Smooth local solutions to weingarten equations and $\sigma_k$-equations

In this paper, we study the existence of smooth local solutions to Weingarten equations and $\sigma_k$-equations. We will prove that, for $2 \leq k \leq n$, the Weingarten equations and the $\sigma_k$-equations always have smooth local solutions regardless of the sign of the functions in the right-hand side of the equations. We will demonstrate that the associate linearized equations are uniformly elliptic if we choose the initial approximate solutions appropriately.


Introduction
Weingarten hypersurfaces are the hypersurfaces whose principal curvatures satisfy some algebraic equations. Specifically, let Ω be a domain in R n and u be a function defined in Ω. Suppose κ 1 , · · · , κ n are the principal curvatures of the graph (x, u(x)). Then the general Weingarten hypersurfaces are given by (1.1) f (κ 1 , · · · , κ n ) = ψ(x), where f : R n → R is a given function. If f (κ 1 , · · · , κ n ) = κ 1 + · · ·+ κ n , then (1.1) reduces to the prescribed mean curvature equation where ψ(x) is the mean curvature of the graph (x, u(x)), usually denoted by H. If f (κ 1 , · · · , κ n ) = κ 1 ·· · ··κ n , then (1.1) reduces to the prescribed Gauss curvature equation where ψ(x) is the Gauss curvature of the graph (x, u(x)), usually denoted by K. It is well known that (1.2) is always elliptic regardless the sign of ψ and that (1.3) is elliptic if ∇ 2 u is positive definite and hence ψ is positive. Caffarelli, Nirenberg and Spruck [3] studied the Dirichlet problem for (1.1) for a class of functions f and positive ψ in strictly convex domains. The corresponding equation is elliptic. The function f in [3] includes as the special cases the k-th elementary symmetric functions σ k . We will refer the corresponding equation The second author acknowledges the support of the NSF Grant DMS-1404596. 1 as the prescribed σ k -curvature equation. We note that k = 1 corresponds to the prescribed mean curvature equation and that k = n corresponds to the prescribed Gauss curvature equation. Refer to [14] for the elliptic σ k -equations and [13] for elliptic Weingarten equations.
In this paper, we study the local solutions of the prescribed σ k -curvature equation (1.4) and σ k -equations.
For n = k = 2, an equation similar to (1.3) appears in the form of the Darboux equation , where g is a smooth 2-dimensional Riemannian metric. Darboux showed that g admits a smooth isometric embedding in R 3 if and only if (1.5) admits a smooth solution u with |∇ g u| < 1. The equation (1.5) is elliptic if K is positive and hyperbolic if K is negative. In [11] and [12], Lin proved the existence of the sufficiently smooth isometric embedding for the following two cases: K(0) = 0 and K nonnegative in a neighborhood of 0 ∈ R 2 , or K(0) = 0 and dK(0) = 0. Han, Hong and Lin in [8] proved the sufficiently smooth isometric embedding if K is nonpositive and satisfies some nondegeneracy condition. In [6], Han gave an alternative proof of the result by Lin [12].
For the general dimension, the σ n -equation ( The linearized equation for small w is a perturbation of This is elliptic if ψ(0) > 0 and hyperbolic if ψ(0) < 0. Hence we can prove the existence of a solution to (1.3) in a neighborhood of the origin if ψ(0) = 0. If ψ(0) = 0, the situation is quite complicated. Hong and Zuily in [10] considered (1.3) if ψ ≥ 0. Following [11], they considered They showed that, by adding some appropriate terms, the modified linearized equations can be made degenerately elliptic. Based on this, they were able to prove the existence of sufficiently smooth solutions (1.3) in the general case ψ ≥ 0 and the existence of smooth solutions if, in addition, ψ does not vanish to infinite order or the zero set of ψ has a simple structure. Han in [7] discussed (1.3) if ψ changes sign. He proved the existence of sufficiently smooth solutions if K changes sign cleanly, i.e., ψ(0) = 0 and ∇ψ(0) = 0. In this case, the modified linearized equations are of the Tricomi type, elliptic in one side of the hypersurface and hyperbolic in another side.
These results clearly demonstrate how ψ determines the type of the σ n -equation (1.3). It is reasonable to expect a similar pattern for the general σ k -equation (1.4), for 2 ≤ k ≤ n − 1. However, this is not the case. In order to find a local solution of (1.4), we can always rewrite it as a perturbation of a linear elliptic equation if we choose the initial approximation appropriately, regardless of the sign of ψ in (1.4). Therefore, we can always find a local solution of (1.4), with no extra assumptions on ψ.
The main result in this paper is the following theorem.
A similar result holds for σ k -equations. Now we describe the method of proof. We consider a solution u of the form In order to have an initial approximation to an actual solution, it is reasonable to require If k = n, this reduces to If ψ(0) = 0, one of µ i has to be zero. This is the reason that x 2 n is missing in (1.6). However, for 2 ≤ k ≤ n − 1, we can choose nonzero µ i for all i and also require the linearized differential equation to be elliptic. In this way, the sign of ψ has no effect on the type of the equation (1.4).
The paper consists of three sections including the introduction. In Section 2, we prove some algebraic inequalities concerning the σ k -function. These inequalities play an important role in the proof of Theorem 1.1. In Section 3, we use the implicit function theorem to prove Theorem 1.1.
For an n × n symmetric matrix A, we let λ(A) be the collection of eigenvalues of A and treat it as a vector in R n . We also write σ k (A) = σ k λ(A) .
Lemma 2.1. Let 1 ≤ k ≤ n, B = (b ij ) be an n × n symmetric matrix and D = diag(µ 1 , · · · , µ n ) be a diagonal matrix, for some µ = (µ 1 , · · · , µ n ) ∈ R n . Then, Proof. Note that σ k (D + B) is a polynomial in µ of degree k. The homogeneous part of degree k is obviously σ k (µ). We only need to find the homogeneous part of degree k − 1. Recall To find the homogeneous part in µ of degree k − 1 in σ k (D + B), we need only identify in det(λI + D + B) the following term det(λI + D + B) = · · · + n p,q=1 where c pq (µ) is a homogeneous polynomial in µ of degree k − 1. The easiest way to do this is to keep b ij for fixed (i, j) and let all other b pq be zero in det(λI + D + B). Then we expand det(λI + D + B) as a polynomial of λ and b ij , and identify the corresponding coefficient of λ n−k and b ij . For (i, j) with i = j, by letting all other b pq be zero, we have This implies c ii (µ) = σ k−1 (µ|i). Similarly, for (i, j) with i = j, by letting all other b pq be zero, we have This implies c ij = 0 for i = j.
By a similar method as in the proof, we can in fact identify all terms in σ k (D + B). For example, the homogeneous part in µ of degree k − 2 is given by We will use Lemma 2.1 in the following way. For a C 2 -function w = w(x), we set We will choose constants µ 1 , · · · , µ n and a sufficiently small function w appropriately such that u defined above is the desired solution. To this end, we need to analyze the linearization of σ k (∇ 2 u) with respect to w at w = 0. Note where D = diag(µ 1 , · · · , µ n ). Hence, the linearization of σ k (∇ 2 u) with respect to w at w = 0 is given by We now demonstrate that we can always make this operator elliptic by choosing µ 1 , · · · , µ n appropriately.
To prove Theorem 1.1, we will construct solutions u as perturbations of the initial approximation where µ 1 , · · · , µ n are from Lemma 2.2. Hence, local behaviors of u such as the convexity coincide with those of u 0 in (2.5), if none of µ i is zero. The proof of Lemma 2.2 gives one choice of µ 1 , · · · , µ n satisfying (2.1) and (2.2), which may not result in solutions u with good geometric properties. In some cases, better choices of µ 1 , · · · , µ n are available. For example, if M > 0, we can choose µ 1 = · · · = µ n = N for some appropriate N > 0. Then, the corresponding u 0 is convex and so will be the resulting solution u.

Proof of the Main Theorem
In the present section, we discuss the linearization of the σ k -equation and prove Theorem 1.1.
To proceed, we temporarily replace x ∈ R n byx ∈ R n and write ∂ i instead of ∂x i . Consider (3.4)F(u) = σ k (κ 1 , · · · ,κ n ) − ψ(x), whereκ = (κ 1 , · · · ,κ n ) is the collection of all principal curvatures of the graph (x, u(x)). Now we set, for ε > 0,x = ε 2 x, and We evaluateF (u) in terms of w by setting We first note that∂ Then the matrix A = (a ij ) in (3.1) has the form where b ij is a smooth function in its arguments. By Lemma 2.1, we have where c is a smooth function in its arguments. By substituting in (3.4), we havẽ By the mean value theorem (applied to ψ) and (3.5), we obtain where f is a smooth function in its arguments. Our goal is to solve F(w, ε) = 0 for sufficiently small ε.
Proof of Theorem 1.1. Let F be as given in (3.6). Then We note that F : X × (−ε 0 , ε 0 ) → Y is a well-defined map for sufficiently small ε 0 and that L : X → Y is an isomorphism by the classical Schauder theory. Then by the implicit function theorem, for sufficiently small ε, there exists a w = w(ε) ∈ X such that F(w, ε) = 0. Note that F is a fully nonlinear elliptic operator for sufficiently small ε. The classical Schauder theory implies that w ∈ C ∞ (B 1 ).