Two problems related to prescribed curvature measures

Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. In surprise, that methods has also brought us to promote Ivochkina's $C^2$ estimates for prescribed curvature equation in \cite{I1, I}.

There is a difficulty issue around equation (1.2): the lack of some appropriate a priori estimates for admissible solutions due to the appearance of gradient term at right side. That problem has been open for many years [4]. More recently, Guan-Li-Li [10] have obtained the C 2 a priori estimates for the admissible k-convex starshaped solutions to prescribing (n-k)th curvature measures for 1 k < n.
In this paper, we are interested in to consider the following problems where 2 ≦ k ≦ n.
Our first motivation is from the existence of convex body with prescribed curvature measures. Equation (1.3) is the equation of prescribing (n − k)th curvature measure for p = 1. In particular, Guan-Li-Li [10] has given a open problem for most general problem in remark 3.5. our result may implies that their conjecture is correct. Caffarelli-Nirenberg-Spruck [1] had been considered some kind of curvature equation including (1.3) for p = 0, their C 1 estimates depends on barrier conditions. However, that is not case for our problem. Thus, we consider (1.3) for p = 0. Moreover, we can obtain C 1 estimates for a class of curvature equations including (1.3) and quotient curvature equations, its idea is from [9]. Now, we can state the main theorem.
Then there is a unique smooth admissible hypersurface M satisfying (1.3).
Our second motivation is to generalize Ivochkina's C 2 estimates for prescribed curvature equation in [12,13] by making use of those methods. Ivochkina [12,13] had considered the generalized type of curvature equation, see also [2,3,4,19,21,22] and their references, where A and λ = (λ 1 , · · · , λ n )denote respectively the second fundamental form and the principle curvatures of the graph For doing C 2 estimates of (1.4), She needed her condition (1.5) in [12](see also (8.28) in [13]),which is We consider mainly a kind of model from Takimoto [20], which is also seen as translating solution of curvature flow.
Ivochkina's conditions (1.5) in [12] needs q 0. However, we can generalize Ivochkina's C 2 estimate to q 1. Theorem 1.3. Suppose g ∈ C 4 (Ω) C 2 (Ω) is an admissible solution of (5.2) for q 1. Then the second fundamental form A of graph u satisfies sup where C depends only on n, g C 1 (Ω) , and Ω × [inf This paper is organized as follows: The C 0 and C 1 bounds and some elementary formulas were listed in section 2, the important C 2 −estimates are derived in section 3, which is by using Guan-Li-Li's innovative methods. In the last section, we can generalize Ivochkina's C 2 estimates for prescribed curvature equation in [12,13].

Some elementary formals and C 0 -C 1 boundness
The standard basis of R n+1 will be denoted by E 1 , E 2 , · · · E n+1 , and the components of the position vector X in this basis will be denoted by X 1 , X 2 , · · · , X n+1 . We choose an orthonormal frame such that e 1 , e 2 , · · · e n are tangent to M and ν is normal.
The second fundamental form of M is given by and some fundamental formulas are well known for hypersurfaces M ∈ R n+1 as [9].
We have the following gradient estimate for general curvature measure equation which included (1.3) and curvature quotient equations. As in Guan-Lin-Ma [9], the result will be obtained without any barrier condition that was imposed in [1]. Moreover, our result holds for any non zero p, i.e. 0 = p ∈ (−∞, +∞).
for F with homogeneous of degree t > 0, and for 0 = p ∈ (−∞, +∞), then there exist a constant C depending only on n, t, p, min S n φ, |φ| C 1 such that We use the method of Guan-Lin-Ma [9]. The gradient bound is equivalent to u = X, ν C > 0 if the lower and upper bound of the solution holds. Setting where the function γ(s) is to be determined. Assume P (X) attains its maximum at point X 0 ∈ M. We choose the smooth local orthonormal frame e 1 , · · · , e n ∈ T X0 M such that X, e i = 0, i 2 Thus |X| 2 = X, e 1 2 + X, ν 2 . If X, e 1 2 is also zero, then |X| 2 = X, ν 2 , then the bounded from below of X, ν is from the bound of |X|. We now consider X, e 1 2 > 0, one has It is easy to know that we only fix e 1 in above process of choosing local orthonormal frame field e 1 , · · · , e n , here we adjust e 2 , · · · , e n , such that Differentiating equation (2.6) with respect to e 1 , Thus we obtained We may assume X, ν 2 C|X| 2 for some C > 0, otherwise the lemma holds. we claim firstly that by taking γ(s) properly. We check it by three case of p < 0, or p > t, and 0 < p t We taking which implies (2.10) for α > 0 large enough. Case (ii) 0 < p t : we have which imply inequality (2.10). Combing (2.9) with (2.10), On the other hand, 12) which is from h 11 0 and then F 11 c 0 n i=1 F ii . We test the case of F = σ k σ l (F = σ k see [9]), from (25) in [15] F 11 C(n, l, k) Thus X, ν C is from (2.11) and (2.12), that is to say there exists a constsnt that depends only on n, t, min S n φ, |φ| C 1 such that max S n |∇ρ| C.
The following lemma is key in our proof for C 2 estimate, which is from Guan-Li-Li's important lemma [5,10].
Lemma 3.1. For any α > 0, one has the following inequality Proof. From Krylov [14], for any α > 0, σ1 This implies we have proved this lemma.
we denote X, ν by u in what follows. Taking test function σ1 u , then at its maximal point P which is equivalent to On the other hand, we have We also compute the following by lemma 2.1, Then , combing (3.4) with (3.5), (3.6), Then with lemma 3.1, and (σ k )s one has the C 2 estimate if (p − 1)[p − (α + 1)p + (α − 1)] 0, which is satisfied by taking p 1 and α = 1 1−p > 0.
From the above certificate process, we know the key point for proving C 2 estimates is the concavity of [ σ k σ1 ] For t = 0, X = (C k n ) − 1 k−p is a solution of (4.1), i.e. I is not empty. Moreover, The a prior estimates lemma 2.2, lemma 2.2 and theorem 3.2 and Evans-Krylov theorem imply the closeness of I. we prove the following proposition that is to say I is open. Proposition 4.1. Assume F (λ) and φ(x, ρ, ∇ρ) satisfying homogeneity property: Then the linearized operator L of F (λ) = φ(x, ρ, ∇ρ) has no non zero kernel, which is from lemma 2.5 in [9].
Lastly, the uniqueness result of such problem is same as lemma 2.4 in [9]. We have complete the proof of theorem 1.2.

some discuss about Ivochkina's problem
Ivochkina [12,13] had considered the generalized type of curvature equation, see also [2]v and their references, where λ = (λ 1 , · · · , λ n ) is the principal curvatures of the graph , she need her condition (1.5) in [12](see also (8.28) in [13]) to do C 2 estimate. An example is for s k 2 . Please note that there is a misprint at page 334 in [12] for that example, that is no her (1.5) for s 1/2. The author want to thank for Ivochkina's mention for that.
Here we give a C 2 estimate for a special case φ(x, u, Du), we consider the example which is similar to (2.6) and (2.7) in [20].
for this example, Ivochkina need Proof of Theorem 1.3: Proof. As [19], taking a local orthonormal frame field e 1 , · · · , e n defined on M = {(x, u(x))|x ∈ Ω} in a neighbourhood of the point at which we are computing and the upward unit normal vector field is We consider test function W = wσ 1 h( w 2 2 ) for h(t) > 0 to be determined, and it attain its interior maximum at X 0 .
By Lemma 2.1 in [22], By (2.9) in [22], this combines with (5.5), one has Taking From (5.2), we set where H(x, g) does not impact the process of proof in the follows, we may consider the following special case for simplifying the denotation, for σ 1 >> 1.
Remark 5.1. h ′ w 2 h F ij h mi h mj is a good term for our estimate in (5.8). Ivochkina has used it in [12,13]. one may use it to control the term like − w −(2+q) |∇w| 2 σ1 and refine theorem 1.3. [20] had used a priori estimates of the second derivatives of u in his (2.6) and (2.7) for 1 q k − 1 at page 368. So his result is incomplete. Remark 5.3. In the end, an interesting problem is what we can generalize Ivochkinas' C 2 estimates in [12,13] to the following quotient curvature equations?

Remark 5.2. Takimoto
where 0 ≦ l < k ≦ n. Of course, it is also interesting for generalized Guan-Li-Li's results to quotient equations.
Remark 5.4. More recently, we have found that Chuanqiang Chen had obtained C 2 estimates for our problem in case of k = 2 by using different methods :"Chuanqiang Chen. A Minimal Value Problem and the Prescribed σ 2 Curvature Measure Problem, arXiv:1104.4283"