INTERIOR C^{1,1} REGULARITY

OF THE DEGENERATE MONGE-AMPÈRE EQUATION

Zbigniew Błocki

IMUJ PREPRINT 2001/02

Abstract. We study interior C^{1,1} regularity of generalized solutions of the Monge-Ampére equation det D^{2}u = ψ, ψ ≥ 0, on a bounded convex domain Ω in ℝ^{n} with u = φ on ∂Ω. We prove in particular that u ∈ C^{1,1}(Ω) if either i) φ = 0 and ψ^{1/(n−1)} ∈ C^{1,1}(Ω) or ii) Ω is C^{1,1} strongly convex, φ ∈ C^{1,1}(\overline{Ω}), ψ^{1/(n−1)} ∈ C^{1,1}(\overline{Ω}) and ψ > 0 on U ∩ Ω, where U is a neighborhood of ∂Ω. The main tool is an improvement of a well known Pogorelov’s C^{1,1} estimate so that it can be applied to the degenerate case.

1. Introduction

For an arbitrary convex function u one can well define a nonnegative Borel measure M(u) such that

M(u) = det D^{2}u dλ

for smooth, and even W_{loc}^{2,n} functions (see [RT] for details). The Dirichlet problem for M is solvable in a fairly general situation: let Ω be an arbitrary bounded convex domain in ℝ^{n} and φ ∈ C(∂Ω) be such that it is convex on any line segment in ∂Ω (we will call such a φ admissible). Then for any nonnegative Borel measure μ with μ(Ω) < ∞ the Dirichlet problem

(1.1)  \{ \begin{matrix}u continuous and convex on \overline{Ω} \\ M(u) = μ in Ω \\ u = φ on ∂Ω \end{matrix}

Typeset by AMS-TEX

Partially supported by KBN Grant #2 P03A 028 19

has a unique solution. (This was proven for example in [RT] for strictly convex Ω where all continuous φ are admissible, and the general case easily follows from this - see Proposition 2.1 below.)

We will primarily consider measures μ with continuous, nonnegative densities ψ in Ω. Unless otherwise stated, u will always denote the solution of (1.1) (with μ = ψdλ), whereas v will be the solution of the corresponding homogeneous problem:

(1.2)  \{ \begin{matrix}v continuous and convex on \overline{Ω} \\ M(v) = 0 in Ω \\ v = φ on ∂Ω. \end{matrix}

Below we list several known regularity results for solutions of (1.1):

1) ϕ = 0, ψ ∈ C^{∞}(Ω), ψ > 0 ⇒ u ∈ C^{∞}(Ω) (Cheng, Yau [CY1], see also [CY2]);

2) Ω is C^{1,1} strongly convex, φ ∈ C^{1,1}(\overline{Ω}) ⇒ v ∈ C^{0,1}(\overline{Ω}) ∩ C^{1,1}(Ω) (Trudinger, Urbas [TU]);

3) Ω is C^{1,1} strongly convex, φ ∈ C^{1,1}(\overline{Ω}), ψ ∈ C^{1,1}(\overline{Ω}), ψ > 0 ⇒ u ∈ C^{1,1}(Ω) (Trudinger, Urbas [TU]);

4) Ω is C^{∞} strongly convex, φ ∈ C^{∞}(∂Ω), ψ ∈ C^{∞}(\overline{Ω}), ψ > 0 ⇒ u ∈ C^{∞}(\overline{Ω}) (Krylov [Kr] and Caffarelli, Nirenberg, Spruck [CNS]);

5) Ω is C^{3,1} strongly convex, φ ∈ C^{3,1}(\overline{Ω}), ψ^{1/(n−1)} ∈ C^{1,1}(\overline{Ω}) ⇒ u ∈ C^{1,1}(\overline{Ω}) (Guan, Trudinger, Wang [GTW]).

Pogorelov had claimed to prove 1) earlier but his proof had many gaps. However, his arguments were enough to get u ∈ C^{1,1}(Ω) in 1). The following results on the local regularity of the Monge-Ampère operator M are also known:

6) u is strictly convex, ψ ∈ C^{∞}, ψ > 0 ⇒ u ∈ C^{∞} (this is in fact equivalent to 1));

7) Either u ∈ W_{loc}^{2,p} for some p > n(n − 1)/2 or u ∈ C^{1,α} for some α > 1 − 2/n, ψ ∈ C^{∞}, ψ > 0 ⇒ u ∈ C^{∞} (Urbas [Ur]).

In this paper we prove two more regularity results:

Theorem 1.1. If φ = 0 and ψ^{1/(n−1)} ∈ C^{1,1}(Ω) then u ∈ C^{1,1}(Ω).

Theorem 1.2. Suppose that Ω is a C^{1,1} strongly convex domain, φ ∈ C^{1,1}(\overline{Ω}) and ψ^{1/(n−1)} ∈ C^{1,1}(\overline{Ω}). Assume moreover that

(1.3)  ψ > 0 on U ∩ Ω

for some neighborhood U of ∂Ω. Then u ∈ C^{1,1}(Ω).

Note that in Theorem 1.1 Ω is an arbitrary bounded convex domain in ℝ^{n}. By 7), Theorem 1.1 is a generalization of 1). We also immediately get the following local regularity which generalizes 6):

Theorem 1.3. If u is strictly convex and ψ^{1/(n−1)} ∈ C^{1,1} then u ∈ C^{1,1}.

This paper was mostly motivated by recent articles [Gu] and [GTW], where global a priori estimates for second derivatives of u depending on ||ψ^{1/(n−1)}||_{C^{1,1}(\overline{Ω})} were established. The main tool in proving Theorems 1.1 and 1.2 will be a corresponding interior a priori estimate (Theorem 3.1). The importance of the exponent 1/(n − 1) is that, by [Wa, Example 3], it is optimal in all of the above results.

We believe that the assumption (1.3) in Theorem 1.2 is superfluous. However, at the end of section 4 we give an example which shows that the application of our methods only does not allow to drop this assumption. Namely, what we really prove is that under the assumptions of Theorem 1.2 except for (1.3), we have u ∈ C^{1,1}({u < v}) (Theorem 4.2). Moreover, {u < v} = Ω if (1.3) is satisfied (Proposition 4.1), but without (1.3) it may happen that {u < v} ≠ Ω.

Terminology and notation. We say that a convex domain Ω (resp. convex function u) is strictly convex if ∂Ω (resp. graph u) contains no line segment. A convex function v in Ω is called maximal if for any D ⋐ Ω and u continuous on \overline{D}, convex in D, we have that u ≤ v on ∂D implies u ≤ v in D. This condition is in fact equivalent to M(v) = 0 (see [RT]) . A function u is called strongly convex if locally there exists λ > 0 such that u − λ|x|^{2} is convex. We say that a bounded domain Ω is strongly convex if there exists w, a strongly convex function in a neighborhood of \overline{Ω} such that Ω = \{w < 0\} and Dw ≠ 0 near ∂Ω (that is there exists ε > 0 such that for any supporting hyperplane H = graph L of w at a point near ∂Ω we have |∇L| ≥ ε). A domain is called C^{k,α} strongly convex if one can find a C^{k,α} defining function w. If we write f ∈ C^{k,1}(\overline{Ω}) then we mean that f ∈ C^{k,1}(Ω) and |D^{k+1}f| is globally bounded in Ω. Then

||f||_{C^{k,1}(\overline{Ω})} = ∑_{i=0}^{k+1} sup_{Ω} |D^{i}f|

and the values of f at ∂Ω are uniquely determined. Finally, \overline{B}(x, r) will denote a closed ball centered at x with radius r and [x, y] will stand for the line segment joining x and y.

2. Existence of generalized solutions

In this section we will show how to generalize the solution of the Dirichlet problem (1.1) in [RT] from strictly convex to arbitrary convex domains.

Proposition 2.1. Assume that Ω is an arbitrary bounded convex domain in ℝ^{n}. Let φ ∈ C(∂Ω) be admissible (that is φ is convex on any line segment in ∂Ω) and μ be a nonnegative Borel measure in Ω with μ(Ω) < ∞. Then the problem (1.1) has a unique solution.

Proof. The uniqueness follows from the comparison principle (see e.g. [RT]). If Ω is strictly convex then the proof can be found for example in [RT]. We shall use this as well as other results from [RT] in the proof.

We first solve the homogeneous problem (1.2) using the Perron method. Define

v := sup\{w convex and continuous on \overline{Ω}, w ≤ φ on ∂Ω\}.

Then v is convex in Ω and we have to show that v continuously extends to ∂Ω and equals φ there. Fix x_{0} ∈ ∂Ω and ε > 0. Since φ is convex on any line segment contained in ∂Ω and passing through x_{0}, we can find an affine function L such that L(x_{0}) ≥ φ(x_{0}) − ε and L ≤ φ on ∂Ω. Let h be a solution of the classical Dirichlet problem

h continuous on \overline{Ω} h harmonic in Ω h = φ on ∂Ω.

Then we have L ≤ v ≤ h in Ω and

φ(x_{0}) − ε ≤ lim inf_{x→x_{0}} v(x) ≤ lim sup_{x→x_{0}} v(x) ≤ φ(x_{0}).

This shows that v is continuous on \overline{Ω} and v = φ on ∂Ω. The definition of v implies that v is maximal and thus M(v) = 0 there. This solves (1.2).

Next we solve (1.1) with φ = 0. This has already been done in [Bł, Theorem 4.1] but we include it for the convenience of the reader. Let Ω_{j} be strictly convex domains such that Ω_{j} ↑ Ω as j ↑ ∞. Let u_{j} be continuous and convex on \overline{Ω}_{j} and such that M(u_{j}) = μ in Ω_{j} and u_{j} = 0 on ∂Ω_{j}. By the comparison principle the sequence u_{j} is decreasing. Moreover, [RT, Lemma 3.5] gives

(2.1)  (−u_{j}(x))^{n} ≤ c_{n}(diam Ω)^{n−1}dist(x, ∂Ω)μ(Ω), x ∈ Ω_{j},

where c_{n} is a constant depending only on n. This implies that u_{j} converges to a convex ũ uniformly on \overline{Ω}. By the continuity theorem for M ([RT, Proposition 3.1]) we have M(ũ) = μ in Ω. This solves (1.1) for φ = 0.

Now let μ and φ be arbitrary. Again, we approximate Ω by strictly convex domains Ω_{j} from inside. There we can find u′_{j}, continuous and convex on \overline{Ω}_{j}, such that M(u′_{j}) = μ in Ω_{j} and u′_{j} = v on ∂Ω_{j}. [RT, Proposition 3.3] gives M(ũ+v) ≥ μ and thus

ũ + v ≤ u′_{j+1} ≤ u′_{j} ≤ v in Ω_{j}

by the comparison principle. It now easily follows that u′_{j} decreases to a function u which solves (1.1). □

3. A priori estimate for second derivatives

In this section we will generalize the Pogorelov estimate (see [Po, p. 73-76]) so that it can be applied to the degenerate case. We will modify Pogorelov’s method using some ideas from [Gu]. We shall prove:

Theorem 3.1. Let u ∈ C^{4}(Ω) ∩ C^{1,1}(\overline{Ω}) be a strongly convex solution of

(3.1)  det D^{2}u = ψ

in a bounded domain Ω in ℝ^{n}. Assume that w is a C^{2} convex function in Ω such that u ≤ w in Ω and lim_{x→∂Ω}(w(x) − u(x)) = 0. Then in Ω we have

(w − u)^{α}|D^{2}u| ≤ C,

where α = α(n) is any number such that

α(n)\{ \begin{matrix}≥ n − 1 & for n ≥ 3, \\ > 1 & for n = 2, \end{matrix}

and C depends only on n, α and on upper bounds of diam Ω, ||ψ^{1/(n−1)}||_{C^{1,1}(\overline{Ω})} and ||w − u||_{C^{0,1}(\overline{Ω})}.

Proof. We may assume that Ω ⊂ B(0, R), where R ≤ diam Ω. We will use the standard notation: u_{i} = ∂u/∂x_{i}, (u_{ij}) = D^{2}u and (u^{ij}) = (D^{2}u)^{−1}. First, we differentiate the logarithm of both sides of (3.1) twice with respect to x_{p}. We get

(3.2)  u^{ij}u_{pij} = (log ψ)_{p}
(3.3)  u^{ij}u_{ppij} = (log ψ)_{pp} + u^{ik}u^{jl}u_{pij}u_{pkl}.

Consider the auxiliary function

h = (w − u)^{α}e^{β|x|^{2}/2}|D^{2}u|,

where α, β > 0 will be specified later. By the assumptions on u and w, h attains a maximum at some y ∈ Ω. After rotation of variables we may assume that (u_{ij}) is diagonal at y. Since |D^{2}u| is equal to the maximal eigenvalue of D^{2}u, we may also assume that at y we have |D^{2}u| = u_{11}. Thus the function

h̃ = (w − u)^{α}e^{β|x|^{2}/2}u_{11}

also attains a maximum at y and h̃(y) = h(y).

We may assume that w < u at y. We will use the fact that Dh̃ = 0 and D^{2}h̃ ≤ 0 at y. Therefore at y we have

(3.4)  0 = (log h̃)_{i} = α\frac{w_{i} − u_{i}}{w − u} + βx_{i} + \frac{u_{11i}}{u_{11}}

for every i = 1, …, n and

u^{ii}(log h̃)_{ii} ≤ 0.

Now we analyze the term where the third and fourth derivatives of u appear. Using (3.2), (3.3), (3.4) and denoting g := ψ^{1/(n−1)} we compute at y

u^{ii}(log u_{11})_{ii} = \frac{1}{u_{11}} ∑_{i} \frac{u_{11ii}}{u_{ii}} − \frac{1}{(u_{11})^{2}} ∑_{i} \frac{(u_{1ii})^{2}}{u_{ii}}

= \frac{(log ψ)_{11}}{u_{11}} + \frac{1}{u_{11}} ∑_{i,j} \frac{(u_{1ij})^{2}}{u_{ii}u_{jj}} − \frac{1}{(u_{11})^{2}} ∑_{i} \frac{(u_{1ii})^{2}}{u_{ii}}

≥ \frac{(log ψ)_{11}}{u_{11}} + \frac{1}{u_{11}} ∑_{i≥2} \frac{(u_{1ii})^{2}}{(u_{ii})^{2}} + \frac{1}{(u_{11})^{2}} ∑_{i≥2} \frac{(u_{11i})^{2}}{u_{ii}}

≥ \frac{(log ψ)_{11}}{u_{11}} + \frac{1}{(n − 1)u_{11}} ((log ψ)_{1} − \frac{u_{111}}{u_{11}} )^{2} + \frac{1}{(u_{11})^{2}} ∑_{i≥2} \frac{(u_{11i})^{2}}{u_{ii}}

≥ \frac{(n − 1)g_{11}}{g u_{11}} − \frac{2g_{1}u_{111}}{g(u_{11})^{2}} + \frac{1}{(u_{11})^{2}} ∑_{i≥2} \frac{(u_{11i})^{2}}{u_{ii}}

= \frac{(n − 1)g_{11}}{g u_{11}} + \frac{2g_{1}}{g u_{11}} (α\frac{w_{1} − u_{1}}{w − u} + βx_{1}) + ∑_{i≥2} \frac{1}{u_{ii}} (α\frac{w_{i} − u_{i}}{w − u} + βx_{i})^{2} .

We thus obtain at y

0 ≥ u^{ii}( log h̃)_{ii}

= \frac{α}{w − u} ∑_{i} \frac{w_{ii}}{u_{ii}} − \frac{α n}{w − u} − \frac{α}{(w − u)^{2}} ∑_{i} \frac{(w_{i} − u_{i})^{2}}{u_{ii}} + β ∑_{i} \frac{1}{u_{ii}}

+ u^{ii}(log u_{11})_{ii}

≥ −\frac{α n}{w − u} − \frac{α(w_{1} − u_{1})^{2}}{(w − u)^{2}u_{11}} + \frac{(n − 1)g_{11}}{g u_{11}} + \frac{2g_{1}}{g u_{11}} (α\frac{w_{1} − u_{1}}{w − u} + βx_{1})

+ β ∑_{i} \frac{1}{u_{ii}} + ∑_{i≥2} \frac{1}{u_{ii}} (\frac{α(α − 1)(w_{i} − u_{i})^{2}}{(w − u)^{2}} + \frac{2αβ(w_{i} − u_{i})}{w − u} + β^{2}x_{i}^{2})

≥ −\frac{α n}{w − u} − \frac{α(w_{1} − u_{1})^{2}}{(w − u)^{2}u_{11}} + \frac{(n − 1)g_{11}}{g u_{11}} + \frac{2g_{1}}{g u_{11}} (α\frac{w_{1} − u_{1}}{w − u} + βx_{1})

+ (β − \frac{β^{2}R^{2}}{α − 1}) ∑_{i≥2} \frac{1}{u_{ii}} .

An optimal choice for β is therefore (α − 1)/(2R^{2}). Multiplying both sides of the obtained inequality by 4R^{2}(w − u)^{n}g u_{11}/(n − 1) and observing that the inequality between arithmetic and geometric means gives

∑_{i≥2} \frac{1}{u_{ii}} ≥ \frac{(n − 1)(u_{11})^{1/(n−1)}}{g} ,

we get at y

(α − 1) ((w − u)^{n−1}u_{11})^{n/(n−1)} − C_{1}(w − u)^{n−1}u_{11} − C_{2} ≤ 0,

where C_{1}, C_{2} are positive constants under control. From this the desired estimate easily follows. □

Note, that in fact in Theorem 3.1 we could replace ||ψ^{1/(n−1)}||_{C^{1,1}(\overline{Ω})} with ||ψ^{1/(n−1)}||_{C^{0,1}(\overline{Ω})} and the maximal eigenvalue of −D^{2}(ψ^{1/(n−1)}) in Ω.

4. C^{1,1} regularity of generalized solutions

First we prove Theorem 1.1:

Proof of Theorem 1.1. We may assume that ψ ∈ C^{1,1}(\overline{Ω}) - otherwise shrink Ω a little. Fix ε > 0. Let Ω_{j} be a sequence of C^{∞} strongly convex domains such that Ω_{j} ↑ Ω as j ↑ ∞. We can find ψ_{j} ∈ C^{∞}(\overline{Ω}_{j}) such that ψ_{j} > 0, ψ_{j} tends uniformly to ψ in Ω and

||ψ_{j}^{1/(n−1)}||_{C^{1,1}(\overline{Ω}_{j})} ≤ C_{1}.

(By C_{1}, C_{2}, … we will denote constants independent of j.) By 4) one can find u_{j} ∈ C^{∞}(\overline{Ω}_{j}), convex in Ω_{j} such that det D^{2}u_{j} = ψ_{j} in Ω_{j} and u_{j} = 0 on ∂Ω_{j}.

From the comparison principle we can easily get that u_{j} tends uniformly to u. In particular,

sup_{Ω_{j}} |u_{j}| ≤ C_{2}.

Since u_{j} is convex,

|Du_{j}(x)| ≤ \frac{−u_{j}(x)}{dist(x, ∂Ω_{j})}, x ∈ Ω_{j}.

Thus

|Du_{j}| ≤ C_{3} in {u_{j} < −ε}.

Theorem 3.1 now gives for some α

(ε − u_{j})^{α}|D^{2}u_{j}| ≤ C_{4} in {u_{j} < −ε}.

Hence

|D^{2}u_{j}| ≤ C_{5} in {u_{j} < −2ε}

and u ∈ C^{1,1}({u < −2ε}). □

Theorem 1.2 will be a direct consequence of 2) and the next two results.

Proposition 4.1. Assume that ψ is continuous and let φ be admissible and such that v ∈ C^{1,1}(Ω). Then u < v on the domain consisting of {ψ > 0} ∩ Ω and the connected components of {ψ = 0} which are relatively compact in Ω.

Proof. Since v is a maximal convex function, it is enough to show that u < v on ψ > 0. As the problem is now purely local, we may assume that ψ ≥ a > 0 and |D^{2}v| ≤ M < ∞ on \overline{B}(x_{0}, r). We will proceed in the same way as in [TU, p. 329]. For ε > 0 define

w(x) := ε(|x − x_{0}|^{2} − r^{2}).

Then

det D^{2}(v + w) ≤ c_{n} ∑_{i=1}^{n} ε^{i}M^{n−i} ≤ a

if ε is small enough. From the comparison principle we get u ≤ v+w on \overline{B}(x_{0}, r). □

Remark. If n = 2 then the assumption that v ∈ C^{1,1}(Ω) in Proposition 4.1 is superfluous. For by an old result of Aleksandrov [Al] u is then strictly convex in {ψ > 0} ∩ Ω. If x_{0} ∈ {ψ > 0} ∩ Ω and H = graph L is a supporting hyperplane of graph v at x_{0} then by [TU, Lemma 2.1] x_{0} belongs to the convex hull of the set {v = L} ∩ ∂Ω. We can thus find y_{1}, y_{2} ∈ ∂Ω such that v is affine on [y_{1}, y_{2}] and x_{0} ∈ [y_{1}, y_{2}]. From the strict convexity of u near x_{0} it now follows that u(x_{0}) < v(x_{0}) which proves the claim.

However, the following example due to Pogorelov shows that this assumption cannot be dropped if n ≥ 3. Set

u(x) = (1 + x_{1}^{2})(x_{2}^{2} + ⋯ + x_{n}^{2})^{1−1/n}.

Then in a neighborhood of the origin u is convex (if n ≥ 3), M(u) ∈ C^{∞}, M(u) > 0. But if Ω is a small ball centered at the origin, we will always have u = v = 0 on {x_{2} = ⋯ = x_{n} = 0} ∩ Ω.

We shall now show that to get u ∈ C^{1,1}({u < v}) one needs only very mild assumptions on Ω and φ:

Theorem 4.2. Assume that Ω is strongly convex and let φ ∈ C(∂Ω) be such that v ∈ C^{0,1}(\overline{Ω}). Then, if ψ^{1/(n−1)} ∈ C^{1,1}(\overline{Ω}), we have u ∈ C^{1,1}(\{u < v\}).

Proof. Regularizing the defining function for Ω we get C^{∞} strongly convex domains Ω_{j} = {w_{j} < 0} such that

(4.1)  D^{2}w_{j} ≥ \frac{1}{C_{1}} I on \overline{Ω}_{j},

and, since convex functions are locally Lipschitz,

(4.2)  |Dw_{j}| ≤ C_{2}, on ∂Ω_{j},

where C_{1}, C_{2}, … are positive constants independent of j and I is the unitary matrix. Regularizing v we get v_{j} ∈ C^{∞}(\overline{Ω}_{j}), convex, converging locally uniformly to v, and such that

(4.3)  |Dv_{j}| ≤ C_{3} on \overline{Ω}_{j}.

We can also find ψ_{j} ∈ C^{∞}(\overline{Ω}_{j}), ψ_{j} > 0, ψ_{j} tending uniformly to ψ in Ω and such that

||ψ_{j}^{1/(n−1)}||_{C^{1,1}(\overline{Ω}_{j})} ≤ C_{4}.

Let u_{j} ∈ C^{∞}(\overline{Ω}_{j}) be convex in Ω_{j} and such that det D^{2}u_{j} = ψ_{j} in Ω_{j}, u_{j} = v_{j} on ∂Ω_{j}. We could choose ψ_{j} so that u_{j} < v_{j} in Ω_{j}. Of course u_{j} tends uniformly to u. By (4.1)

v_{j} + C_{5}w_{j} ≤ u_{j} ≤ v_{j} on \overline{Ω}_{j}.

By (4.2) and (4.3) we thus get

|Du_{j}| ≤ C_{6} on \overline{Ω}_{j}.

We may now use Theorem 3.1 to get

(v_{j} − u_{j})^{α}|D^{2}u_{j}| ≤ C_{7} on \overline{Ω}_{j}.

The required result follows if we let j ↑ ∞. □

Finally, we want to show that if we drop the assumption (1.3) in Theorem 1.2 then it may happen that {u = v} ∩ Ω ≠ ∅. Let Ω be the unit ball B and set φ(x) := x_{1}^{2} for x ∈ ∂B, so that v(x) = x_{1}^{2} for x ∈ \overline{B}. Let ψ be such that ψ^{1/(n−1)} is smooth and supp ψ ⊂ {x_{1} > 0} ∩ B. For ε > 0 let u_{ε} denote the solution of (1.1) with μ = εψ dλ. Then u_{ε} ↑ v uniformly as ε ↓ 0. For ε sufficiently small we thus have u_{ε} ≥ 0 on supp ψ and then one can easily show that u_{ε} = v on {x_{1} ≤ 0} ∩ B.

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Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland, Currently at the Mathematical Institute of the Polish Academy of Sciences

E-mail address: blocki@im.uj.edu.pl