SECOND ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS. 1. Introduction

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1 SECOND ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS BO GUAN Abstract. We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under structure conditions which are close to optimal. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary. These estimates yield regularity and existence results some of which are new even for equations in Euclidean space. Mathematical Subject Classification (2010): 35J15, 58J05, 35B45. Keywords: Fully nonlinear elliptic equations on Riemnnian manifolds; a priori estimates; Dirichlet problem; subsolutions; strict concavity property. 1. Introduction Let (M n, g) be a compact Riemannian manifold of dimension n 2 with smooth boundary M, M := M M, and χ a smooth (0, 2) tensor on M. In this paper we consider fully nonlinear equations of the form (1.1) f(λ[ 2 u + χ]) = ψ in M where 2 u denotes the Hessian of u C 2 (M) and λ[ 2 u + χ] = (λ 1,, λ n ) are the eigenvalues of 2 u + χ with respect to the metric g, and f is a smooth symmetric function defined in a symmetric open and convex cone Γ R n with vertex at the origin and boundary Γ, (1.2) Γ n {λ R n : each component λ i > 0} Γ. The study of fully nonlinear equations of form (1.1) were pioneered by Ivochkina [22] who treated some special cases, and Caffarelli-Nirenberg-Spruck [5] in R n (with χ = 0) Research supported in part by NSF grants. 1

2 2 BO GUAN and have received extensive attention since then. The standard and fundamental structure conditions on f in the literature include (1.3) f i = f λi f > 0 in Γ, 1 i n, λ i (1.4) f is a concave function in Γ, and (1.5) δ ψ,f inf ψ sup f > 0 M where sup f sup lim sup f(λ). Γ λ 0 Γ λ λ 0 Condition (1.3) ensures that equation (1.1) is elliptic for solutions u C 2 (M) with λ[ 2 u + χ] Γ; we shall call such functions admissible, while condition (1.4) implies the function F defined by F (A) = f(λ[a]) to be concave for A S n n with λ[a] Γ, where S n n is the set of n by n symmetric matrices; see [5]. The most typical equations of form (1.1) are given by f = σ 1 k k and f = (σ k /σ l ) 1 k l, 1 l < k n defined on the cone Γ k = {λ R n : σ j (λ) > 0 for 1 j k}, where σ k is the k-th elementary symmetric function σ k (λ) = λ i1 λ ik, 1 k n. i 1 < <i k These functions satisfy (1.3)-(1.4) and have other important properties which have been widely used in study of the corresponding equations; see e.g. [5], [7], [22], [23], [24], [26], [28], [29], [32], [36]. Geometric quantities of the form 2 u + χ appear naturally in many different subjects. For example, Ric u = 2 u + Ric, where Ric is the Ricci tensor of (M n, g), is the Bakry-Emery Ricci tensor of the Riemannian measure space (M n, g, e u dvol g ) (see e.g. [37] and references therein), while the gradient Ricci soliton equation takes the form Γ 2 u + Ric = λg which plays important roles in the theory of Ricci flow and has received extensive study. When χ = g equation (1.1) was studied by Li [28] and Urbas [35] on closed manifolds.

3 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 3 A central issue in order to solve equation (1.1) is to derive a priori C 2 estimates for admissible solutions; the Evans-Krylov theorem then yields C 2,α bounds by assumptions (1.4) and (1.5) which prevents equation (1.1) from becoming degenerate and guarantees that equation (1.1) becomes uniformly elliptic once a priori C 2 bounds are established for admissible solutions. Our main concern in this paper is the estimates for second derivatives of admissible solutions (1.6) 2 u C in M where C may depend on u C 1 ( M). Such estimates have been established under various assumptions on f in addition to (1.3)-(1.5) as well as conditions on the geometry of M, with significant contributions from Caffarelli-Nirenberg-Spruck [5], Ivochkina [23], Li [28], Trudinger [33], Chou-Wang [7], Urbas [35], and Ivochkina-Trudinger-Wang [25] who considered the degenerate case, among others (see also [10], [13]), Our primary goal in this paper is to establish (1.6) under conditions (in addition to (1.3)-(1.5)) which are essentially optimal, on general Riemannian manifolds with (smooth, compact but otherwise) arbitrary boundary. (In a forthcoming paper we shall come back to discuss possibilities to weaken (1.4), the concavity condition.) In order to state our result we first introduce some notation. For σ > sup Γ f, define Γ σ = {λ Γ : f(λ) > σ}, and we shall only consider the case Γ σ. By conditions (1.3) and (1.4) we see that the level hypersurface Γ σ = {λ Γ : f(λ) = σ}, i.e. boundary of Γ σ, is a smooth convex hypersurface in R n. Define for µ Γ \ Γ σ S σ µ = {λ Γ σ : µ T λ Γ σ } where T λ Γ σ denotes the tangent plane of Γ σ at λ, and C + σ = V σ Γ σ where V σ = {µ Γ \ Γ σ : S σ µ is nonempty and compact}. As we shall see (Lemma 2.8) C σ + is an open subset of Γ. For convenience we call C σ := C σ + the tangent cone at infinity of Γ σ. Our first main result may be stated as follows.

4 4 BO GUAN Theorem 1.1. Let ψ C 2 (M) C 1 ( M) and u C 4 (M) C 2 ( M) be an admissible solution of (1.1). Assume (1.3)-(1.5) hold and that there exists a function u C 2 ( M) satisfying for all x M (1.7) λ[ 2 u + χ](x) C + ψ(x). Then ( (1.8) max 2 u C max M M 2 u ). where C 1 depends on u C 1 ( M) (but not on δ 1 ψ,f ). In particular, if M is closed, i.e. M = then (1.9) 2 u C 1 e C 2(u inf M u) on M where C 2 is a uniform constant (independent of u). Equation (1.1) with χ = g was first studied by Li [28] who derived (1.6) on closed manifolds of nonnegative sectional curvature, followed by Urbas [35] who removed the nonnegative curvature assumption. In addition to (1.3)-(1.5), both of these papers needed extra assumptions which exclude the case f = (σ k /σ l ) 1/(k l). This case is covered by Theorem 1.1; see Lemma In [14] it is shown that the hypotheses in [28] imply (1.7) for χ = g and any constant u. From Theorem 1.1 and the Evans-Krylov theorem we obtain the following regularity result by approximation. Theorem 1.2. Let (M n, g) be a closed Riemannian manifold and ψ C 1,1 (M). Under conditions (1.3)-(1.5) and (1.7), any weak admissible solution (in the viscosity sense) u C 0,1 (M) of (1.1) belongs to C 2,α (M), 0 < α < 1 and (1.9) holds. Higher regularities follow from the classical Schauder theory for linear elliptic equations. In particular, u C (M) if ψ C (M). Remark 1.3. If λ(χ) C σ + for all σ (e.g. if χ = ag, a > 0 and 0 C σ + ), any constant u satisfies (1.7). For f = σ 1/k k (k 2), Γ n C σ + for any σ > 0. Corollary 1.4. Let (M, g) be a closed Riemannian manifold and ψ C 1,1 (M). In addition to (1.3)-(1.5), suppose χ C + σ for all σ (sup Γ f, sup M ψ]. Then any admissible weak solution u C 0,1 (M) of (1.1) belongs to C 2,α (M), 0 < α < 1, and (1.9) holds.

5 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 5 We now turn to the second order boundary estimate (1.10) max M 2 u C 3 when M. Even for domains in R n this is very subtle and usually requires extra assumptions on f and M. In their original work [5], Caffarelli-Nirenberg-Spruck derived (1.10) for the Dirichlet problem (with χ = 0) in a bounded domain M R n satisfying the curvature condition: there exists R > 0 such that (1.11) (κ 1,..., κ n 1, R) Γ on M where (κ 1,..., κ n 1 ) are the principal curvatures of M (with respect to the interior normal), assuming (1.3)-(1.5) and the following hypotheses on f: for every C > 0 and compact set K in Γ there is a number R = R(C, K) such that (1.12) f(λ 1,..., λ n 1, λ n + R) C for all λ K, (1.13) f(rλ) C for all λ K. Their result was extended by Li [28] to the general case where χ is a symmetric matrix, and by Trudinger [33] who removed condition (1.12). It was shown in [5] that if a domain Ω of type 2 in R n satisfies (1.11) then Ω is connected; see [5] for details. In particular, for Γ = Γ n, (1.11) implies that M R n is strictly convex. In this paper we wish to derive the boundary estimate (1.10) on a general Riemannian manifold M without imposing any geometric restrictions to M beyond being smooth and compact. The first effort in this direction was made by the author [10] where we had to assume (1.12) and the existence of an admissible strict subsolution; Trudinger [33] later showed that one can replace (1.12) by (1.13). In this paper we were able to prove the following result. Theorem 1.5. Let ψ C 1 ( M), ϕ C 4 ( M) and u C 3 (M) C 2 ( M) be an admissible solution of (1.1) with u = ϕ on M. Assume f satisfies (1.3)-(1.5) and (1.14) fi λ i 0 in Γ {inf M ψ f sup M ψ}. Suppose that there exists an admissible subsolution u C 0 ( M) in the viscosity sense: { f(λ[ 2 u + χ]) ψ in M, (1.15) u = ϕ on M, and that u is C 2 and satisfies (1.7) in a neighborhood of M. Then the boundary estimate (1.10) holds with C 3 depending on u C 1 ( M) and δ 1 ψ,f.

6 6 BO GUAN Remark 1.6. An admissible subsolution u C 2 ( M) will automatically satisfy (1.7) provided that Γ σ C σ + which is equivalent to (1.16) Γ σ C + σ =, σ [ inf M ψ, sup ψ ]. M Condition (1.16) excludes the linear function f = σ 1 which corresponds to the Poisson equation, but is satisfied by a wide class of concave functions including f = σ 1/k k, k 2 and f = (σ k /σ l ) 1/(k l) for all 1 l < k n; see Lemma Note that condition (1.16) holds if Γ σ is strictly convex at infinity, i.e. outside a compact set. Remark 1.7. For bounded domains in R n and any given smooth boundary data it was shown in [5] that (1.11) and (1.13) imply the existence of admissible strict subsolutions which satisfy condition (1.7) automatically. Remark 1.8. The hypothesis (1.14), which is clearly implied by (1.13), can be dropped when M is a bounded domain in R n. Since it requires different arguments in several places of the proof, we shall treat it elsewhere ([14]). Remark 1.9. If M and there is a strictly convex function v C 2 ( M), then u = Av satisfies (1.7) on M for A sufficiently large. The Dirichlet problem for equation (1.1) in R n was treated by Caffarelli-Nirenberg- Spruck [5], followed by [28], [10], [33], [25] among others. The important special case f = σ 1 k k has also received extensive study from different aspects, including [27], [23], [36], [34], [7]. Applying Theorems 1.1 and 1.5 one can prove the following existence result by the standard continuity method. Theorem Let ψ C ( M) and ϕ C ( M). Suppose f satisfies (1.3)-(1.5), (1.14) and that there exists an admissible subsolution u C 2 ( M) satisfying (1.15) and (1.7) for all x M. Then there exists an admissible solution u C ( M) of the Dirichlet problem for equation (1.1) with boundary condition u = ϕ on M, provided that (i) Γ = Γ n, or (ii) the sectional curvature of (M, g) is nonnegative, or (iii) f satisfies (1.17) f j δ 0 fi (λ) if λ j < 0, on Γ σ σ > sup Γ f. Remark For bounded domains in R n, Theorem 1.10 (which holds without assumption (1.14)) extends the previous results of Caffarelli-Nirenberg-Spruck [5], Trudinger [33] and the author [10]. The assumptions (i)-(iii) are only needed to derive gradient estimates; see Proposition 5.1.

7 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 7 Remark When Γ = Γ n, (1.14) and (1.16) hold automatically; see Lemma Corollary Let f = σ 1/k k, k 2 or f = (σ k /σ l ) 1 k l, 1 l < k n. Given ψ C ( M), ψ > 0 and ϕ C ( M), suppose that there exists an admissible subsolution u C 2 ( M) satisfying (1.15). Then there exists an admissible solution u C ( M) of equation (1.1) with u = ϕ on M. For f = (σ k /σ l ) 1 k l, 0 l < k n, which satisfies (1.17), Corollary 1.13 is new even when M is a bounded domain in R n. It would be worthwhile to note that in Theorem 1.10, since there are no geometric restrictions to M being made, the Dirichlet problem is not always solvable without the subsolution assumption. In the classical theory of elliptic equations, a standard technique is to use the distance function to the boundary to construct local barriers for boundary estimates. So one usually needs require the boundary to possess certain geometric properties; see e.g. [31] for the prescribed mean curvature equation and [4], [3] for Monge-Ampère equations; see also [9] and [5]. In our approach we use u u to replace the boundary distance function in deriving the second order boundary estimates. This idea was first used by Haffman-Rosenberg-Spruck [21] and further developed in [16], [15], [11], [12] to treat the real and complex Monge-Ampère equations in general domains as well as in [10], [13] for more general fully nonlinear equations. The technique has found useful applications; see e.g. [2], [6], [16], [17], [18], [20], [30]. We shall also make use of u u in the proof of the global estimate (1.8). This is one of the notable features of the paper; see the proof in Section 3. Note that in Theorem 1.1 the function u is not necessarily a subsolution. On a closed manifold, an admissible subsolution of equation (1.1) must be a solution if there is a solution at all, and any two admissible solutions differ at most by a constant. This is a consequence of the concavity condition (1.4) and the maximum principle. The rest of this paper is organized as follows. In Section 2 we discuss some consequences of the concavity condition. Our proof of the estimates heavily depends on results in Section 2 where we also give a brief proof that (1.16) holds for f = σ 1/k k, k 2 and f = (σ k /σ l ) 1 k l, 1 l < k n, and therefore Corollary 1.13 follows from Theorem In Sections 3 and 4 we derive the global and boundary estimates for second derivatives, respectively. In Section 5 we derive the gradient estimates needed to prove Theorem 1.10.

8 8 BO GUAN The author wishes to thank Jiaping Wang for helpful discussions on the proof of Theorem 2.17 and related topics, and the referees for their insightful comments and suggestions. The author is also pleased to thank Shujun Shi and Zhenan Sui who carefully read the manuscript and pointed out many errors. Finally, we remark that the ideas in this article can be used to significantly weaken the concavity assumption (1.4) and treat general fully nonlinear elliptic equations of the form (1.18) F ( 2 u + χ) = ψ. This will be addressed in forthcoming papers. 2. The concavity condition In this section we examine Γ σ and the associated cone C + σ. The main purpose is to prove Theorem 2.17 below on which all the estimates in this paper will heavily depend. For the reader s convenience we shall include some elementary properties of the related geometric objects. Let σ > sup Γ f and assume Γ σ := {f > σ}. Then Γ σ is a smooth convex noncompact complete hypersurface contained in Γ. For λ Γ σ let ν λ = Df(λ)/ Df(λ) denote the unit normal vector to Γ σ at λ. For convenience we shall often implicitly identify ν λ with its image G(λ) S n under the Gauss map G : Γ σ S n of Γ σ. Recall that S σ µ for µ Γ \ Γ σ, V σ and C + σ = V σ Γ σ are defined in Section 1. Lemma 2.1. Let µ Γ σ. Then S σ µ = Γ σ T µ Γ σ. Consequently, µ C + σ if and only if Γ σ T µ Γ σ is compact. Proof. This clearly follows from the fact that Γ σ is globally convex. Lemma 2.2. Let µ Γ σ C + σ. Then for any t > 0, the part of Γ σ Σ t = Σ t (µ) := {λ Γ σ : (λ µ) ν µ t} is a convex spherical cap with smooth boundary on the plan tν µ + T µ Γ σ.

9 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 9 Proof. Consider the distance function h(λ) = (λ µ) ν µ, λ Γ σ from Γ σ to T µ Γ σ. It is easy to see that the gradient h of h on Γ σ satisfies h(λ) 2 = 1 ν λ ν µ 2. Indeed h(λ) can be identified in a natural way with ν µ (ν λ ν µ )ν λ. Therefore, λ Γ σ is a critical point of h if and only if ν λ ν µ = 1. Since Γ σ is convex, ν λ ν µ = 1 if and only if λ Γ σ T µ Γ σ. By (1.3), on the other hand, ν λ ν µ = Df(λ) Df(µ) Df(λ) Df(µ) > 0. This shows that any t > 0 is a regular value of h and therefore the level set of h Σ t := {λ Γ σ : h(λ) = t} is a smooth convex (and compact) hypersurface in the plane tν µ + T µ Γ σ. Lemma 2.3. Let µ Γ σ C + σ. Its Gauss map image G(µ) = ν µ is an interior point of G( Γ σ ) in S n. Proof. Write N = G(µ) and consider any E S n with E N = 0. For θ small let P θ be the plane through µ normal to N θ = N cos θ + E sin θ. (So P 0 = T µ Γ σ.) Since Γ σ is smooth, the set Λ θ := {λ Γ σ : (λ µ) N θ < 0} is nonempty for all small θ 0. It is clear, on the other hand, that (λ µ) N θ > 0 on Σ 1, and therefore Λ θ Σ 1 if θ 0 is sufficiently small. Geometrically this just means that P θ cuts off a (compact) convex spherical cap Λ θ from Γ σ. Now for all θ 0 sufficiently small there is a point ζ θ Σ 1 such that (ζ θ µ) N θ = min ζ Γ σ (ζ µ) N θ < 0. Thus G(ζ θ ) = N θ. This completes the proof. Corollary 2.4. For µ V σ, G(S σ µ) is contained in the interior of G( Γ σ ). Note that G(S σ µ) is compact for µ V σ since G is continuous (smooth). Consequently, dist(g(s σ µ), G( Γ σ )) > 0, µ V σ.

10 10 BO GUAN For λ, µ Γ, let h h(µ, λ) = (µ λ) ν λ denote the (signed) distance from µ to the tangent plane at λ of Γ f(λ). Thus h is smooth in both µ and λ with gradients: More explicitly, since ν λ = Df/ Df, D µ h = ν λ, D λ h = ν λ + (µ λ) Dν λ. h = f i λ i Df + µ j λ j Df We also note that S σ µ = {λ Γ σ : (µ λ) ν λ = 0}. ( δ jk f jf k Df 2 ) f ik. Lemma 2.5. Let µ C + σ \ Γ σ. Then Γ σ \ S σ µ consists of two components: Γ σ \ S σ µ = {λ Γ σ : (µ λ) ν λ > 0} {λ Γ σ : (µ λ) ν λ < 0}. Proof. These two sets are obviously disconnected in Γ σ since (µ λ) ν λ is continuous in λ Γ σ (as well as in µ). Lemma 2.6. Let µ V σ. Then H µ (R) := min λ Γ σ B R (0) (µ λ) ν λ is positive and nondecreasing in R for R > max λ S σ µ λ. Proof. Let R > R > max λ S σ µ λ and let λ Γ σ B R (0) satisfy H µ (R ) = (µ λ ) ν λ. By Lemma 2.5 we see that H µ (R) > 0 since Γ σ B R (0) is compact. Let η S σ µ such that (η λ ) ν λ = min(ζ λ ) ν λ ζ Sµ σ > 0. Consider the Gauss map G : Γ σ S n. Note that G(S σ µ) is contained in the interior of G( Γ σ B R (0)) while G(λ ) is not. (G(λ ) may belong to G( Γ σ B R (0)).) So there exists λ 0 G( Γ σ B R (0)) such that G(λ 0 ) = G(λ ) cos θ + G(η) sin θ for some θ [0, π/2).

11 We have ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 11 (µ λ 0 ) ν λ0 = (µ λ ) ν λ cos θ + (λ λ 0 ) ν λ cos θ + (µ η) ν η sin θ + (η λ 0 ) ν η sin θ (µ λ ) ν λ cos θ since (λ λ 0 ) ν λ 0, (η λ 0 ) ν η 0 by the convexity of Γ σ, and (µ η) ν η = 0. Therefore H µ (R) H µ (R ). For µ V σ we define Γ σ (µ) := {λ Γ σ : (µ λ) ν λ 0}, C σ (µ) := {tλ + (1 t)µ : λ Γ σ (µ)}, 0 t 1}. Note that Γ σ (µ) is compact. Therefore its Gauss map image is also compact and contained in the interior of G( Γ σ ). Geometrically, C σ (µ) is the region bounded by Γ σ (µ) and the tangent cone to Γ σ with vertex at µ. Lemma 2.7. Let µ V σ. Then C σ (µ) V σ and C σ (µ ) C σ (µ) for µ C σ (µ). Proof. Let t [0, 1], λ Γ σ (µ) and denote µ t = tλ + (1 t)µ. Then (µ t ζ) ν ζ = (1 t)(µ ζ) ν ζ + t(λ ζ) ν ζ > t(λ ζ) ν ζ 0, ζ Γ σ \ Γ σ (µ) since (µ ζ) ν ζ > 0 and Γ σ is convex. This shows Sµ σ t Γ σ (µ). So Sµ σ t and µ t V σ. Clearly C σ (µ t ) C σ (µ). is compact Lemma 2.8. The cone C + σ is open. Proof. Let µ V σ so Γ σ (µ) is compact. Let R > 0 such that Γ σ (µ) Γ σ B R where B R is the (open) ball of radius R centered at the origin in R n. Then α := 1 2 min (µ ζ) ν ζ > 0. n ζ Γ σ B R Therefore, (µ α1 λ) ν λ α1 ν λ + min ζ Γ σ B R (µ ζ) ν ζ nα > 0, λ Γ σ B R where 1 = (1,..., 1). This proves that Sµ α1 σ Γ σ B R and hence µ α1 V σ. On the other hand, (µ α1 λ) ν λ α1 ν λ = α fi (λ) < 0, λ Γ σ (µ). Df(λ) So C σ (µ) C σ (µ α1) V σ. Clearly C σ (µ α1) contains a ball centered at µ.

12 12 BO GUAN Lemma 2.9. Let K be a compact subset of V σ = C + σ \ Γ σ. Then sup µ K max λ <. λ Sµ σ Proof. Suppose this is not true. Then for each integer k 1 there exists µ k K and λ k Sµ σ k with λ k k. By the compactness of K we may assume µ k µ K as k. Thus lim (µ λ k) ν λk = lim (µ µ k ) ν λk = 0. k k On the other hand, by Lemma 2.6 there exists δ > 0 such that This is a contradiction. (µ λ k ) ν λk δ, k > max λ. λ Sµ σ Remark When K Γ σ there is a more geometric proof which does not use Lemma 2.6. Indeed, if µ k Γ σ for each k and µ k µ K as k, then µ Γ σ and ν µk ν µ as k. We may assume in addition that E k := (λ k µ k )/ λ k µ k E S n. Now for any t > 0, since µ k max ζ K ζ < for all k, µ k + te k Sµ σ k when k is large so that λ k µ k > t, and lim (µ k + te k ) = µ + te T µ Γ σ. k Therefore µ + te Sµ σ for all t > 0. This contradicts the compactness of Sµ. σ Let µ Γ σ and λ Γ σ. By the convexity of Γ σ, the open segment (µ, λ) := {tµ + (1 t)λ : 0 < t < 1} is completely contained in either Γ σ or Γ σ by condition (1.3). Therefore, f(tµ + (1 t)λ) > σ, 0 < t < 1 unless (µ, λ) Γ σ which is equivalent to S σ µ = S σ λ. For R > µ, let Θ R (µ) := inf max f(tµ + (1 t)λ) σ 0. λ B R (0) Γ σ 0 t 1 Clearly Θ R (µ) = 0 if and only if (µ, λ) Γ σ for some λ B R (0) Γ σ, since the set B R (0) Γ σ is compact. Lemma For µ Γ σ, Θ R (µ) is nondecreasing in R. Moreover, if Θ R0 (µ) > 0 for some R 0 µ then Θ R > Θ R for all R > R R 0.

13 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 13 Proof. Write Θ R = Θ R (µ) when there is no possible confusion. Suppose Θ R0 (µ) > 0 for some R 0 µ. Let R > R R 0 and assume λ R B R (0) Γ σ such that Θ R = max f(tµ + (1 t)λ R ) σ. 0 t 1 Let P be the (two dimensional) plane through µ, λ R and the origin of R n. There is a point λ R B R (0) which lies on the curve P Γ σ. Note that µ, λ R and λ R are not on a straight line, for (µ, λ R ) can not be part of (µ, λ R ) since Θ R0 > 0 and Γ σ is convex. We see that max 0 t 1 f(tµ + (1 t)λ R) σ < Θ R by condition (1.3). This proves Θ R < Θ R. Lemma Let µ Γ σ C + σ. Then Θ R (µ) > 0 for any R > max λ S σ µ λ. Proof. This follows from Lemma 2.2. Corollary Let µ Γ σ. The following are equivalent: (a) µ / C + σ ; (b) µ C + σ ; (c) S σ µ Γ σ C + σ ; (d) Γ σ C + σ contains a ray through µ; (e) Θ R (µ) = 0 for all R > µ ; (f) G(µ) G( Γ σ ). Proof. By definition Γ σ C σ +. So (a) and (b) are equivalent. Since Sλ σ = Sσ µ for all λ Sµ, σ we have Sµ σ C σ + if µ C σ +. This shows (b) and (c) are equivalent. Next, Sµ σ is convex and therefore contains a ray through each point in it when it is noncompact. We see (c) implies (d) which clearly implies (e). By Lemma 2.12, (e) in turn implies (a). Finally (d) also implies (f) since T µ Γ σ is tangential to Γ σ and therefore G = G(µ) on the ray. It follows from Corollary 2.4 that (f) implies (a). Lemma For µ Γ σ C σ + let N µ = 2 max λ S σ µ λ. Then for any λ Γ σ, when λ N µ, (2.1) fi (λ)(µ i λ i ) Θ Nµ (µ) > 0.

14 14 BO GUAN Proof. By the concavity of f, fi (λ)(µ i λ i ) max 0 t 1 f(tµ + (1 t)λ) σ 0, µ, λ Γσ. So Lemma 2.14 follows from Lemma 2.11 and Lemma Lemma Let K be a compact subset of Γ σ C + σ. Define N K := sup N µ, Θ K = inf Θ N K (µ). µ K µ K Then for any λ Γ σ, when λ N K, (2.2) fi (λ)(µ i λ i ) Θ K > 0, µ K. Proof. From Lemma 2.9 we see that N K < and consequently, Θ K := inf Θ N K (µ) = inf inf max f(tµ + (1 t)λ) σ > 0 µ K µ K λ B NK (0) Γ σ 0 t 1 by the continuity of f. Now the first inequality in (2.2) follows from Lemma The following results will play key roles in our proof of both global and boundary second order estimates in the next two sections. Theorem Let µ C σ + and 0 < ε < 1dist(µ, 2 C+ σ ). There exist positive constants θ µ, R µ such that for any λ Γ σ, when λ R µ, (2.3) fi (λ)(µ i λ i ) θ µ + ε f i (λ). Proof. We first note that µ ε := µ ε1 C + σ. So if µ ε Γ σ then Θ R0 (µ ε ) > 0 for some R 0 > 0. By Lemma 2.11 fi (λ)(µ i ε λ i ) Θ R0 (µ ε ), if λ R 0. So (2.3) holds. We now consider the case µ ε C σ + \ Γ σ. Recall that C σ (µ ε ) denotes the tangent cone to Γ σ with vertex µ ε and Γ σ (µ ε ) is the compact component of Γ σ bounded by Sµ σ = ε Γσ C σ (µ ε ). For any λ Γ σ \ Γ σ (µ ε ), The segment (µ ε, λ) goes through Γ σ (µ ε ) at a point λ ε. By the concavity of f and Lemma 2.15 applied to K = Γ σ (µ ε ), we obtain fi (λ)(µ i ε λ i ) f i (λ)(λ ε i λ i ) Θ K > 0 when λ N K. This proves (2.3) for R µ = max{r 0, N K } and θ µ = min{θ R0, Θ K }.

15 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 15 Theorem Let K be a compact subset of C σ + and 0 < ε < 1dist (K, 2 C+ σ ). There exist constants θ K, R K > 0 such that for any λ Γ σ, when λ R K, (2.4) fi (λ)(µ i λ i ) θ K + ε f i (λ), µ K. Furthermore, for any closed interval [a, b] ( sup Γ f, sup Γ f ), θ K and R K can be chosen so that (2.4) holds uniformly in σ [a, b]. Proof. Let K 1 = {µ K : µ 3ε/2 Γ σ }, K 2 = {µ K : µ 3ε/2 V σ }. By the concavity of f and compactness on K 1 we have (2.5) f i (λ)(µ i ε λ i ) f(µ ε ) f(λ) min ζ K 1 f(ζ ε ) σ > 0, µ K 1, λ Γ σ. Next, let K := {µ 3ε/2 : µ K 2 }, W := ζ K Γ σ (ζ). By Lemma 2.9, R 0 := sup ζ <. ζ W So W is a compact subset of C σ + Γ σ. Applying Lemma 2.15 to W combined with the concavity of f we obtain for any λ Γ σ with λ 2R 0, (2.6) fi (λ)(µ i 3ε ) 2 λ i min fi (λ)(ζ i λ i ) Θ ζ W W, µ K 2 since the segment (µ 3ε/2, λ) must intersect W. Now (2.4) follows from (2.5) and (2.6). Finally, we note that θ K and R K can be chosen so that they continuously depends on σ. This can be seen from that fact that the hypersurfaces { Γ σ : σ [a, b]} form a smooth foliation of the region bounded by Γ a and Γ b, which also implies that the distance function dist(µ, C σ + ) also depends continuously on σ (as well as on µ). Theorem 2.17 can not be used directly in the proofs of (1.8) and (1.10) in the next two sections. So we modify it as follows. Let A be the set of n by n symmetric matrices A = {A ij } with eigenvalues λ[a] Γ. Define the function F on A by F (A) f(λ[a]). Throughout this paper we shall use the notation F ij (A) = F A ij (A), F ij,kl (A) = 2 F A ij A kl (A).

16 16 BO GUAN The matrix {F ij } has eigenvalues f 1,..., f n and is positive definite by assumption (1.3), while (1.4) implies that F is a concave function of A ij [5]. Moreover, when A is diagonal so is {F ij (A)}, and the following identities hold F ij (A)A ij = f i λ i, F ij (A)A ik A kj = f i λ 2 i. Theorem 2.17 can be rewritten as follows. Theorem Let [a, b] ( sup Γ f, sup Γ f ). For any σ [a, b] and K A such that λ[k] := {λ(a) : A K} is a compact subset of C σ +, there exist positive constants θ K, R K depending only on d := dist (λ[k], C σ + ) and sup A K λ(a) (continuously), such that for any B A with λ(b) Γ σ, when λ(b) R K, (2.7) F ij (B)(A ij B ij ) θ K + d 2 F ii (B). Namely, (2.7) holds uniformly in σ [a, b]. Proof. This follows immediately from Theorem 2.17 and Lemma 6.2 in [5]. Indeed, by Lemma 6.2 in [5] we see that F ij (B)A ij min f i (λ(b))λ π(i) (A) π where the minimum is taken for all permutations π of {1,..., n}. Note also that λ(a εi) = λ(a) ε1. Theorem 2.18 now follows from Theorem We next present some results which are taken from [19] with minor modifications (and simplification of proofs). For f = σ 1 k k and f = (σ k /σ l ) 1 k l, 1 l < k n they were proved earlier by Ivochkina [24] and Lin-Trudinger [29], respectively. We shall need these results when we derive the boundary estimate (1.10) in Section 4. Proposition Let A = {A ij } A and set F ij = F ij (A). There is an index r such that (2.8) F ij A il A lj 1 f i λ 2 i. 2 l<n i r

17 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 17 Proof. Let B = {b ij } be an orthogonal matrix that simultaneously diagonalizes {F ij } and {A ij }: F ij b li b kj = f k δ kl, A ij b li b kj = λ k δ kl. Then (2.9) l<n F ij A il A lj = l<n Suppose for some r that b 2 nr > 1 2 Therefore This proves (2.8). i r f i λ 2 i b 2 li = f i λ 2 i (1 b 2 ni). (otherwise we are done). Then b 2 ni < 1 2. i r l<n F ij A il A lj i r f i λ 2 i (1 b 2 ni) > 1 f i λ 2 i. 2 Lemma Suppose f satisfies (1.3), (1.4) and f i λ i K 0 for some constant K 0 0. Then (2.10) f i λ 2 i 1 fi λ 2 i nk2 0 n + 1 n + 1 min 1, if λ r < 0. 1 i n f i Proof. Suppose λ 1 λ n and λ r < 0. By the concavity condition (1.4) we have f n f i > 0 for all i and in particular f n λ 2 n f r λ 2 r. By (1.14), K 0 + i n i r f i λ i f n λ n = f n λ n. It follows from Schwarz inequality that, fnλ 2 2 n 1 + ɛ K0 2 + (1 + ɛ) f i f i λ 2 i ɛ i n i n 1 + ɛ K0 2 + (1 + ɛ)(n 1)f n f i λ 2 i ɛ i n = nk0 2 + nf n f i λ 2 i i n if we take ɛ = 1. Therefore, n 1 f i λ 2 i f i λ 2 i 1 n + 1 i r i n completing the proof. i n f i λ 2 i + 1 n + 1 f nλ 2 n K2 0 f n

18 18 BO GUAN Corollary Suppose f satisfies the assumptions of Lemma 2.20 For any index r and ɛ > 0, (2.11) fi λ i ɛ f i λ 2 i + C fi + Q(r) ɛ i r where Q(r) = f(λ) f(1) if λ r 0, and Proof. By the concavity of f, Therefore, if λ r 0 then Q(r) = ɛnk 2 0 min 1 i n 1 f i, if λ r < 0. f(1) f(λ) f i (1 λ i ). f r λ r f(λ) f(1) + f i + λ i <0 f i λ i ɛ f i λ 2 i + C fi + f(λ) f(1). 2 ɛ λ i <0 Suppose λ r < 0. By Lemma 2.20 we have fi λ i ɛ fi λ 2 i + n + 1 n + 1 4ɛ ɛ i r fi f i λ 2 i + C ɛ fi + ɛnk 2 0 min 1 i n 1 f i. This proves (2.11). We end this section by noting the fact that Γ σ is strictly convex, and therefore (1.16) holds for f = σ 1/k k, k 2 and f = (σ k /σ l ) 1 k l, 1 l < k n. Consequently Corollary 1.13 follows from Theorem Lemma For f = σ 1/k k, k 2 or f = (σ k /σ l ) 1 k l, 1 l < k n, Γ σ = {f = σ} is strictly convex and, in particular, Γ σ C + σ =, σ > 0. Proof. This was probably noticed before and may be seen in many ways; here we note that Γ σ does not contain any line segment. Consider f = (σ k /σ l ) 1 k l, 0 l < k n (and k 2) where σ 0 = 1. Suppose µ+at Γ σ, i.e. f(µ+at) σ, for some µ Γ k, a R n, and t ( ε, ε). Then σ k (µ + at) σ k l σ l (µ + at) 0 for all t R n since it is a polynomial. This is impossible unless a = 0 as Γ k (k 2) does not contains whole straight lines and f = 0 on Γ k.

19 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 19 From the above proof one can draw similar conclusions for analytic functions, and in particular if f is a hyperbolic polynomial (see [5]). Lemma Suppose f satisfies (1.3) and (1.4) in Γ = Γ n. Then Γ σ C σ + any σ (sup Γ f, sup Γ f). for Proof. It is easy to see that Γ σ does not contain any (straight) rays. Indeed, suppose µ + at Γ σ, i.e. f(µ + at) σ, for some µ Γ n, a = (a 1,..., a n ) R n, and all t 0. Then d dt f(µ + at) t=0 = f i (µ)a i = 0. This implies a / Γ n and therefore a = 0. For otherwise the ray γ(t) = µ + at, t > 0 would intersect Γ n which contradicts the assumption f(µ + at) σ > sup Γ f. 3. Global bounds for the second derivatives The goal of this section is to prove (1.8) under the hypotheses (1.3)-(1.5) and (1.7). We start with a brief explanation of our notation and basic formulas needed. Throughout the paper denotes the Levi-Civita connection of (M n, g). The curvature tensor is defined by R(X, Y )Z = X Y Z + Y X Z + [X,Y ] Z. Let e 1,..., e n be local frames on M n and denote g ij = g(e i, e j ), {g ij } = {g ij } 1, and i = ei, ij = i j i e j, etc. Define R ijkl, R i jkl and Γk ij respectively by R ijkl = R(e k, e l )e j, e i, R i jkl = g im R mjkl, i e j = Γ k ije k. For a differentiable function v defined on M n, we identify v with the gradient of v, and 2 v denotes the Hessian of v which is given by ij v = i ( j v) Γ k ij k v. Recall that ij v = ji v and (3.1) ijk v jik v = R l kij l v, (3.2) ijkl v ikjl v = R m ljk im v + i R m ljk m v, (3.3) ijkl v jikl v = R m kij ml v + R m lij km v. From (3.2) and (3.3) we obtain (3.4) ijkl v klij v = R m ljk im v + i R m ljk m v + R m lik jm v + R m jik lm v + R m jil km v + k R m jil m v.

20 20 BO GUAN Let u C 4 (M) be an admissible solution of equation (1.1). Under orthonormal local frames e 1,..., e n, equation (1.1) is expressed in the form (3.5) F (U ij ) := f(λ[u ij ]) = ψ where U ij = ij u + χ ij. For simplicity, we shall still write equation (1.1) in the form (3.5) even if e 1,..., e n are not necessarily orthonormal, although more precisely it should be F (γ ik U kl γ lj ) = ψ where {γ ij } is the square root of {g ij }: γ ik γ kj = g ij ; as long as we use covariant derivatives whenever we differentiate the equation it will make no difference. We now begin the proof of (1.8). Let W = max x M max ( ξξu + χ(ξ, ξ))e η ξ T xm n, ξ =1 where η is a function to be determined. Suppose W > 0 and is achieved at an interior point x 0 M for some unit vector ξ T x0 M n. Choose smooth orthonormal local frames e 1,..., e n about x 0 such that e 1 (x 0 ) = ξ and {U ij (x 0 )} is diagonal. We may also assume that i e j = 0 and therefore Γ k ij = 0 at x 0 for all 1 i, j, k n. At the point x 0 where the function log U 11 + η (defined near x 0 ) attains its maximum, we have for i = 1,..., n, (3.6) i U 11 U 11 + i η = 0, (3.7) ii U 11 U 11 ( i U 11 U 11 ) 2 + ii η 0. Here we wish to add some explanations which might be helpful to the reader. First we note that U 1j (x 0 ) = 0 for j 2 so {U ij (x 0 )} can be diagonalized. To see this let e θ = e 1 cos θ + e j sin θ. Then U e θ e θ(x 0) = U 11 cos 2 θ + 2U 1j sin θ cos θ + U jj sin 2 θ has a maximum at θ = 0. Therefore, d dθ U e θ e θ(x 0) = 0. θ=0 This gives U 1j (x 0 ) = 0. Next, at x 0 we have (3.8) i (U 11 ) = i U 11,

21 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 21 that is e i (U 11 ) = i U 11 3 u(e 1, e 1, e i ) + χ(e 1, e 1, e i ), and (3.9) ij (U 11 ) = ij U 11. One can see (3.8) immediately if we assume Γ k ij = 0 at x 0 for all 1 i, j, k n. In general, we have i (U 11 ) = i U Γ k i1u 1k = i U Γ 1 i1u 11 as U 1k (x 0 ) = 0. On the other hand, since e 1,... e n are orthonormal, and Thus g( k e i, e j ) + g(e i, k e j ) = 0 g( i e 1, j e 1 ) + g(e 1, i j e 1 ) = 0. (3.10) Γ j ki + Γi kj = 0 and Γ k i1γ k j1 + i (Γ 1 j1) + Γ k j1γ 1 ik = 0. This gives Γ 1 i1 = 0 and i (Γ 1 j1) = 0. So we have (3.8). For (3.9) we calculate directly, ij (U 11 ) = i ( j (U 11 )) Γ k ij k (U 11 ) = i ( j U Γ k j1u 1k ) Γ k ij k U 11 = ij U 11 + Γ k ij k U Γ k i1 j U 1k + 2 i (Γ k j1)u 1k + 2Γ k j1 i U 1k + 2Γ k j1γ l i1u lk + 2Γ k j1γ l iku 1l Γ k ij k U 11 = ij U Γ k i1 j U 1k + 2Γ k j1 i U 1k + 2Γ k i1γ k j1u kk 2Γ k i1γ k j1u 11 by (3.10) and i (Γ 1 j1) = 0. Therefore we have (3.9) if Γ k ij = 0 at x 0. We now continue our proof of (1.8). Differentiating equation (3.5) twice, we obtain at x 0, (3.11) F ij k U ij = k ψ, for all k, (3.12) F ii 11 U ii + F ij,kl 1 U ij 1 U kl = 11 ψ. Here and throughout rest of the paper, F ij = F ij ({U ij }). By (3.4), (3.13) F ii ii U 11 F ii 11 U ii + 2F ii R 1i1i ( 11 u ii u) C F ii F ii 11 U ii C(1 + U 11 ) F ii.

22 22 BO GUAN Here we note that C depends on the gradient bound u C 0 ( M). From (3.7), (3.12) and (3.13) we derive (3.14) U 11 F ii ii η E 11 ψ + C(1 + U 11 ) F ii where E F ij,kl 1 U ij 1 U kl + 1 U 11 F ii ( i U 11 ) 2. To estimate E let 0 < s < 1 (to be chosen) and J = {i : U ii su 11 }, K = {i > 1 : U ii > su 11 }. It was shown by Andrews [1] and Gerhardt [8] that F ij,kl 1 U ij 1 U kl i j F ii F jj U jj U ii ( 1 U ij ) 2. Therefore, F ij,kl 1 U ij 1 U kl 2 i 2 F ii F 11 U 11 U ii ( 1 U i1 ) 2 (3.15) 2 F ii F 11 ( 1 U i1 ) 2 U 11 U ii i K 2 (F ii F 11 )( 1 U i1 ) 2 (1 + s)u 11 i K 2(1 s) (F ii F 11 )[( i U 11 ) 2 C/s]. (1 + s)u 11 i K We now fix s 1/3 and hence 2(1 s) s From (3.15) and (3.6) it follows that (3.16) Let E 1 F ii ( i U 11 ) 2 + C F ii + U 11 U 11 i J i K CF 11 U 11 ( i U 11 ) 2 U 11 F ii ( i η) 2 + C F ii + CU 11 F 11 ( i η) 2. U 11 i J i/ J η = φ( u 2 ) + a(u u) i/ J

23 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 23 where φ is a positive function, φ > 0, and a is a positive constant. We calculate Therefore, (3.17) (3.18) and by (3.11), (3.19) i η = 2φ k u ik u + a i (u u) = 2φ (U ii i u χ ik k u) + a i (u u), ii η = 2φ ( ik u ik u + k u iik u) + 2φ ( k u ik u) 2 + a ii (u u). F ii ( i η) 2 8(φ ) 2 F ii ( k u ik u) 2 + Ca 2 F ii, i J i J i J ( i η) 2 C(φ ) 2 U C(φ ) 2 + Ca 2 i/ J F ii ii η φ F ii U 2 ii + 2φ F ii ( k u ik u) 2 + af ii ii (u u) Cφ ( 1 + F ii ). Let φ(t) = b(1 + t) 2 ; we may assume φ 4(φ ) 2 = 2b(1 8φ) 0 in any fixed interval [0, C 1 ] by requiring b > 0 sufficiently small. Combining (3.14), (3.16), (3.17), (3.18) and (3.19), we obtain φ F ii Uii 2 + af ii ii (u u) Ca 2 F ii + C((φ ) 2 U a 2 )F 11 i J (3.20) 11ψ + C (1 + ) F ii. U 11 Suppose U 11 (x 0 ) > R sufficiently large and apply Theorem 2.18 to A = { ij u+χ ij } and B = {U ij } at x 0. We see that F ii ii (u u) = F ii [( ii u + χ ii ) U ii ] θ (1 + ) F ii. Plug this into (3.20) and fix a sufficiently large. We derive (3.21) φ F ii U 2 ii Ca 2 i J F ii + C((φ ) 2 U a 2 )F 11. Note that (3.22) F ii U 2 ii F 11 U i J F ii Uii 2 F 11 U s 2 U11 2 F ii. Fixing b sufficiently small we obtain from (3.21) a bound U 11 Ca/ b. This implies (1.8), and (1.9) when M is closed. i J

24 24 BO GUAN 4. Boundary estimates In this section we establish the boundary estimate (1.10) under the assumptions of Theorem 1.5. Throughout this section we assume the function ϕ C 4 ( M) is extended to a C 4 function on M, still denoted ϕ. For a point x 0 on M, we shall choose smooth orthonormal local frames e 1,..., e n around x 0 such that when restricted to M, e n is normal to M. Let ρ(x) denote the distance from x to x 0, ρ(x) dist M n(x, x 0 ), and M δ = {x M : ρ(x) < δ}. Since M is smooth we may assume the distance function to M d(x) dist(x, M) is smooth in M δ0 for fixed δ 0 > 0 sufficiently small (depending only on the curvature of M and the principal curvatures of M.) Since ij ρ 2 (x 0 ) = 2δ ij, we may assume ρ is smooth in M δ0 and (4.1) {δ ij } { ij ρ 2 } 3{δ ij } in M δ0. The following lemma which crucially depends on Theorem 2.18 plays key roles in our boundary estimates. Lemma 4.1. There exist some uniform positive constants t, δ, ε sufficiently small and N sufficiently large such that the function (4.2) v = (u u) + td Nd2 2 satisfies v 0 on M δ and (4.3) F ij ij v ε (1 + ) F ii in M δ. Proof. We note that to ensure v 0 in M δ we may require δ 2t/N after t, N being fixed. Obviously, (4.4) F ij ij v = F ij ij (u u) + (t Nd)F ij ij d NF ij i d j d C 1 (t + Nd) F ii + F ij ij (u u) NF ij i d j d. Fix ε > 0 sufficiently small and R R A so that Theorem 2.18 holds for A = { ij u+χ ij } and B = {U ij } at every point in M δ0. Let λ = λ[{u ij }] be the eigenvalues of {U ij }. At a fixed point in M δ we consider two cases: (a) λ R; and (b) λ > R.

25 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 25 In case (a) there are uniform bounds (depending on R) 0 < c 1 f i (λ) C 1 and therefore F ij i d j d c 1 since d 1. We may fix N large enough so that (4.3) holds for any t, ε (0, 1], as long as δ is sufficiently small. In case (b) by Theorem 2.18 and (4.4) we may further require t and δ small so that (4.3) holds for some different (smaller) ε > 0. We now start the proof of (1.10). Consider a point x 0 M. Since u u = 0 on M we have (4.5) αβ (u u) = n (u u)π(e α, e β ), 1 α, β < n on M where Π denotes the second fundamental form of M. Therefore, (4.6) αβ u C, 1 α, β < n on M. To estimate the mixed tangential-normal and pure normal second derivatives we note the following formula ij ( k u) = ijk u + Γ l ik jl u + Γ l jk il u + ij e k u. By (3.11), therefore, F ij ij k (u ϕ) 2F ij Γ l ik jl u + C (1 + ) F ii (4.7) C (1 + f i λ i + ) f i. Let (4.8) Ψ = A 1 v + A 2 ρ 2 A 3 β (u ϕ) 2. By (4.7) we have (4.9) β<n F ij ij β (u ϕ) 2 = 2F ij β (u ϕ) ij β (u ϕ) + 2F ij i β (u ϕ) j β (u ϕ) F ij U iβ U jβ C (1 + f i λ i + ) f i. For fixed 1 α < n, by Lemma 4.1, Proposition 2.19 and Corollary 2.21 we see that (4.10) F ij ij (Ψ ± α (u ϕ)) 0, in M δ

26 26 BO GUAN and Ψ ± α (u ϕ) 0 on M δ when A 1 A 2 A 3 1. principle we derive Ψ ± α (u ϕ) 0 in M δ and therefore By the maximum (4.11) nα u(x 0 ) n Ψ(x 0 ) + nα ϕ(x 0 ) C, α < n. It remains to derive (4.12) nn u(x 0 ) C. We show this by proving that there are uniform constants c 0, R 0 such that for all R > R 0, (λ [{U αβ (x 0 )}], R) Γ and f(λ [{U αβ (x 0 )}], R) ψ(x 0 ) + c 0 where λ [{U αβ }] = (λ 1,, λ n 1) denotes the eigenvalues of the (n 1) (n 1) matrix {U αβ } (1 α, β n 1). Suppose we have found such c 0 and R 0. By Lemma 1.2 of [5], from estimates (4.6) and (4.11) we can find R 1 R 0 such that if U nn (x 0 ) > R 1, f(λ[{u ij (x 0 )}]) f(λ [{U αβ (x 0 )}], U nn (x 0 )) c 0 2. By equation (1.1) this gives a desired bound U nn (x 0 ) R 1 for otherwise, we would have a controdiction: f(λ[{u ij (x 0 )}]) ψ(x 0 ) + c 0 2. For a symmetric (n 1) by (n 1) matrix {r αβ } with (λ [{r αβ }], R) Γ when R > 0 is sufficiently large, define F [r αβ ] lim inf R Following an idea of Trudinger [33] we consider Note that F is concave, and that m min x M f(λ [{r αβ }], R). F [U αβ (x)] ψ(x). c inf M ( F [U αβ ] F [U ij ]) > 0. We wish to show m c 0 for some uniform constant c 0 > 0. Suppose that m is achieved at a point x 0 M. Choose local orthonormal frames around x 0 as before. By the concavityof F αβ, there exists a symmetric matrix { F 0 } such that (4.13) F αβ 0 (r αβ U αβ (x 0 )) F [r αβ ] F [U αβ (x 0 )],

27 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 27 for any symmetric matrix {r αβ } with (λ [{r αβ }], R) Γ; if F is C 1 at {U αβ (x 0 )} then In particular, F αβ 0 = F r αβ [U αβ (x 0 )]. (4.14) F αβ 0 U αβ ψ By (4.5) we have on M, F αβ 0 U αβ (x 0 ) + ψ(x 0 ) F [U αβ ] ψ m 0 on M. (4.15) U αβ = U αβ n (u u)σ αβ where σ αβ = α e β, e n ; note that σ αβ = Π(e α, e β ) on M. It follows that Consequently, if n (u u) F αβ 0 σ αβ (x 0 ) = then m c/2 and we are done. Suppose now that n (u u)(x 0 ) n (u u)(x 0 ) F αβ 0 (U αβ (x 0 ) U αβ (x 0 )) F [U αβ (x 0 )] F [U αβ (x 0 )] = F [U αβ (x 0 )] ψ(x 0 ) m c m. F αβ 0 σ αβ (x 0 ) c/2 F αβ 0 σ αβ (x 0 ) > c 2, and let η F αβ 0 σ αβ. Note that (4.16) η(x 0 ) c/2 n (u u)(x 0 ) 2ɛ 1 c for some uniform ɛ 1 > 0. We may assume η ɛ 1 c on M δ by requiring δ small. Define in M δ, Φ = n (u ϕ) + 1 αβ F 0 ( αβ ϕ + χ αβ U αβ (x 0 )) ψ ψ(x 0) η η n (u ϕ) + Q. We have Φ(x 0 ) = 0 and Φ 0 on M near x 0 by (4.14) since αβ u = αβ ϕ n (u ϕ)σ αβ on M, while by (4.7), (4.17) F ij ij Φ F ij ij n u + C F ii C (1 + f i λ i + f i ).

28 28 BO GUAN Consider the function Ψ defined in (4.8). Applying Lemma 4.1, Proposition 2.19 and Corollary 2.21 as before for A 1 A 2 A 3 1 we derive Ψ + Φ 0 on M δ and (4.18) F ij ij (Ψ + Φ) 0 in M δ. By the maximum principle, Ψ + Φ 0 in M δ. Thus Φ n (x 0 ) n Ψ(x 0 ) C. This gives nn u(x 0 ) C. So we have an a priori upper bound for all eigenvalues of {U ij (x 0 )}. Consequently, λ[{u ij (x 0 )}] is contained in a compact subset of Γ by (1.5), and therefore m m R f(λ [U αβ (x 0 )], R) ψ(x 0 ) > 0 when R is sufficiently large. This completes the proof of (1.10). 5. The gradient estimates and proof of Theorem 1.10 By Theorems and Evans-Krylov theorem, one only needs to derive a prior C 1 estimates in order to prove Theorem 1.10 using the continuity method. It seems an interesting question whether one can prove gradient estimates under assumption (1.7). We wish to come back to the problem in future work. Here we only list some conditions for gradient estimates that were more or less known to Li [28] and Urbas [35]. Proposition 5.1. Let u C 3 ( M) be an admissible solution of equation (1.1) where ψ C 1 ( M). Suppose f satisfies (1.3)-(1.5). Then (5.1) max M u C ( 1 + max u ) M where C depends on u C 0 ( M), under any of the following additional assumptions: (i) Γ = Γ n ; (ii ) (1.7) and that (M, g) has nonnegative sectional curvature; (iii ) (1.14), (1.15) and (1.17) for λ sufficiently large. Proof. Case (i): Γ = Γ n. For fixed A > 0 suppose Au + u 2 has a maximum at an interior point x 0 M. Then A i u + 2 k u ki u = k u(aδ ki + ki u) = 0 at x 0 for all 1 i n. This implies u(x 0 ) = 0 when A is sufficiently large. Therefore, ( ) sup u 2 A sup u inf u + sup u 2. M M M M Case (iii ) was proved by Urbas [35] under the additional assumption (5.2) fi (λ) δ σ, λ Γ σ, inf ψ σ sup ψ,

29 ESTIMATES FOR FULLY NONLINEAR ELLIPTIC EQUATIONS 29 which is in fact implied by (1.14). Indeed, by the concavity of f and (1.14)-(1.15), A f λi (λ) f λi (λ)λ i +f(a1) f(λ) f(a1) σ f(a1) sup f(λ( 2 u+χ)) M for any λ Γ, f(λ) = σ. Fixing A sufficiently large gives (5.2). Case (ii ) is a slight improvement of the gradient estimates derived by Li [28]. So we only outline a modification of the proof in [28]. Suppose u 2 e φ achieves a maximum at an interior point x 0 M. Then at x 0, 2 k u ik u u 2 + i φ = 0, 2F ij ( k u jik u + ik u jk u) + u 2 F ij ( ij φ i φ j φ) 0. Following [28] we use the nonnegative sectional curvature condition to derive (5.3) u F ij ( ij φ i φ j φ) C F ii + C Now let φ = A(1 + u u + sup(u u)) 2 and fix A > 0 sufficiently small. From (5.3), 2AF ij ij (u u) + 2A(1 2φ)F ij i (u u) j (u u) C F ii + C u u. By (1.7) and Theorem 2.18 we derive a bound u(x 0 ) C if λ[ 2 u + χ](x 0 ) R for R sufficiently large. Suppose λ[ 2 u + χ](x 0 ) R. By (1.3) and (1.5) there exists C 1 > 0 depending on R such that at x 0, Then g 1 C 1 {F ij } C 1 g 1. 2A(1 2A)C 1 1 (u u) 2 C u. We derive a bound for u(x 0 ) again. By the maximum principle we have u u h where h C 2 ( M) is the solution of h + trχ = 0 in M with h = ϕ on M. This gives bounds for u C 0 ( M) and u on M. The proof of Theorem 1.10 using the continuity method is standard and therefore omitted here.

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