliminary

In this section,we give some basic properties of elementary symmetric functions and some notations. First,we denote bySk(λ|i)the symmetric function withλi=0 andSk(λ|ij)the symmetric function withλi=λj=0.

Proposition 2.1.Let λ=(λ1,···,λn)∈Rn and k=1,···,n,then

We also denote bySk(W|i)the symmetric function withWdeleting thei-row andicolumn andSk(W|ij)the symmetric function withWdeleting thei,j-rows andi,j-columns.Then we have the following identities.

Proposition 2.2.Suppose A=(aij)n×n is diagonal and k is a positive integer,then

Furthermore,suppose defined as in(1.2)is diagonal,then

Proof.For(2.4),see a proof in[18].Note that

Using(1.3)and(1.4),then(1.5)and(2.5)are immediate consequences of(2.4).

Recall that the G?arding's cone is defined by

Proposition 2.3.Let λ ∈Γk and k ∈{1,···,n}.Suppose that λ1≥···≥λk ≥···≥λn,then we have

Proof.All the properties are well-known. For example,see[18]or[15]for a proof of(2.7),[5]or[14]for(2.8)and[2]for(2.9).

The Newton-Maclaurin inequality is as follows:

Proposition 2.4.For λ∈Γk and k＞l ＞0,we have

Proof.See[28]for a proof of(2.10).For(2.11),we use(2.10)and Proposition 2.1 to get

This completes the proof of the proposition.

We define

It is well known that there exists a small positive universal constantμ0such thatd(x)∈C4(Ωμ),?0＜μ≤μ0,provided?Ω∈C4.As in Simon-Spruck[27]or Lieberman[17](p.331),we can extendνbyν=?Ddin Ωμand note thatνis avector field. As mentioned in the book[17],we also have the following formulas

3 The zero-order and first-order estimates

As proved in[6],we have the following theorem.

Theorem 3.1.LetΩ?Rn(n ≥3)be a bounded domain with C3boundary,and f ∈C1(Ω)bea positive function and Suppose thatis a k-admissible solutionof the Neumann problem(1.6).Then there exists a constant C1depending only on k,m,n,|f|C1,|φ|C3andΩ,such that

Proof.See Theorem 3.1 in[6]for the zero-order estimate.See Theorem 4.2 and Theorem 4.4 in[6]for the first-order estimate.The proof of the gradient estimates could also be found in[7].

4 Global second order derivatives estimates

Generally,the double normal estimates are the most important and hardest parts for the Neumann problem.As in[21]and[22],we construct sub and super barrier functions to give lower and upper bounds foruννon the boundary.Then we give the global second order estimates.

In this section,we establish the following global second order estimate.

Theorem 4.1.SupposeΩ?Rn(n≥3)is a bounded strictly convex domain with C4boundary.Denote κmax(x)(κmin(x))the maximum(minimum)principal curvature at x∈?Ωsuch that

Let f(x,z)∈C2(Ω×R)be a positive function andbe decreasing with respect(n?1)-convex solution of the Neumann problem

Then we have

where C depends only on n,minΩ,withM0=supΩ|u|.

Throughout the rest of this paper,we always admit the Einstein's summation convention.All repeated indices come from 1 to n.We will always denoteF(D2u)=det(W)and

From(1.3)and(2.5)in Proposition 2.2 we have,for any 1≤i≤n,

Throughout the rest of the paper,we will also denoteis a strictly(n?1)-convex solution of the Neumann problem(4.1)

for simplicity.

4.1 Reduce the global second derivative estimates into double normal derivatives estimates on boundary

Using the method of Lions-Trudinger-Urbas[21],we can reduce the second derivative estimates of the solution into the boundary double normal estimates.

Lemma 4.1.LetΩ?Rn be a bounded strictly convex domain with C4boundary. Assumehave

where C0depends only on n,minΩ,withM0=supΩ|u|.

Proof.Write Eq.(4.1)in the form of

whereSinceλ(W)∈Γn ?Γ2in Rn,we have

wherec(n)is a universal number independent ofu.It is sufficiently to prove(4.5)for any directionξ ∈Sn?1,that is

We consider the following auxiliary function in Ω×Sn?1,

where

withand

K1,K2are positive constants to be determined.By direct computations,we have

Denoteand

and

sinceis a linear combination ofuij,1≤i,j≤n.Differentiating Eq.(4.6)twice,we have

withAt the maximum pointx0∈Ω ofv,we can assumeis diagonal.It follows that,by the Cauchy-Schwartz inequality,

where

Assumeu11≥u22···≥unn,and denoteλ1≥λ2≥···≥λnthe eigenvalues of the matrixIt is easy to see

Then we have,by(2.5)in Proposition 2.2 and(2.9)in Proposition 2.3,

We can assumeuξξ ≥0,otherwise we have(4.8). Plug(4.19)into(4.18)and use the Cauchy-Schwartz inequality,then

ChoosingandK1=C(K2+2)+1,it follows that

Case1.is a tangential vector atx0∈?Ω.

We directly have,ν=?Dd,andAs in[17],we define

and it is easy to see thatcijDjis a tangential direction on?Ω. We compute atFrom the boundary condition,we have

It follows that

then we obtain

We useφz ≤0 in the last inequality.We assumeit is easy to get the bound foru1i(x0)fori ＞1 from the maximum ofin thedirection.In fact,we can assumeThen we have

consequently,

Similarly,we have for?i＞1,

Thus we have,by

On the other hand,we have from the Hopf lemma,(4.10)and(4.26),

Then we get

Case2.ξ0is non-tangential.

We can find a tangential vectorτ,such thatwithα2+β2=1.Then we have

By the definition of

Thus,and

In conclusion,we have(4.8)in both cases.

First,we denoted(x)=dist(x,?Ω),and define

The constantK3will be determined later.Then we give the following key lemma.

Lemma 4.2.SupposeΩ?Rn is a bounded strictly convex domain with C2boundary.Denoteκmax(x)(κmin(x))the maximum(minimum)principal curvature at x ∈?Ωstrictly(n?1)-convex and h(x)is defined as in(4.29)3,a sufficiently large number depending only on n,γ,minf andΩ,such that,

where

Fij is defined by(4.3),andμ0is mentioned in(2.13).As γ tends to 1,K3tends to infinity.

Proof.Forx0∈Ωμ,there existsy0∈?Ω such that|x0?y0|=d(x0).Then,in terms of the principal coordinate system aty0,we have(see[8],Lemma 14.17)

Observe that

Denote,andμn=2K3for simplicity.Then we define

and assumeλ1≥···≥λn?1≥λn,it is easy to see that

if we chooseK3sufficiently large andIt is also easy to see thathis strictly convex.

We now consider the functionAs above,we definethe eigenvalues of the HessianD2w,and

Bythe concavity ofdenotewe have

for a large enoughThen we get

This completes the proof of the lemma.

Following the line of Qiu-Ma[22]and Chen-Ma-Zhang[4],we construct the sub barrier function as

with

whereA,σ,andβare positive constants to be determined.We have the following lemma.The curvature condition in Theorem 4.1 is only used here.

Lemma 4.3.Fix σ,if we select β large,μsmall,A large,and assume N large,then

Furthermore,we have

where C depends only on n,|f|C2and|φ|C2.

Proof.We assumeP(x)attains its minimum pointx0in the interior of Ωμ.DifferentiatePtwice to obtain

By a rotation of coordinates,we may assume that(uij)n×nis diagonal atx0,so areWand(Fij)n×n,withFijdefined by(4.3).

We choosewhereis defined in Lemma 4.2 andis a small positive number to be determined,such thatIt follows that

Sincehi=?(1?2K3d)di,we also have

By a straight computation,using Lemma 4.2,we obtain

where

We divide indexesI={1,2,···,n}into two sets in the following way,

whereκmin(κmax)is the minimum(maximum)principal curvature of the boundary.Fori∈G,byPi(x0)=0,we get

Because(4.44)and(4.45),we have

Then letA≥3βC2,we have

for?i ∈G.We chooseβ≥2n?κmin+1 to letfori ∈B.Because|Dd|=1,there is ai0∈G,sayi0=1,such that

We have

Plug(4.50)and(4.51)into(4.46)to get

Denoteu22≥···≥unn,and

andλ2≥···≥λn ＞0 the eigenvalues of the matrixW.AssumeN ＞1,from(4.5)we see that

Then

Ifu11≤u22,we see thatλm1=λn.Then

it follows that

if we chooseand

In the following cases,we always assumeu11＞u22.

Case1.unn ≥0.It follows from

and(4.80)that

if we chooseA＞βC1+2κmaxnmaxf/γκ0.

We can choose a sufficiently small?=?(n,γ,κmax,κmin)to get

We now chooseA＞βC1+1 andβ≥4nκmaxto get

It follows that

if we assumeThis contradicts to thatu11＞u22.

In conclusion,we choose a small

andIfandwe obtainwhich contradicts to thatPattains its minimum in the interior of Ωμ. This implies thatPattains its minimum on the boundary?Ωμ.

On?Ω,it is easy to see

On?Ωμ∩Ω,we have

if we takeFinally the maximum principle tells us that

Supposewe have

Then we get

This completes the proof of the lemma.

In a similar way,we construct the super barrier function as

We also have the following lemma.

Lemma 4.4.Fix σ,if we select β large,μsmall,A large,then

Furthermore,we have

where C depends only on n,|u|C1,|?Ω|C2|f|C2and|φ|C2.

Proof.We assumeattains its maximum pointx0in the interior of Ωμ.Differentiatetwice to obtain

As before we assume that(uij)is diagonal atx0,so areWand(Fij),withFijdefined by(4.3).We chooseμsmall enough such thatIt follows that

Recall thathi=?(1?2K3d)di,we also have

By a straight computation,using Lemma 4.7,we obtain

where

We divide indexesI={1,2,···,n}into two sets in the following way,

whereκmin(κmax)is the minimum(maximum)principal curvature of the boundary.Fori∈G,bywe get

Then letA≥3βC2,we have

for?i ∈G.We chooseβ ≥nκmin+1 to letfori ∈B.Because|Dd|=1,there is ai0∈G,sayi0=1,such that

We have

Plug(4.78)and(4.79)into(4.74)to get

Denoteu22≥···≥unn,and

andλ1≥λ2≥···≥λn ＞0 the eigenvalues of the matrixW.AssumeN ＞1,from(4.5)we see that

Then

Sinceu11≤u22,we see thatλm1=λn.Then

it follows that

if we chooseandThis contradicts to thatattains its maximum in the interior of Ωμ.This contradiction implies thatattains its maximum on the boundary?Ωμ.

On?Ω,it is easy to seewe have

Supposewe have

Then we get

This completes the proof of the lemma.

Then we prove Theorem 4.1 immediately.

Proof of Theorem 4.1.We choosein Lemmas 4.3 and 4.4,then

Combining(4.88)with(4.5)in Lemma 4.1,we obtain(4.2).

5 Existence of the Neumann boundary problem

We use the method of continuity to prove the existence theorem for the Neumann problem(1.1).

Proof of Theorem 1.1.Consider a family of equations with parametert,

From Theorems 3.1 and 4.1,we get a globalC2estimate independent oftfor Eq.(5.1).It follows that Eq.(5.1)is uniformly elliptic.Due to the concavity ofwith respect

whereCdepends only onn,,minf,|φ|C3and Ω.It is easy to see thatis a strictly(n?1)-convex solution to(5.1)fort=0.Applying the method of continuity(see[8],Theorem 17.28),the existence of the classical solution holds fort=1.By the standard regularity theory of uniformly elliptic partial differential equations,we can obtain the higher regularity.The uniqueness is easy to get from maximum principle.toD2u(see[2]),we can get the global H?lder estimates of second derivatives following the discussions in[19],that is,we can get

Acknowledgments

The author would like to thank Professor Xi-Nan Ma,his supervisor,for his constant encouragement and guidance.The research of the author is partly supported by NSFC No.11721101 and No.11871255.