Nonlinear Degenerate Elliptic Differential Equations

Pengfei Guan

Elliptic differential equations play important roles in mathematics, physics and other fields of sciences. One often encounters such equations with nonlinearility and degeneracy. There is a general theory for nonlinear elliptic equations, while study of degenerate linear elliptic equations is also satisfactory. But not much is known about nonlinear degenerate elliptic equations. There are many basic questions need to be answered. Here we will discuss some recent progress in nonlinear degenerate elliptic equations. We will restrict ourselves in the existence and regularity of the following three topics: (i) nonlinear oblique derivative problem, (ii) degenerate quasilinear elliptic equations, (iii) degenerate Monge-Ampere equations.

In the treatments of these problems, we employ some microlocal analysis techniques. Namely, the calculus of pseudodifferential operators, a standard tool in study of degenerate linear equations. To adapt them for the nonlinear degenerate cases, we have to deal with pseudodifferential operators with limited smoothness. In most cases, the crucial step is to control the commutators of these type of operators.

1. Nonlinear Oblique Derivative Problem

The oblique derivative problem is the following boundary value problem:

\{ \begin{matrix}P (D^{2}u, Du, u, x) = 0 & in Ω \\ ℓ^{→} u = g & on ∂Ω \end{matrix} (1.1)

where P is an elliptic differential operator on R^{n}, Ω is a bounded domain in R^{n} and ℓ^{→} is a unit vector field on R^{n}.

The problem was first proposed by Poincaré in [22] at the beginning of this century. If ℓ^{→} is everywhere transversal to the boundary, then the problem is an elliptic boundary value problem. When ℓ^{→} is permitted to become tangent to ∂Ω, the situation is significantly different. The existence and regularity of the problem are closely related to the behavior of the integral curves of the vector field near the boundary. (e.g. [19], [9], and references in [16]) In the case the P and ℓ^{→} are linear and smooth, if ∂Ω is also smooth, the problem is well understood (see [16]). When P is nonlinear, in general, we still don’t know how to solve the problem. Here we try to establish some existence and regularity results when the prescribed data is small.

1991 Mathematics Subject Classification. Primary 35J25, 35J70; Secondary 35B65, 53C45.

Supported in part by NSERC grant OGP0046732.

In order to understand nonlinear versions of this problem, we first have to consider the linear equations where P and ℓ^{→} in (1.1) have limited smoothness. The key tool is the symbol splitting technique of pseudodifferential operators with limited smoothness, which enable us to obtain appropriate a priori estimates.

Let Λ^{s} be the Hölder spaces. Let Ω be a connected open set in R^{n+2} with C^{λ+3} boundary ∂ Ω = Σ. Let

P (x, D) = ∑_{i,j=1}^{n+2} a_{ij}(x)\frac{∂}{∂x_{i}} \frac{∂}{∂x_{j}} + ∑_{i=1}^{n} b_{i}(x)\frac{∂}{∂x_{i}} + c(x)

be an elliptic second order real differential operator on R^{n+2} with C^{λ} coefficients, i.e. there are C, δ > 0 such that

C|ξ|^{2} ≥ ∑_{i,j=1}^{n+2} a_{ij}(x)ξ_{i}ξ_{j} ≥ δ|ξ|^{2}, ∀x, ξ ∈ R^{n+2},

where a_{ij}, b_{i} and c are C^{λ} on R^{n+2}. Let ℓ⃗(x) be a real C^{λ+2} unit vector field in R^{n+2} with

ℓ⃗|_{Σ} = T⃗ + a(x)n⃗,

where n⃗ is the unit outward normal to Σ at x and T⃗ is tangential to Σ. Finally let N = {x ∈ Σ : a(x) = 0} ≠ ϕ. We assume throughout the following restriction (R_{int}) on T⃗ :

No integral curve of T⃗ in Σ lies in N for an infinite interval of time. (1.2)

By the necessary conditions on local solvability of pseudodifferential operators, one can only expect the existence of the problem in the following two cases:

Case (I) a(x) ≥ 0 for all x ∈ Σ or a(x) ≤ 0 for all x ∈ Σ.

Case (II) a(x) takes on both positive and negative values and there is an ndimensional C^{λ+2} submanifold Γ (possibly with several components) contained in N and open subsets Σ^{+} and Σ^{−} of Σ such that Σ is a disjoint union of Σ^{+}, Σ^{−} and Γ and

(i) a(x) ≥ 0 for x ∈ Σ^{+} (ii) a(x) ≤ 0 for x ∈ Σ^{−} (iii) ∂Σ^{+} = ∂Σ^{−} = Γ (iv) ℓ⃗ is transversal to Γ and points in the direction of Σ^{+}. For case (I) the correct setting of the problem is:

\{ \begin{matrix}P u & = & f & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \end{matrix} , (1.3)

and for case (II) the problem is

\{ \begin{matrix}P u & = & f & in Ω \\ ℓ^{→} u & = & g & on ∂Ω \\ u & = & h & on Γ \end{matrix} . (1.4)

The following theorem [17] is a generalization of classical Agmon-Douglis-Nirenberg results [1] for the degenerate oblique derivative problem. Fix 0 < μ < min \{\frac{λ}{λ+1}, \frac{λ+2}{2(λ+3)}\}.

Theorem 1.1. Suppose that P, Ω and ℓ⃗ are as at the beginning of this section with λ > 0.

(i) If ℓ⃗ satisfies case (I), then for all 2 < s ≤ λ + 2, there is a subspace F of finite codimension in Λ^{s−2}(Ω)× Λ^{s}(∂Ω) such that for (f, g) ∈ F , there exists u satisfying (1.3) and u ∈ Λ^{s}(Ω). Moreover, if u ∈ Λ^{s′}(Ω) for some s′ > 1 satisfies (1.3) with f, g as above, then u ∈ Λ^{s}(Ω), and there is C_{s} such that

∥u∥_{Λ^{s}(Ω)} ≤ C_{s} (∥f∥_{Λ^{s−2}(Ω)} + ∥g∥_{Λ^{s}(∂Ω)} + ∥u∥_{Λ^{s−μ}(Ω)}) .

Furthermore, if the zero order term c (x) in the operator P is negative, then for every (f, g) ∈ Λ^{s−2}(Ω)× Λ^{s}(∂Ω), there is a unique solution u ∈ Λ^{s}(Ω). (ii) If ℓ⃗ satisfies case (II), then for all 2 < s ≤ λ + 2, there is a subspace F of finite codimension in Λ^{s−2}(Ω)× Λ^{s}(∂Ω)× Λ^{s}(Γ) such that for (f, g, h) ∈ F , there exists u satisfying (1.4) and u ∈ Λ^{s}(Ω). Moreover, if u ∈ Λ^{s′}(Ω) for some s′ > 1 satisfies (1.4) with f, g, h as above, then u ∈ Λ^{s}(Ω), and there is C_{s} such that

∥u∥_{Λ^{s}(Ω)} ≤ C_{s} (∥f∥_{Λ^{s−2}(Ω)} + ∥g∥_{Λ^{s}(∂Ω)} + ∥h∥_{Λ^{s}(Γ)} + ∥u∥_{Λ^{s−μ}(Ω)}) .

Furthermore, if the zero order term c (x) in the operator P is nonpositive, then for every (f, g, h) ∈ Λ^{s−2}(Ω)× Λ^{s}(∂Ω)× Λ^{s}(Γ), there is a unique solution u ∈ Λ^{s}(Ω).

Above theorem and the standard implicit function theorem yield the following existence result for nonlinear problem (1.1).

Theorem 1.2. Let Ω and ℓ^{→} be as in Theorem 1.1. Suppose that λ > 0 is nonintegral and that

\begin{matrix}P (x, u, ξ, ζ) ∈ C^{λ} (\overline{Ω} × R × R^{n+2} × R^{(n+2)^{2}}) , \\ ∂_{u}P, ∂_{ξ}P, ∂_{ζ}P ∈ C^{λ} (\overline{Ω} × R × R^{n+2} × R^{(n+2)^{2}}) . \end{matrix}

Moreover, suppose P (x, 0, 0, 0) = 0 and that there is C > 0 such that

\begin{matrix}C|η|^{2} ≥ ∑_{i,j=1}^{n+2} \frac{∂}{∂ζ_{ij}} P (x, 0, 0, 0) η_{i}η_{j} ≥ C^{−1}|η|^{2}, \\ −C ≤ ∂_{u}P (x, 0, 0, 0) ≤ −C^{−1}, \end{matrix} for all x ∈ \overline{Ω}.

(i) If ℓ^{→} satisfies case (I), there is c_{0} > 0 such that if ∥f∥_{Λ^{λ}(Ω)} and ∥g∥_{Λ^{λ+2}(∂Ω)} ≤ c_{0}, then there is u ∈ Λ^{λ+2} (Ω) satisfying

\{ \begin{matrix}P (x, u, ∇u, ∇^{2}u) = f (x) & in Ω \\ ℓ^{→} u = g & on ∂Ω \end{matrix} .

(ii) If ℓ^{→} satisfies case (II), then there is c_{0} > 0 such that if ∥f∥_{Λ^{λ}(Ω)}, ∥g∥_{Λ^{λ+2}(∂Ω)} and ∥h∥_{Λ^{λ+2}(Γ)} ≤ c_{0}, then there is u ∈ Λ^{λ+2} (Ω) satisfying

\{ \begin{matrix}P (x, u, ∇u, ∇^{2}u) = f (x) & in Ω \\ ℓ^{→} u = g & on ∂Ω \\ u = h & on Γ \end{matrix} .

2. Quasilinear Degenerate Elliptic Equations

There have been some works on degenerate nonlinear elliptic equations in connection to Bony’s theory of paradifferential operators [5]. But, most of the regularity results for degenerate quasilinear elliptic equations are based on the aprior high initial smoothness assumption on the solutions. Here, we present some regularity results for a class of quasilinear degenerate elliptic equations with the C^{0,1} initial smoothness assumption on the solution u. They are related to degenerate Monge-Amperé equations, which is our main motivation.

The quasilinear degenerate elliptic equations we want to consider is the following:

∑_{i,j} ∂_{j}(a_{ij}(x, u)∂_{i}u) = f, (2.1)

where a_{ij}(x, u) ∈ C^{∞}(R^{n} × R), semi-positive.

Unlike in elliptic case, Schauder theory is not available for the degenerate equations in the form (2.1). We are interested to know when the weak solutions of (2.1) are smooth. We establish that a weak solution u of equation (2.1) is C^{∞} if u is in C^{0,1}, and the linearized equation satisfies some subellipticity conditions. Again, calculus of pseudodifferential operators with limit smoothness plays the crucial role here. We prove a commutator lemma which signals out the first order terms of Calderón commutator with a careful control of the coefficients. The difficulty is the limited smooth assumption. The proof of the lemma makes use of symbol decomposition of pseudodifferential operators with limited smoothness. A similar calculus was also used in solving nonlinear oblique derivative problem.

Let Ω ⊂ R^{n} be a domain. Let (a_{ij}(x)) be a nonnegatively definite matrix function in Ω. Definition We say the matrix A(x) = (a_{ij}(x)) satisfies subunit condition in Ω if there is constant C such that,

|A_{x_{k}}(x) · ξ|^{2} ≤ C^{t}ξAξ,

∀ ξ ∈ R^{n}, and ∀ x ∈ Ω. If L be a linear operator of the form

Lu = ∑_{i,j}^{n} ∂_{i}(a_{ij}(x)∂_{j}u) (2.2)

with a_{ij} ∈ L^{∞}(Ω), (a_{ij}) ≥ 0 and tr(a_{ij}(x)) > 0 for all x in Ω. We say L is αsubelliptic, if there is α > 0, such that, f_{0}, F ∈ L^{∞}(Ω), if v is a weak solution of the equation Lv = f_{0} + ∇_{λ}F, then v ∈ C^{α}(Ω), and for each K ⊂⊂ Ω, ∥v∥_{C^{α}(K)} ≤ C where C depends only on K, ∥f_{0}∥_{L^{∞}(Ω)}, ∥F ∥_{L^{i}nfty(Ω)}, ∥v∥_{L^{2}(Ω)}, and ∥T_{j}∥_{C^{1}}.

The following is a sample of regularity results (see [12] and [13]) for weak solutions of (2.1).

Theorem 2.1. Suppose u ∈ C^{0,1}(Ω), u is a weak solution of equation

∑_{i,j}^{n} ∂_{i}(a_{ij}(x, u)∂_{j}u) = f(x)

with a_{ij} ∈ C^{∞}(Ω × R), (a_{ij}) ≥ 0, f ∈ C^{∞}(Ω). Then u ∈ C^{∞}(Ω), if the linear operator

Lv = ∑_{i,j}^{n} ∂_{i}(ã_{ij}(x)∂_{j}v),

where ã_{ij}(x) = a(x, u(x)), is subunit and α-subelliptic for some α > 0.

Remark By De Giorgi-Nash-Moser theory if (a_{ij}(x)) in (2.2) is positive definite, L is α-subelliptic for some α > 0. Subellipticity is much weaker notion than ellipticity. If (a_{ij}(x)) is nonnegative definite, and the coefficients of L in (2.2) are only assumed in L^{∞}, there are some sufficient conditions of subellipticity (see [10]). In smooth coefficients case, the subellipticity of the operator L can be completely characterized by Fefferman-Phong condition in terms of the geometry of the associated metric.

The following key commutator lemma in [13] is the key in the proof of Theorem 2.1. We believe that it will be useful in other places. Definition Let Λ^{s}, s ∈ R be Hölder-Zygmund spaces. We denote C_{∗}^{s} = Λ^{s} if s > 0, s ∉ Z, and C_{∗}^{ℓ} = C^{ℓ,0}, if ℓ ∈ Z^{+}. If B is a linear operator, B is bounded from Λs+m_{comp}(R^{n}) → Λs_{loc}(R^{n}) for some m ∈ R, and ∀ 0 < s ≤ t, then we say B ∈ O_{t}^{m}; if B is bounded from C_{∗}^{s} → C_{∗}^{s+m} for some m ≤ s ∈ R, and ∀ 0 < s + m ≤ t, then we say B ∈ O_{∗t}^{m}.

Let ρ(t) ∈ C_{0}^{∞}(R), ∀ ϵ > 0, let ρ_{δ}(x) = δ^{−n} ∏_{i} ρ(\frac{x_{i}}{δ}). We also denote the ρ_{δ} be the convolution operator:

ρ_{δ}(g)(x) = ∫ ρ_{δ}(x − y)g(y)dy,

Lemma 2.2. Let a(x, u) ∈ C^{∞}(Ω × R), suppose u ∈ C_{∗}^{t}(Ω) for some t ≥ 1. (i) Let ρ_{δ} as above, there are operators B_{δ,j} ∈ O_{∗t}^{−1}, i = 1, 2, …, n+1, and B_{δ,0} ∈ O_{∗t}^{−2}, with bounds independent of δ, such that,

[ρ_{δ}, a(x, u(x))] = ∑_{j=1}^{n} a_{x_{j}}(x, u(x))B_{δ,j} + a_{u}(x, u(x))B_{δ,n+1} + B_{δ,0}.

Moreover, for any first order differential operator with constant coefficients D, there are operators B̄_{δ,j} ∈ O_{∗t−1}^{0}, i = 1, 2, …, n + 1, and B̄_{δ,0} ∈ O_{∗t}^{−1}, with bounds independent of δ, such that,

D[ρ_{δ}, a(x, u(x))] = ∑_{j=1}^{n} a_{x_{j}}(x, u(x))B̄_{δ,j} + a_{u}(x, u(x))B̄_{δ,n+1} + B̄_{δ,0}.

(ii) Let |D|^{s} = (1 − Δ)^{s/2}, where Δ = ∂_{1}^{2} + ⋯ + ∂_{n}^{2}. Then, we have ∀ ε > 0, there are operators B_{j} ∈ O_{t−s}^{−1+ε}, i = 1, 2, …, n + 1, and B_{0} ∈ O_{t−s}^{s−2+ε}, such that

[|D|^{s}, a(x, u(x))] = ∑_{j=1}^{n} a_{x_{j}}(x, u(x))B_{j} + a_{u}(x, u(x))B_{n+1} + B_{0}.

There are some interesting works (see [25] and [26]) on quasilinear degenerate elliptic equations and systems of the form

∑_{i,j=1}^{m} A_{ij}(x, u, Xu)X_{i}X_{j}u + B(x, u, Xu) = 0,

where X = (X_{1}, ⋯, X_{m}) is a system of real smooth vector fields satisfying the Hörmander condition, A_{ij}, B ∈ C^{∞} and A_{ij} positive definite. There the degeneracy occurs in the vector fields X_{j} (which are assumed to be smooth and subelliptic by Hörmander’s condition), while the matrix A is assumed non-degenerate. For these type of subelliptic equations and systems, there are some parallel existence and regularity results as in the classical elliptic cases.

3. Degenerate Monge-Ampére Equations

Monge-Ampére equations are connected naturally with many geometric problems. They play important roles in solving some problems in geometric analysis. From analysis point of view, people want to understand the Dirichlet problem for Monge-Ampère equation:

\{ \begin{matrix}det(u_{ij}) = f  in Ω ⊂ R^{n} \\ u|_{∂Ω} = ϕ . \end{matrix} (3.1)

The Dirichlet problem for elliptic Monge-Ampère equations (i.e., f(x) > 0) has received considerable study. When f is only assumed non-negative, the equation is degenerate. In this case, in general one can only expect the solutions to be C^{1,1}. Thus, much attention has been directed to obtain C^{2} a priori estimates of the problem. There are some works in this direction (e.g., [6], [7] and [23]), under the assumptions f^{\frac{1}{n}} ∈ C^{1,1} with homogeneous boundary condition or f = 0. These assumptions are not natural, in particular from geometric point view. We ask,

Question: Is the solution of equation (3.1) in C^{1,1} if f is nonnegative and smooth?

In the following, we present some partial answers (see [14] and [18]).

Theorem 3.1. Let Ω be a strongly convex domain in R^{n} with ∂Ω ∈ C^{3,1}, g ∈ C^{3,1}(∂Ω), and let f be a non-negative function in Ω such that f^{1/(n−1)} ∈ C^{1,1}(\overline{Ω}), then there is an unique convex solution u ∈ C^{1,1}(\overline{Ω}) of the Dirichlet problem (3.1) for Monge-Ampére equation. Consequently, any generalized solution of the above problem in C^{0}(\overline{Ω}) must belong to C^{1,1}(\overline{Ω}). And

∥u∥_{C^{1,1}(\overline{Ω})} ≤ C

where C depends only on Ω, ∥g∥_{C^{3,1}}, ∥f^{\frac{1}{n−1}} ∥_{C^{1,1}}.

Remark. Above result was obtained by Krylov under assumption that f^{\frac{1}{n}} ∈ C^{1,1}(\overline{Ω}). The result in the present form is due to Guan-Trudinger-Wang [18]. Examples show that the above result (with exponent \frac{1}{n−1}) is optimal. If the problem is homogeneous (i.e., g = 0 in (3.1)), we have the following more general result (which answers the question in cases n = 2 or 3).

Theorem 3.2. Suppose Ω be a bounded strictly convex domain in R^{n} with C^{2,1} boundary. If f ≥ 0, f^{\frac{1}{n−1}} ∈ C^{0,1} near the boundary of Ω and Δf^{\frac{1}{n−1}} (x) ≥ −(n − 1)^{−1}A for all x ∈ Ω, then, the Monge-Ampère equation (3.1) with the homogeneous boundary condition ϕ ≡ 0, has an unique C^{1,1} convex solution u, with

∥u∥_{C^{2}(Ω̄)} ≤ C,

where the constant C depends only on Ω, A, δ, sup(f), and modulus continuity of f in Ω. In particular, in the case n = 2, if f ∈ C^{1,1} is nonnegative; or in the case n = 3, if f ∈ C^{3,1} is nonnegative, then then u ∈ C^{1,1}(Ω̄).

For problems related to differential geometry, one expects better regularity of the solutions. We want to understand the higher regularity of the solutions of Monge-Ampére equations:

det(u_{ij}) = k, (3.2)

where k ≥ 0. When k > 0, the regularity of solution u of (3.2) is well-understood. In the degenerate case k ≥ 0, we have already C^{1,1} regularity for the solutions of (3.2) under some reasonable assumptions. We need to impose some conditions on k in order to get better regularity than C^{1,1} as indicated in the following two examples.

Example a. Let u(x, y) = max \{(max \{(x^{2} − \frac{1}{2}) , 0\})^{2} , (max \{(y^{2} − \frac{1}{2}) , 0\})^{2}\},

ϕ(θ) = (cos^{2} θ − sin^{2} θ)^{2}. u satisfies

u_{xx}u_{yy} − u_{xy}^{2} = 0 in x^{2} + y^{2} < 1 .

And

u|_{x^{2}+y^{2}=1} = ϕ(θ) .

Example b. Let u(x) = |x|^{2+ \frac{1}{n}} − 1, x ∈ R^{n}. u satisfies

\{ \begin{matrix}det(u_{ij}) = c_{n}|x|^{2} & in \{|x| < 1\} \\ u|_{|x|=1} = 0 \end{matrix}

for some c_{n} > 0.

In Example a, k = 0. So, one needs to control the degeneracy of k near its null set. While in Example b, x = 0 is a plannar point of the hypersurface S = (x, u(x)). This indicates that we can only expect higher regularity of u away from the plannar points of S = (x, u(x)). When n = 2, it turns out these are in fact the two main obstacles for the higher regularity of u.

Theorem 3.3. If u satisfies degenerate Monge-Ampère equation

u_{xx}u_{yy} − u_{xy}^{2} = k(x, y)

near the origin. Suppose k ≥ 0, k ∈ C^{∞}(R × R), and u ∈ C^{1,1}. Suppose that near the origin, u_{yy} ≥ C_{0} > 0 and that there exist constants A > 0 and positive integers l, such that

Ax^{2l} ≤ k(x, y),

Then, u ∈ C^{∞} near the origin.

To prove the Theorem, we use a partial Legendre transformation. At the origin, we may assume ∇u(0) = 0. We let

\{ \begin{matrix}s = x \\ t = u_{y} . \end{matrix}

The change of variables T (x, y) = (s, t) is C^{0,1} near the origin by the assumption of u ∈ C^{1,1}. If u_{yy} ≥ C_{0} > 0 near the origin, T and T^{−1} are C^{0,1} diffeomorphisms near the origin.

The next lemma relates regularity problem of equation (3.2) to a degenerate quasilinear equation of the form (2.1). The Theorem is then a consequence of regularity results for solutions of degenerate quasilinear elliptic equations in dimension n = 2. Lemma x(s, t), y(s, t) ∈ C^{0,1}(T B_{R}) is a weak solution of the equation

z_{ss} + ∂_{t}(k(x(s, t), y(s, t))∂_{t}z) = 0 .

And u ∈ C_{∗}^{2+β}(B_{r}) for 0 < r < R, β ≥ 0 if and only if y(s, t) ∈ C_{∗}^{1+β}(T B_{r}).

We now briefly discuss complex degenerate Monge-Ampère equations. For strongly pseudoconvex domains, we refer [3], [4] and [6]. But in many geometric applications, the assumption of strongly pseudoconvex boundary is not natural. For example, the problem concerning extremal function of Chern-Levine-Nirenberg intrinsic norm (see [15]), the pluri-complex Green functions (see [21] and [11]) and extremal metric problem in Kahler metrics (see [8]). In these cases, some part of the boundary is pseudoconvcave and equation is degenerate.

For example, the extremal function associated to Chern-Levine-Nirenberg intrinsic norm satisfies the following homogeneous complex Monge-Ampère equation:

\{ \begin{matrix}(dd^{c}u)^{n} = 0 & in Ω \\ u|_{Γ_{1}=1} \\ u|_{Γ_{0}=0}, \end{matrix} (3.3)

where d^{c} = i(¯∂ − ∂), Ω a domain is certain complex manifold, and Γ_{1} and Γ_{0} are the pseudoconvex and pseudoconcave patrts of the boundary of Ω respectively.

The following is one sample of the regularity results for degenerate complex Monge-Ampère equations.

Theorem 3.4. There is a unique C^{1,1}(Ω̄) solution u of (3.3). In fact, there is C > 0 and there is a sequence {u_{k}} ∈ C^{∞}(Ω̄) such that

∥u_{k}∥_{C^{2}(Ω̄)} ≤ C, ∀ k,  lim_{k→∞} sup(dd^{c}u_{k})^{n} = 0.

The regularity in above is optimal by example in [2]. Similar regularity results also hold for other type of degenerate Monge-Ampère equations arising from several complex analysis and geometry, we refer [11], [8] and [15] for further details.

Some Open Problems

Problem 1, Theorem 1.2 is for small data, is the result still ture for general data?

Problem 2, Can we generalize Theorem 2.1 to the case where ∇u is also involved in a_{ij}? It is also interesting to consider Dirichlet problem for the equation.

Problem 3, Give an affirmative answer to Question in section 3 for higher dimensions (n ≥ 4), or find a counter-example. The smoothness assumption on f is necessary, see examples in [24].

Problem 4, Are there any natural sufficient conditions to ensure higher regularity (e.g., C^{2,α} or better) for degenerate Monge-Ampére equations in dimension n ≥ 3?

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Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada

E-mail address: guan@icarus.math.mcmaster.ca