$C^\infty$ local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$

In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.

In this work, we study the existence of C ∞ -local solution of the following 2-Hessian equation in R 3 , where we also have S 2 [u] = u 11 u 22 − u 2 12 + u 22 u 33 − u 2 23 + u 11 u 33 − u 2 13 . We have proved the following results.

Moreover, the equation (1.3) is uniformly elliptic with respect to the above local solutions.
For the local solution, Hong and Zuily [5] obtained the existence of C ∞ local solutions to arbitrary dimensional Monge-Ampére equation, in which f is not only nonnegative but also satisfies a variant of Hörmander rank condition. Lin [8] proved the existence of a local H s solution in R 2 with f ≥ 0. We will follow the ideas of [5] and [8,9], the existence of the local solution can be obtained by a perturbation of polynomial-typed solution for S 2 [u] = a where a is a constant, so that our solution is in the form u(y) = 1 2 3 j=1 τ j y 2 j + ε 5 w(ε −2 y), τ = (τ 1 , τ 2 , τ 3 ) ∈ R 3 .
The significance of theorem 1.1 is our results break away from the framework of Gårding cone. The sign of f is permitted to change in case (1). For the case (2), we say that it is a degenerate 2-Hessian equation if f (Z 0 ) = 0(see [10]). The non-convex solution in (1) and (2) never occurs for Monge-Ampére equation. There is also many works about the convexity of solution to Hessian equation, see [11] and reference therein. Besides, these results seems to be strange. However, that is because the relationship between the sign of f and the ellipticity of the nonlinear k-Hessian equation may not be close.
The rest of this paper is arranged as follow: in Section 2, we will give definitions and some known results. Section 3 is devoted to the proof of Theorem 1.1.

Preliminaries
In this section, we collect some definitions and known results of k-Hessian equations. Firstly some algebraic properties of Gårding cone. Proposition 2.1 (See [12]). Using the notations introduced in Section 1, (2) Maclaurin's inequalities, for any λ ∈ Γ k , 1 ≤ l ≤ k, then p ≥ k and When n = 3, we see that σ 3 (λ) > 0 cannot occur for λ ∈ ∂Γ 2 (λ), therefore we can express ∂Γ 2 as two parts ∂Γ 2 (λ) = P 1 ∪ P 2 , Next, we will recall that what condition can lead to the ellipticity. As for the framework of ellipticity, we follow the ideas of [6] and [7]. Denote Sym(n) as the set of symmetric real n × n matrix. Through the matrix language, we recall the direct condition which leads to the elliptic k-Hessian operator. The ellipticity set of the k-Hessian operator, k = 1, 2, . . . , n, is E k = S ∈ Sym(n) : S k (S + tξ × ξ) > S k (S ) > 0, |ξ| = 1, t ∈ R + and the Gårding cones where the definition of S k (S ) is given in (1.2). It is easy to show that E k = Γ k only for k = 1, n and the example in [7] assures that Γ k ⊂ E k and mess(E k \ Γ k ) > 0 when 1 < k < n. Ivochkina, Prokofeva and Yakunina [7] point out that the ellipticity of (1.1) is independent of the sign of f .
Besides, σ 3 (τ) < 0 imply that τ i 0 and τ i can not be positive at the same time. Then property (4) of Proposition 2.1 implies We also have the following elliptic results for τ ∈ Γ 1 \Γ 2 .
For the linearized operators of k-Hessian equation, we have the following results, the general version of which can be found in section 2, [2]. Lemma 2.4. The matrix S i j 2 (r(w)) and (r i j (w)) can be diagonalized simultaneously, that is, for any smooth function w, we can find an orthogonal matrix T (x, ε) satisfying where Id is the identity matrix.
Now we set (r i j ) can be diagonalized by T , Thus, we have, when s t Now for is a diagonal matrix, our proof was done. Indeed, when s t, we have T si T t j (r 11 + r 22 + r 33 ) = 0.
When ε = 0, S i j 2 [r(w)] and (r i j (w)) are diagonal, thus, T can be the identity matrix Id. From the view above, when k = 2 and f < 0, the corresponding Hessian operator is possible to be uniformly elliptic. In this paper, we will study some uniformly elliptic 2-Hessian equations which have non-positive right-hand functions f .

Existence of C ∞ local Solutions for uniformly elliptic case
From now on, we fixed n = 3, k = 2, by a translation y −→ y − y 0 and replacing u by u − u(0) − y · Du(0), we can assume Z 0 = (0, 0, 0) in Theorem 1.1. We prove now the following results, in the neighborhood of y 0 = 0, w C 4,α ≤ 1 and ε > 0 very small.
Moreover, the equation (1.3) is uniformly elliptic with respect to the solution (3.1).
Remark that, in Theorem 3.1 the function f is permitted to change sign. It is well known that, for Monge-Ampere operator, the type of equation is determined by the sign of f (y, u, Du), it is elliptic if f > 0, hyperbolic if f < 0 and degenerate elliptic or hyperbolic if f vanishes; it is of mixed type if f changes sign [4]. So that Theorem 3.1 never occurs in Monge-Ampére case.

Lemma 3.2.
Assume that τ ∈ P 2 and w C 2 (B 1 (0)) ≤ 1, then the operator L G (w) is a uniformly elliptic operator if ε is small enough.
We follows now the idea of Hong and Zuily [5] to prove the existence and a priori estimates of solution for linearized operator. In our case, although L G (w) is uniformly elliptic, the existence and a priori Schauder estimates of classical solutions are not directly obtainable, because we do not know whether the coefficient a of au in (3.3) is non-positive. If we can prove the existence (Lemma 3.3), we can employ Nash-Moser procedure to prove the existence of local solution for (1.3) in Hölder space rather than Sobolev space. One goal is to see how the procedure depends on the condition w k C 4,α ≤ A. We shall use the following schema: .
It is pointed out on page 107, [3] that, if the operator L G does not satisfy the condition a ≤ 0, as is well known from simple examples, the Dirichlet problem for L G (w)ρ = g no longer has a solution in general. Notice a in (3.9) has the factor ε 4 , we will take advantage of smallness of a to obtain the uniqueness and existence of solution for Dirichlet problem (3.9) and then uniformly Schauder estimates of its solution follows.
By virtue of (3.3), we write (3.7) as Notice that for ∂S 2 (r(w)) ∂r i j , a i = a i (x, w(x), Dw(x)), a = a(x, w(x), Dw(x)) and g m = −G(w m ) = g m (x, w(x), Dw(x), D 2 w(x)) by (3.6), we regard them as the functions with variable x. In a word, we regard that all of the coefficients and non-homogeneous term in (3.9) are functions of variable x. For example, When we regardf ε as a function of variable x, usually f C 3,α is denoted as f C 3,α (B 1 (0)) , but it maybe cause confusion because it must be involved in D α w, 0 ≤ |α| ≤ 3 as above. Therefore, here and after, we denote the norm as f ε C 3 , f ε C 3,α as above, by dropping B 1 (0).
It follows from standard elliptic theory (see Theorem 6.17, [3] and Remark 2, [1]) and an iteration argument that we obtain. a solution of (1.3), and the linearized operators with respect to u, is uniformly elliptic, then u ∈ C ∞ (Ω).

Proof.
Let v be a function on Ω and denote by e l , l = 1, 2, 3 the unit coordinate vector in the y l direction. We define the difference quotient of v at y in the direction e l by and Taylor expansion give f (y + he l , u(y + he l ), Du(y + he l )) − f (y, u(y), Du(y)) Taking the difference quotients of both sides of the equation Since u ∈ C 2,α (Ω), then all the coefficients a i j , b i , c and inhomogeneous term g are in C α (Ω), from the interior estimates of Corollary 6.3 in [3], we can infer Letting h → 0, we see ∂ l u ∈ C 2,α (Ω), l = 1, 2, 3 and 3 i, j=1 Repeating the above proof, we obtain u ∈ C ∞ (Ω).  (3.6). Suppose that w j C 4,α ≤ A for j = 1, 2, . . . , k. Then we have , where C is some positive constant depends only on τ ,A and f C 4,α . In particular, C is independent of k.
Proof. Applying Taylor's expansion with integral-typed remainder to (3.2), we have where Q k is the quadratic error of G which consists of S 2 and f .
Since S 2 ((r(w))) is a second-order homogeneous polynomial with variable r i j (r(w)) and f ε (x, w, Dw) is independent of r i j , we see that Thus, and O(ε 9 ), repectively. Therefore and where C depends on A and f C 4,α . And I 3 C 2,α and I 4 C 2,α can be estimated similarly. Accordingly, where C is independent of k but dependent of A and f C 4,α . Thus, by the interpolation inequalities, we have where C is independent of k. By Schauder estimates of Lemma 3.3, we have ρ k C 4,α ≤ C g k C 2,α .
Combining the estimates above, we obtain (3.13). Proof is done.
Since C is independent of k, more exactly, A, τ and f C 4,α are independent of k. So here and after, we can assume A = 1.
The C ∞ regularity of solution is given by Corollary 3.4. We have then proved Theorem 3.1.