TRANSLATING SOLUTIONS FOR GAUSS CURVATURE FLOWS WITH NEUMANN BOUNDARY CONDITION

OLIVER C. SCHNÜRER AND HARTMUT R. SCHWETLICK

Abstract. We consider the evolution of strictly convex hypersurfaces driven by the non-parametric logarithmic Gauß curvature flow subject to a Neumann boundary condition. In particular, solutions exist globally and converge smoothly to translating solutions.

1. Introduction

In this paper we consider the evolution of non-parametric strictly convex hypersurfaces in a cylinder Ω × ℝ ⊂ ℝ^{n+1} driven by the non-parametric logarithmic Gauß curvature flow subject to a Neumann boundary condition. For this flow equation we show that solutions to the corresponding initial value problem exist for all time and converge smoothly to a solution that moves by translation.

To be more precisely, we address to the following slightly more general problem: Let Ω ⊂ ℝ^{n} be a smooth strictly convex bounded domain. (In this paper we use strictly convex provided the respective principal curvatures are all positive.) Let f : \overline{Ω} × ℝ^{n} → ℝ be a smooth positive function. At the boundary we have a given smooth function φ : ∂Ω → ℝ. Assume that u_{0} : \overline{Ω} → ℝ is a smooth strictly convex function that satisfies the boundary condition

(u_{0})_{ν} (x) ≡ D_{ν}u_{0}(x) = φ(x) ∀ x ∈ ∂Ω,

where ν is the inner unit normal to Ω and indices will here and later denote partial derivatives. We will prove the following

Theorem 1.1. For Ω, f, φ, ν, and u_{0} as introduced above, there exists a global strictly convex solution u : \overline{Ω} × [0, ∞) → ℝ such that

\{ \begin{matrix}u̇ = & log det D^{2}u − log f(x, Du) & in Ω × [0, ∞) \\ u_{ν} = & φ(x) & on ∂Ω × [0, ∞) \\ u|_{t=0} = & u_{0} & in Ω, \end{matrix}  (1.1)

Date: February 2002.

1991 Mathematics Subject Classification. Primary 53C44; Secondary 35K20, 53C42.

where u is smooth for all times t and approaches u_{0} in C^{2} (\overline{Ω}) as t ↓ 0. Moreover, u(·, t) converges smoothly to a translating solution, i. e. to a solution with constant time derivative.

For the problem mentioned above, where the flow is driven by the Gauß curvature, the flow equation takes the form

u̇ = log det D^{2}u − \frac{n+2}{2} log (1 + |Du|^{2}) .  (1.2)

Similar as in [6] it seems possible to us, to use a more general flow equation. This generalization of the theorem above, however, seems to us to be only of technical interest and furthermore it seems easy to combine the techniques of the mentioned paper and those used here to obtain this generalization. For the proof of our Main Theorem 1.1, however, we assume that we have the more general boundary condition

u_{β} = φ(x),

where β is a unit vectorfield which is C^{1} close to ν, i. e., there exists a small positive constant c_{β} > 0 such that ∥ν − β∥_{C^{1}} ≤ c_{β}. Hence, our theorem remains true for such an oblique boundary condition. This generalization of a Neumann boundary condition is also studied in [7].

Translating solutions can also be considered as elliptic solutions. Thus we obtain (for a proof with elliptic methods see Section 4)

Corollary 1.2. For Ω, f, ν, and φ as introduced above, there exists a strictly convex solution (v, u) ∈ ℝ_{+} ×C^{∞} (\overline{Ω}) to the boundary value problem

\{ \begin{matrix}det D^{2}u = & v · f(x, Du) & in Ω, \\ u_{ν} = & φ(x) & on ∂Ω, \end{matrix}  (1.3)

provided that there exists a smooth strictly convex function u_{0} that fulfills the boundary condition (u_{0})_{ν} = φ(x).

A similar situation to that of Theorem 1.1 is considered for the mean curvature flow in [2]. Hypersurfaces of prescribed Gauß curvature subject to a Neumann boundary condition are found in the pioneering paper [4]. This is extended in [7] to the situation of an oblique boundary value condition. A flow approach is used in [6] to solve these kind of equations for Neumann and second boundary value problems. For a second boundary value problem translating solutions for curvature flows are considered in [5].

It is already mentioned in [5] that solutions to the second boundary value problem converge in a similar situation as in Theorem 1.1 to translating solutions. For the non-parametric logarithmic Gauß curvature flow this follows by adapting methods of [6]. For the Neumann boundary value problem, however, a new proof is needed to show that the oscillation of the function, whose graph is the hypersurface, remains bounded. Therefore we establish in Section 3 a generalization of the (spatial) C^{1}-estimates of [4]. The main difference is that in our situation the C^{0}-bounds and even the oscillation estimates are not uniform before we use this new estimate. There is another essential difference compared to the second boundary value problem. From the maximum principle we get that for both the Neumann and for the second boundary value problem, smooth translating solutions are unique up to translation. Especially the final “vertical” velocity is independent of the initial value u_{0}. However, it is not clear to the authors how this final velocity can be determined easily from the data of the problem in the case of a Neumann boundary condition. For the C^{2}-estimates in the presence of a Neumann boundary condition, we need an auxiliary barrier function. To guarantee that these C^{2}-estimates are uniform in time, we need to shift this auxiliary barrier function so that the vertical difference to our solution remains bounded. This can be obtained easily, when the final velocity is known. We need time-independent C^{2}-estimates to prove smooth convergence to a translating solution.

We proceed as follows. As explained in [6] we can use standard parabolic theory to obtain shorttime existence. Therefore we may assume without loss of generality that we have a smooth solution (up to t = 0) for a short time interval. Within Section 2 and 3 we prove time-independent lower order estimates. In Section 4 we show the existence of a unique (up to an additive constant) translating solution using elliptic methods. As soon as we know the velocity of a translating solution, we can prove time-independent a priori C^{2}-estimates in Section 5. The estimates of Krylov, Safonov and Evans and Schauder theory give uniform control for higher derivatives. Finally we prove smooth convergence to a translating solution.

We wish to thank Jürgen Jost, Stefan Müller and the Max Planck Institute for Mathematics in the Sciences for their hospitality. The first author also wishes to acknowledge that this paper was finished while he was at Harvard University, supported by the Humboldt foundation.

2. u̇-estimates

Notation 2.1. We use indices to denote partial derivatives, f_{p_{i}} denotes a derivative with respect to the gradient. For a vector ξ we define u_{ξ} ≡ ξ^{i}u_{i}. For the logarithm of f we use f̂ ≡ log f. We use the Einstein summation convention and sum over repeated upper and lower indices. The inverse of the Hessian of u is denoted by (u^{ij}) = (u_{ij})^{−1}. We remark that – besides in the case u^{ij} – indices are lifted with respect to the Euclidean metric. The letter c denotes a positive constant and may have a new value at each occurrence. Furthermore we may assume that 0 ∈ Ω.

Lemma 2.2. As long as a smooth convex solution of (1.1) exists, we obtain the estimate

min {min_{t=0} u̇, 0} ≤ u̇ ≤ max {max_{t=0} u̇, 0} .

Proof. Similar to [6] we consider

r := (u̇)^{2}.

We get the evolution equation

ṙ = u^{ij}r_{ij} − 2u^{ij}u̇_{i}u̇_{j} − f̂_{p_{i}}r_{i}.  (2.1)

From the evolution equation (2.1) we see that ṙ ≤ 0 in a maximum of r in Ω × [0, t]. Assume that a maximum of r restricted to \overline{Ω} × [0, t] occurs in (t_{0}, x_{0}) with t_{0} > 0 and x_{0} ∈ ∂Ω. If r is constant, then u is a translating solution and our lemma holds. Otherwise we use the Hopf boundary point lemma and get r_{β}(x_{0}) < 0. This is impossible as we get in this case at x_{0}

0 > r_{β} = (\binom{}{u̇}^{2})_{β} = 2u̇u̇_{β} = 2u̇\frac{d}{dt}φ(x) = 0.

In this formula we interchanged time derivatives and spatial derivatives. To do so, we need that u is sufficiently smooth. This is fulfilled since by assumption t_{0} > 0. □

Integrating the obtained estimate yields

Corollary 2.3. As long as a smooth convex solution of (1.1) exists, we obtain the estimate

|u(x, t)| ≤ sup_{Ω} |u_{0}| + sup_{Ω} |u̇(·, 0)| · t.

3. Ice-cream cone estimate

The name for the next theorem comes from the fact that its proof uses cones and balls similarly to their occurrence when ice-cream is in a cone of waffle. The result generalizes the C^{1}-estimates of [4] and is essential for our proof; namely, the C^{0}-estimates obtained from integrating u̇ are much to weak and stronger oscillation estimates are needed.

We wish to mention oscillation estimates of Urbas [8] that can be combined with the C^{1}-estimates of [4] to prove the following Theorem provided that f(x, Du) has the special structure as in the Gauß curvature flow (1.2). More precisely, f(x, Du) must have an appropriate growth in Du. It is interesting to observe that the oscillation estimate of Urbas uses the convexity of u and the equation of prescribed Gauß curvature, whereas Theorem 3.1 combines convexity and the boundary condition.

Theorem 3.1 (Ice-cream cone estimate). Let Ω ⊂ ℝ^{n} be a smooth bounded domain, u : \overline{Ω} → ℝ a smooth strictly convex function with |u_{β}| uniformly bounded on ∂Ω, where β is a unit vector field on ∂Ω such that ⟨β, ν⟩ ≥ c̃_{β} for a positive constant c̃_{β} > 0 (recall that ν is the inner unit normal to ∂Ω). Then there is a uniform bound for sup |Du| independent of sup |u|.

In view of Lemma 2.2 the result above concludes an estimate for the full C^{1}- norm of solutions u to (1.1). Note that only the estimate for the derivatives of u is uniform in time.

Proof. We argue by contradiction. Assume that there exists a point x_{0}, where |Du| is maximal and equals M which is assumed to be sufficiently large. We will show that this leads to a contradiction if M is larger than a suitably chosen constant M_{0}. As u is strictly convex we see that x_{0} ∈ ∂Ω. At x_{0} we find a tangential direction ξ_{0} such that ⟨Du(x_{0}), ξ_{0}⟩ is maximal compared to all other tangential directions there. Here and later we denote unit vectors as directions. We wish to prove a lower estimate for this quantity in terms of M. Let ξ_{1} be a direction such that ⟨Du(x_{0}), ξ_{1}⟩ = |Du|. Similar to [7] we decompose a direction ξ using β and a tangential vector τ(ξ) as

ξ = τ(ξ) + \frac{⟨ν, ξ⟩}{⟨β, ν⟩}β,  (3.1)

where

τ(ξ) = ξ − ⟨ν, ξ⟩ν − \frac{⟨ν, ξ⟩}{⟨β, ν⟩}β^{T} , β^{T} = β − ⟨β, ν⟩ν.

Note that |τ(ξ)| is a priori bounded. Decomposing ξ_{1} we get

M = ⟨Du, ξ_{1}⟩ = ⟨Du, τ(ξ_{1})⟩ + \frac{⟨ν, ξ_{1}⟩}{⟨β, ν⟩}⟨Du, β⟩
≤ |τ(ξ_{1})| · max_{{τ∈Tx_{0} ∂Ω \atop |τ|=1}}⟨Du, τ⟩ + c
= |τ(ξ_{1})| · ⟨Du(x_{0}), ξ_{0}⟩ + c

and deduce that ⟨Du(x_{0}), ξ_{0}⟩ ≥ \frac{M}{c} for M ≥ M_{0}, provided that M_{0} is sufficiently large. For a direction ξ near ξ_{0}, say |ξ − ξ_{0}| < ε = \frac{1}{2c} < 1, we obtain

⟨Du(x_{0}), ξ⟩ = ⟨Du, ξ_{0}⟩ + ⟨Du, ξ − ξ_{0}⟩ ≥ \frac{M}{c} − M|ξ − ξ_{0}| ≥ εM. (3.2)

From the convexity of u we deduce that ⟨Du(y), ξ⟩ ≥ εM for all points y ∈ \overline{Ω} of the form y = x_{0} + λ · ξ. Here λ > 0 and ξ are chosen such that |ξ − ξ_{0}| ≤ ε and x_{0} + t · λ · ξ ∈ \overline{Ω} for all t ∈ [0, 1].

We deduce from the smoothness of Ω and the uniform boundedness of the principal curvatures of ∂Ω ⊂ ℝ^{n+1} that there exists R > 0 and x_{1} ∈ ∂Ω with |x_{0} − x_{1}| > 2R such that especially |Du| ≥ εM in ∂Ω ∩ B_{R}(x_{1}). Assume moreover that any x ∈ B_{R}(x_{1}) ∩ ∂Ω is of the form x_{0} + λ · ξ as described above with |ξ − ξ_{0}| ≤ ε. Due to our construction we have

min_{x∈B_{R}(x_{1})∩∂Ω} u(x) > u(x_{0}).  (3.3)

Because of the shape of the set of directions ξ satisfying (3.2) together with the ball mentioned above, we call our theorem the ice-cream cone estimate.

Figure 1. Ice-cream cone estimate

Figure 1 shows part of ∂Ω and two cones corresponding to the directions ξ as well as two pairs of concentric balls. The larger ones are the balls B_{R} mentioned above, the smaller ones are introduced in the following.

Now we may proceed iteratively. Note that R and ε can be chosen as fixed constants independent of the point x_{0}. As long as |Du(x_{i})| ≥ Mε^{i} ≥ M_{0} we can find a further point x_{i+1} in the same way as we found x_{1} starting from x_{0}. Thus for M = sup |Du| sufficiently large we can construct a sequence of points {x_{i}}_{i=0,...,N} of arbitrarily large length N satisfying for all i ≥ 1

|Du| ≥ Mε^{i} on ∂Ω ∩ B_{R}(x_{i})

and

min_{x∈B_{R}(x_{i})∩∂Ω} u(x) > u(x_{i−1}).

Since ∂Ω has finite measure and bounded principal curvatures there is an upper bound N_{0}(ρ) on the number of pairwise disjoint restricted balls B_{ρ}(y_{j}) ∩ ∂Ω for fixed ρ > 0 and y_{j} ∈ ∂Ω.

Hence, if M = sup |Du| > M_{0}ε^{−N_{0}(\frac{R}{3})} there will be two points x_{i_{0}}, x_{j_{0}} with i_{0} > j_{0} > 0 such that

B_{\frac{R}{3}} (x_{i_{0}}) ∩ B_{\frac{R}{3}} (x_{j_{0}}) ∩ ∂Ω ≠ ∅.

From x_{j_{0}} ∈ B_{R}(x_{i_{0}}) we deduce a contradiction since

u(x_{j_{0}}) < u(x_{j_{0}+1}) < … < u(x_{i_{0}−1}) < min_{x∈B_{R}(x_{i_{0}})∩∂Ω} u(x) ≤ u(x_{j_{0}})

is clearly impossible. □

4. Existence of translating solutions

This section is devoted to finding solutions to the elliptic equation

\{ \begin{matrix}v = & log det D^{2}u − log f(x, Du) & in Ω, \\ u_{β} = & φ & on ∂Ω. \end{matrix}  (4.1)

The absence of any monotonicity property (wrt. u) both in f as well as in the boundary condition φ limits seriously the existence of solutions.

Step 1: Here we show that for given ε > 0 and v ∈ ℝ there is a unique solution u_{ε,v} of

\{ \begin{matrix}det D^{2}u = & e^{v}f(x, Du)e^{εu} & in Ω, \\ u_{β} = & φ & on ∂Ω. \end{matrix}  (∗_{ε,v})

Note that the dependence on v is continuous and strictly decreasing. In fact we have the explicit relation

u_{ε,v} = u_{ε,0} − v/ε.

To show the unique existence of u_{ε,v} we will derive below an a priori sup bound, then the ice-cream cone estimate gives the C^{1}-bound. Having the full C^{1}-norm controlled we can estimate the C^{2}-norm exactly as in Urbas [7], since the strict monotonicity assumption on the boundary condition therein is not used in this part. A detailed proof for this is given in the next section where we extend the argument to the parabolic case. Bounds for higher C^{k}-norms follow via the estimates due to Krylov, Safonov, Evans, and from Schauder theory.

To get the sup bound we define suitable barriers for (∗_{ε,0}). Recall that u_{0} is a convex function satisfying (u_{0})_{β} = φ on ∂Ω. Define u_{ε}^{±} = u_{0} ± M/ε, where M > 0 will be chosen later. Then

\frac{det D^{2}u_{ε}^{±}}{f(x, Du_{ε}^{±})e^{εu_{ε}^{±}}} = \frac{det D^{2}u_{0}}{f(x, Du_{0})e^{εu_{0}±M}} = g(x)e^{−εu_{0}}e^{∓M},

with c^{−1} < g(x) < c. Hence for ε < 1 there exists a large constant M > 0 such that u_{ε}^{+} is a strict supersolution and u_{ε}^{−} is a strict subsolution of (∗_{ε,0}). This implies, that

u_{ε}^{−} < u_{ε,0} < u_{ε}^{+},  (4.2)

or equivalently,

|u_{ε,v} − (u_{0} − \frac{v}{ε})| < \frac{M}{ε} .

Step 2: Now we consider the limit ε → 0. In general, we cannot expect that the sup bounds for u_{ε,v} can be obtained uniformly in ε. In fact it follows from the maximum principle that only for a unique v there is a solution to (∗_{ε,v}) with ε = 0. Observe that (4.2) implies that

u_{ε,+M} < u_{0} < u_{ε,−M}.

Therefore we can find for every ε > 0 a unique v_{ε} ∈ (−M, M) such that u_{ε,v_{ε}}(0) = u_{0}(0). Note that (4.2) does not suffice to control the oscillation of u_{ε,v_{ε}} uniformly in ε. We employ the ice-cream cone estimate to bound u_{ε,v_{ε}} uniformly in C^{1}. Again, uniform C^{1}-bounds imply uniform higher C^{k}-bounds.

Now we choose a sequence ε_{i} → 0 as i → ∞. Since v_{ε_{i}} is bounded, there is a subsequence (relabeled) such that v_{ε_{i}} → v^{∞} and u_{ε_{i},v_{ε_{i}}} → u_{ell}^{∞} in any C^{k}-norm.

The extension u^{∞}(x, t) := u_{ell}^{∞}(x) + v^{∞}t is a translating solution as by construction u^{∞} satisfies

\{ \begin{matrix}v^{∞} = u̇^{∞} = & log det D^{2}u^{∞} − log f(x, Du^{∞}) & in Ω, \\ u_{β}^{∞} = & φ(x) & on ∂Ω. \end{matrix}

5. Parabolic C^{2}-estimates

We remark that this proof is an adaption of the respective proofs in [4, 6, 7]. It seems important to us for the proof of the a priori C^{2}-estimates to have a translating solution u^{∞} and especially to know its velocity v^{∞}.

We will construct an auxiliary function for these estimates. Assume that u_{ell}^{∞} > u_{0}. We define

φ̃(x, z) = φ(x) + (z − u_{ell}^{∞}).

Due to the uniform estimates for the norm of the gradient of u we can find μ_{0} such that

min{f(x, Du), f(x, Du_{ell}^{∞})} ≥ μ_{0}.

Consider the following boundary value problem for 0 < ρ < 1

\{ \begin{matrix}v^{∞} = & log det D^{2}ψ − log \frac{μ_{0}}{2} & in Ω, \\ ψ_{β} = & φ̃ (x, ψ + ρ · |x|^{2}) − 2ρ ⟨x, β⟩ & on ∂Ω. \end{matrix}  (5.1)

We wish to show uniform a priori C^{2}-estimates for ψ. Lemma 3.1 gives estimates for the gradient and estimates for the second derivatives follow from C^{1}-estimates and [7]. Thus it remains to prove uniform C^{0}-estimates. As φ̃(x, z) → ∞ uniformly as z → ∞, we get an upper bound for ψ, as any strictly convex solution has to fulfill ψ_{β} < 0 somewhere on ∂Ω. The lower bound follows using the maximum principle as follows. For ψ − u_{ell}^{∞} we get the differential inequality

\{ \begin{matrix}0 & > & log det D^{2}ψ − log det D^{2}u_{ell}^{∞} & in Ω, \\ (ψ − u_{ell}^{∞})_{β} & = & ψ − u_{ell}^{∞} + ρ · |x|^{2} − 2ρ ⟨x, β⟩ & on ∂Ω. \end{matrix}

Thus ψ − u^{∞} cannot attain an interior minimum. If a minimum occurs on ∂Ω, we get

0 ≤ (ψ − u^{∞})_{β} = ψ − u^{∞} + ρ · |x|^{2} − 2ρ ⟨x, β⟩.

Thus ψ is uniformly bounded below and we can solve (5.1). Due to the uniform C^{2}-estimates, we can fix λ > 0 such that

ψ_{ij} ≥ λδ_{ij}  (5.2)

in the matrix sense. Further on, these estimates allow to fix ρ > 0 such that \overline{ψ} := ψ + ρ · |x|^{2} satisfies

\{ \begin{matrix}v^{∞} & > & log det D^{2}\overline{ψ} − log μ_{0} & in Ω, \\ \overline{ψ}_{β} & = & φ̃ (x, \overline{ψ}) & on ∂Ω. \end{matrix}

Now we apply the maximum principle and get u_{ell}^{∞} ≤ \overline{ψ}. We extend ψ and \overline{ψ} by setting ψ(x, t) := ψ(x) + t · v^{∞} and \overline{ψ}(x, t) := \overline{ψ}(x) + t · v^{∞} and see that u ≤ u^{∞} ≤ \overline{ψ}, where u^{∞} is defined as in Section 4. So we get

\overline{ψ}_{β} − u_{β} = \overline{ψ}_{β}(·, 0) − φ(x) = (\overline{ψ} − u^{∞}) (·, 0) = \overline{ψ} − u^{∞} ≥ 0

and furthermore for a sufficiently small δ_{0} > 0

(ψ − u)_{β} = (\overline{ψ} − ρ · |x|^{2} − u)_{β} ≥ −2ρ ⟨x, β⟩ ≥ δ_{0} > 0,  (5.3)

provided that β is C^{0}-close to ν. Here we used that 0 ∈ Ω and thus we have ⟨x, ν⟩ < 0. Using these preparations, we are able to prove C^{2}-a priori estimates similar to [6, 7]. For the reader’s convenience, we repeat the arguments.

5.1. Preliminary results. We use τ for a direction tangential to ∂Ω.

Lemma 5.1 (Mixed C^{2}-estimates at the boundary). Let u be a solution of our flow equation (1.1). Then the absolute value of u_{τβ} remains a priori bounded on ∂Ω during the evolution.

Proof. We represent ∂Ω locally as graph ω over its tangent plane at a fixed point x_{0} ∈ ∂Ω such that locally Ω = {(x̂, x^{n}) : x^{n} > ω(x̂)}. We extend β smoothly and differentiate the oblique boundary condition

β^{i}(x̂)u_{i}(x̂, ω(x̂)) = φ(x̂, ω(x̂)), x̂ ∈ ℝ^{n−1},

with respect to x̂^{j}, 1 ≤ j ≤ n − 1,

β_{j}^{i}u_{i} + β^{i}u_{ij} + β^{i}u_{in}ω_{j} = φ_{j} + φ_{n}ω_{j}

and obtain at x_{0} ≡ (x̂_{0}, ω(x̂_{0})) ∈ ∂Ω a bound for β^{i}u_{ij} in view of the gradient estimates and Dω(x̂_{0}) = 0. Multiplying with τ^{j} gives the result. We remark that it is only possible to multiply the equation with a tangential vector as the differentiation with respect to x̂^{j} and so also j correspond to tangential directions. □

Lemma 5.2 (Double oblique C^{2}-estimates at the boundary). For any solution of the flow equation (1.1) the absolute value of u_{ββ} is a priori bounded from above on ∂Ω. (u_{ββ} > 0 also follows from the strict convexity of a solution.)

Proof. We assume the same geometric situation as in the proof of Lemma 5.1 with x_{0} ∈ ∂Ω. From (1.1) we obtain

u̇_{k} = u^{ij}u_{ijk} − (f̂_{k} + f̂_{p_{i}}u_{ik})

and define therefore

Lw := ẇ − u^{ij}w_{ij} + f̂_{p_{i}}w_{i},

where we evaluate the terms by using the function u. From the definition of L it is easy to see that for appropriate extensions of β and φ

|L (β^{k}u_{k} − φ(x))| ≤ c · (1 + tr u^{ij}) .

We define Ω_{δ} := Ω ∩ B_{δ}(x_{0}) for δ > 0 sufficiently small and set

ϑ := d − μd^{2}

for μ ≫ 1 sufficiently large where d denotes the distance from ∂Ω. We will show that Lϑ ≥ \frac{ε}{3}tr u^{ij} for a small constant ε > 0 (depending only on a positive lower bound for the principal curvatures of ∂Ω) in Ω_{δ}.

Lϑ = −u^{ij}d_{ij} + 2μu^{ij}d_{i}d_{j} + 2μu^{ij}dd_{ij} + f̂_{p_{i}}(d_{i} − 2μdd_{i})
≥ −u^{ij}d_{ij} + 2μu^{ij}d_{i}d_{j} − cμd (1 + tr u^{ij}) − c.

We use the strict convexity of ∂Ω, d_{i} ≈ δ_{in}, |u^{kl}| ≤ tr u^{ij}, 1 ≤ k, l ≤ n, and the inequality for arithmetic and geometric means

Lϑ ≥ ε tr u^{ij} + μu^{nn} − cμδ (1 + tr u^{ij}) − c
≥ \frac{n}{3} (det u^{ij})^{\frac{1}{n}} · ε^{\frac{n−1}{n}} · μ^{\frac{1}{n}} + \frac{2}{3}ε tr u^{ij}  (5.4)
− cμδ (1 + tr u^{ij}) − c.

Let us explain more precisely how we used the inequality for arithmetic and geometric means. We get immediately from the inequality for arithmetic and geometric means

\frac{ε tr u^{ij} + μu^{nn}}{3} ≥ \frac{n}{3}ε^{\frac{n−1}{n}} μ^{\frac{1}{n}} ∏_{i=1}^{n} u^{ii}.  (5.5)

We may assume that (u^{ij})_{i,j<n} is diagonal and combine (5.5) with

det u^{ij} = det ( \begin{matrix}u^{11} & 0 & ⋯ & 0 & u^{1n} \\ 0 & ... & ... & ⋮ & ⋮ \\ ⋮ & ... & ... & 0 & ⋮ \\ 0 & ⋯ & 0 & u^{n−1 n−1} & u^{n−1 n} \\ u^{1n} & ⋯ & ⋯ & u^{n−1 n} & u^{nn} \end{matrix} )

= ∏_{i=1}^{n} u^{ii} − ∑_{i<n} |u^{ni}|^{2} ∏_{{j≠i \atop j<n}} u^{jj} ≤ ∏_{i=1}^{n} u^{ii} .  (5.6)

So we get the inequality involving tr u^{ij} and det u^{ij} used above.

As det u^{ij} is a priori bounded from below by a positive constant in view of

det u^{ij} = (det u_{ij})^{−1} = exp (−f̂− u̇) ,

we may choose μ so large that the first term in (5.4) is greater than c + 1. For δ ≤ \frac{1}{cμ} min \{1, \frac{1}{3}ε\} we get

Lϑ ≥ \frac{1}{3}ε tr u^{ij}

and furthermore ϑ ≥ 0 on ∂Ω_{δ} if we choose δ smaller if necessary.

Let l be an affine linear function such that we have ±(β^{i}u_{i} − φ(x)) + l ≥ 0 for t = 0 in Ω_{δ} and l(x_{0}) = 0. For constants A, B > 0 consider the function

Θ := Aϑ + B|x − x_{0}|^{2} ± (β^{i}u_{i} − φ(x)) + l.

We fix B ≫ 1, get Θ ≥ 0 on ∂Ω_{δ}, and deduce for A ≫ B that LΘ ≥ 0 as tr u^{ij} is bounded from below by a positive constant. The maximum principle yields Θ ≥ 0 in Ω_{δ}. As Θ(x_{0}) = 0 we have Θ_{β}(x_{0}) ≥ 0 which in turn gives immediately |u_{ββ}| ≤ c. □

5.2. Remaining C^{2}-estimates. We proceed similar as in [7]. Take T > 0 slightly smaller than the maximal time interval for which a solution exists. We define for (x, τ, t) ∈ \overline{Ω} × S^{n−1} × [0, T] and for positive constants α, γ to be fixed later

w(x, τ, t) := e^{α(ψ−u)+γ·|Du|^{2}} · u_{ττ} .

We assume that w attains its maximum over all points of ∂Ω and all tangential directions τ at a boundary point x_{w}, in a direction we may take to be e_{1} and for t_{0} > 0. We may choose Euclidean coordinates such that (u_{ij})(x_{w}) is diagonal for 1 ≤ i, j ≤ n − 1, where e_{1}, …, e_{n−1} are tangential directions. We decompose e_{1} similar as in (3.1)

e_{1} = τ(e_{1}) + \frac{⟨ν, e_{1}⟩}{⟨β, ν⟩}β,

where

τ(e_{1}) = τ = e_{1} − ⟨ν, e_{1}⟩ν − \frac{⟨ν, e_{1}⟩}{⟨β, ν⟩}β^{T} , β^{T} = β − ⟨β, ν⟩ν.

We differentiate the boundary condition

u_{β} = φ(x) on ∂Ω

and multiply the result with the tangential vector τ. Assume that all involved quantities defined on ∂Ω have been extended smoothly. On ∂Ω we obtain

\frac{2⟨ν, e_{1}⟩}{⟨ν, β⟩}u_{βτ} = \frac{2⟨ν, e_{1}⟩}{⟨ν, β⟩} (φ_{j}τ^{j} − τ^{j}β_{j}^{i}u_{i}) =: χ(x, Du).  (5.7)

We remark that χ as defined above is affine linear in Du. From the definitions of τ and χ we get

u_{11} = u_{ττ} + χ + \frac{⟨ν, e_{1}⟩^{2}}{⟨β, ν⟩^{2}} u_{ββ}.

Since χ(x_{w}, Du) = 0, the function

w̃ := e^{α(ψ−u)+γ|Du|^{2}}(u_{11} − χ)

satisfies for x ∈ ∂Ω

w̃(x) = e^{α(ψ−u)+γ|Du|^{2}}(x) \{u_{ττ} (x) + \frac{⟨ν, e_{1}⟩^{2}}{⟨β, ν⟩^{2}} u_{ββ}(x)\}

≤ \{1 − ⟨ν, e_{1}⟩^{2} (1 − \frac{|β^{T} |^{2}}{⟨β, ν⟩^{2}} ) − \frac{2⟨ν, e_{1}⟩ ⟨β^{T} , e_{1}⟩}{⟨β, ν⟩}\} w̃(x_{w})
+ c⟨ν, e_{1}⟩^{2}e^{α(ψ−u)+γ|Du|^{2}}(x)

≤ \{1 + c⟨ν, e_{1}⟩^{2} − \frac{2⟨ν, e_{1}⟩ ⟨β^{T} , e_{1}⟩}{⟨β, ν⟩} + c \frac{⟨ν, e_{1}⟩^{2}}{max_{{ξ∈Tx∂Ω \atop |ξ|=1}} u_{ξξ}(x)}\} w̃(x_{w})

≤ \{1 + c_{1}⟨ν, e_{1}⟩^{2} − \frac{2⟨ν, e_{1}⟩ ⟨β^{T} , e_{1}⟩}{⟨β, ν⟩}\} w̃(x_{w}).  (5.8)

The proof of (5.8) is different to the proof in [7]. In the respective proof in the cited paper it was not clear to us why the constant could be chosen independent of α and γ. Moreover, for (5.8) we used

Lemma 5.3. For a solution of our flow equation (1.1)

max_{{ξ∈Tx∂Ω \atop |ξ|=1}} u_{ξξ}(x)

is uniformly bounded from below by a positive constant.

Proof. We have already seen that there is a uniform positive lower bound for det D^{2}u. In a fixed boundary point we may choose a coordinate system, such that e_{n} is equal to the inner unit normal of ∂Ω and (u_{ij})_{i, j<n} is diagonal. Similar as in (5.6) we estimate

det u_{ij} = ∏_{i=1}^{n} u_{ii} − ∑_{i<n} |u_{ni}|^{2} ∏_{{j≠i \atop j<n}} u_{jj} ≤ ∏_{i=1}^{n} u_{ii} .  (5.9)

We decompose ν as

ν = \frac{1}{⟨β, ν⟩} (β − β^{T} )

with β^{T} as above and get immediately in view of the Lemmata 5.1 and 5.2

u_{νν} ≤ \frac{1}{⟨β, ν⟩^{2}} (|β^{T} |^{2} · max_{i<n} u_{ii} + c) .

Finally, we combine the last statement with (5.9). The claim follows. □

We may assume that c_{1} in (5.8) is chosen so large and β is so close to ν such that the expression in the last curly brackets in (5.8) is bounded below by \frac{1}{2}.

We use χ as above and define

W = \frac{e^{α(ψ−u)+γ|Du|^{2}}(u_{11} − χ)}{1 + c_{1}⟨ν, e_{1}⟩^{2} − \frac{2⟨ν,e_{1}⟩⟨β^{T} ,e_{1}⟩}{⟨β,ν⟩}}

and consider this quantity in \overline{Ω} × [0, T] for some positive time T. We may assume that W attains its maximum at a positive time at x_{W} ∈ \overline{Ω}.

We first address to the case x_{W} ∈ ∂Ω. Observe that W (x_{W}) ≤ w̃(x_{w}) = W (x_{w}), where we evaluate at appropriate times. So we get W_{β} ≤ 0 at x_{w}, which implies that

u_{11β} + αδ_{0}u_{11} ≤ c(1 + (1 + γ)u_{11}).  (5.10)

We differentiate the boundary condition u_{β} = φ(x) as in Lemma 5.1, use the a priori estimates obtained so far, and the fact that D^{2}u restricted to tangential directions is diagonal

u_{β11} ≥ −c − 2β_{1}^{1}u_{11} − 2β_{1}^{n}u_{n1}.  (5.11)

For β C^{1}-close to ν we see that β_{1}^{1} ≈ −ω_{11} < 0, β_{1}^{n} ≈ 0, so combining (5.10) and (5.11) yields

c(1 + (1 + γ)u_{11}) ≥ αδ_{0}u_{11} − c.

So we see that for α = α(γ) sufficiently large we get an upper bound for u_{11}(x_{w}). This completes the C^{2}-estimates, when W attains its maximum on ∂Ω.

Now we consider the case when W attains its maximum at x_{W} ∈ Ω. We use

Γ = − log (1 + c_{1}⟨ν, e_{1}⟩^{2} − \frac{2⟨ν, e_{1}⟩ ⟨β^{T} , e_{1}⟩}{⟨β, ν⟩})

in the following calculations and remark that Γ is well-defined as the argument of the logarithm is bounded below by a positive constant and its C^{2} (\overline{Ω})-norm is uniformly bounded independent of α and γ. We use that

log W = α · (ψ − u) + γ · |Du|^{2} + log(u_{11} − χ) + Γ

attains its maximum at x_{W}. Of course we may assume that (u_{11} −χ)(x_{W}) ≥ 1. At the point x_{W} we get

0 = \frac{W_{i}}{W} = α(ψ − u)_{i} + 2γu^{k}u_{ki} + \frac{u_{11i} − D_{i}χ}{u_{11} − χ} + Γ_{i},
0 ≥ \frac{W_{ij}}{W} − \frac{W_{i}W_{j}}{W^{2}}
= α(ψ − u)_{ij} + 2γu_{j}^{k}u_{ki} + 2γu^{k}u_{kij}

+ \frac{u_{11ij} − D_{ij}χ}{u_{11} − χ} − \frac{(u_{11i} − D_{i}χ)(u_{11j} − D_{j}χ)}{(u_{11} − χ)^{2}} + Γ_{ij},

0 ≤ Ẇ = α (ψ̇ − u̇) + 2γu^{k}u̇_{k} + \frac{u̇_{11} − \frac{d}{dt}χ}{u_{11} − χ}

where we have used that Γ is time-independent. The second line has to be interpreted in the matrix sense. The notations D_{·} and \frac{d}{dt} indicate that the chain rule has not yet been applied. The remaining calculations refer to the point, where W attains its maximum. We get there

0 ≥ u^{ij}(log W)_{ij} − Ẇ .

Using estimates for the time derivatives of ψ and u, the strict convexity of ψ (5.2), the fact that Γ ∈ C^{2} with uniform bounds and the up to twice differentiated flow equation (1.1) yields

0 ≥ 2γΔu + \frac{1}{u_{11} − χ} (u^{ir}u^{js}u_{ij1}u_{rs1}) − u^{ij} \frac{(u_{11i} − D_{i}χ)(u_{11j} − D_{j}χ)}{(u_{11} − χ)^{2}} (5.12)

+ \frac{1}{u_{11} − χ} (f̂_{p_{i}}u_{i11} − c − c · |D^{2}u|^{2}) + 2γu^{k}f̂_{p_{i}}u_{ik}

+ \frac{1}{u_{11} − χ} (\frac{d}{dt}χ − u^{ij}D_{ij}χ) − c(α + γ) + (αλ − c)tr u^{ij}.

Direct calculations and (1.1) differentiated once yield

\frac{d}{dt}χ − u^{ij}D_{ij}χ ≥ −c (1 + |D^{2}u| + tr u^{ij})

and \frac{W_{i}}{W} = 0 implies

2γu^{k}f̂_{p_{i}}u_{ik} + \frac{f̂_{p_{i}}u_{11i}}{u_{11} − χ} ≥ −c · α − \frac{c · (1 + |D^{2}u|)}{u_{11} − χ} − c.

These estimates are used to simplify (5.12). For a small constant ϑ > 0 to be fixed later on we may assume that

(1 − ϑ)u_{ηη}(x_{W}) ≡ (1 − ϑ) max_{|ξ|=1} u_{ξξ}(x_{W}) ≤ (u_{11} − χ)(x_{W}),

where we have defined a direction η, |η| = 1, implicitly that corresponds to a maximal eigenvalue. Schwarz’s inequality gives

u^{ij}(u_{11i} − D_{i}χ)(u_{11j} − D_{j}χ) ≤ (1 + ϑ)u^{ij}u_{11i}u_{11j} + \frac{c}{ϑ}u^{ij}D_{i}χD_{j}χ.

From the definition of η we get

u^{ir}u^{js}u_{ij1}u_{rs1} ≥ \frac{max_{|ξ|=1} u^{js}u_{ξj1}u_{ξs1}}{u_{ηη}}

≥ \frac{1 − ϑ}{u_{11} − χ}u^{ij}u_{11i}u_{11j}.

We assume that ϑ ≤ \frac{1}{2}, use \frac{W_{i}}{W} = 0 and get using the inequalities above

u^{ir}u^{js}u_{ij1}u_{rs1} − u^{ij} \frac{1}{u_{11} − χ}(u_{11i} − D_{i}χ)(u_{11j} − D_{j}χ)

≥ u^{ir}u^{js}u_{ij1}u_{rs1} − (1 + ϑ) \frac{1}{u_{11} − χ}u^{ij}u_{11i}u_{11j} − \frac{c}{ϑ(1 − ϑ)} \frac{1}{u_{ηη}} u^{ij}D_{i}χD_{j}χ

≥ −ϑ\frac{2}{u_{11} − χ}u^{ij}u_{11i}u_{11j} − \frac{2}{ϑ} \frac{c}{u_{ηη}} (1 + tr u^{ij} + |D^{2}u|)

≥ −cϑ(u_{11} − χ) (tr u^{ij} + α^{2} tr u^{ij} + γ^{2} |D^{2}u|) − \frac{1}{ϑ} \frac{c}{u_{ηη}} (1 + tr u^{ij} + |D^{2}u|) .

Combining this with (5.12) gives

0 ≥ (2γ − c ϑγ^{2} − \frac{c}{ϑ(Δu)^{2}} − c) Δu − c(1 + α + γ)

+ (αλ − c − c ϑα^{2} − \frac{c}{ϑ(Δu)^{2}} − \frac{c}{Δu}) tr u^{ij}

Fixing γ, α = α(γ) sufficiently large and finally ϑ = ϑ(γ, α) sufficiently small gives an upper bound for u_{11}. We remark that the constants have been chosen in the same order as above.

It remains to consider the case when

(1 − ϑ)u_{ηη}(x_{W}) ≥ (u_{11} − χ)(x_{W}).

We may restrict to the non-trivial situation, i. e. we may assume that

(1 − \frac{ϑ}{2}) u_{ηη}(x_{W}) ≥ u_{11}(x_{W}).

We define

\overline{Ω} × S^{n−1} × [0, T] ∋ (x, ξ, t) ↦ W̃ (x, ξ, t) = \frac{e^{α(ψ−u)+γ|Du|^{2}}(u_{ξξ} − χ)}{1 − c_{1}⟨ν, e_{1}⟩^{2} − \frac{2⟨ν,e_{1}⟩⟨β^{T} ,e_{1}⟩}{⟨β,ν⟩}} .

Here we use χ as introduced in (5.7). Assume that W̃ attains its maximum at a positive time at xW̃ for a direction ξ ∈ S^{n−1}. We assume first, that xW̃ ∈ ∂Ω. Using a decomposition of ξ as in (3.1) we obtain for β C^{0}-close to ν

|τ(ξ)|^{2} ≤ 1 + c ∥β^{T} ∥_{C^{0}} .

As a direct consequence of this decomposition it is easy to see that

u_{ξξ} ≤ u_{τ(ξ)τ(ξ)} + c.

In the next estimate we use the choice of x_{w}, x_{W} above (5.10) and remark that the constant c used here may now (in contrast to the inequalities above) also depend on α and γ. We evaluate at a time that corresponds to the choice of x_{·}.

W̃ (xW̃ , ξ) ≤ |τ(ξ)|^{2}W (x_{w}) + c

≤ (1 + c ∥β^{T} ∥_{C^{0}}) W (x_{W}) + c

≤ (1 − \frac{ϑ}{2}) (1 + c ∥β^{T} ∥_{C^{0}}) W̃ (x_{W} , η) + c

≤ (1 − \frac{ϑ}{2}) (1 + c ∥β^{T} ∥_{C^{0}}) W̃ (xW̃ , ξ) + c.

For β^{T} sufficiently small, this is only possible if W̃ (xW̃ , ξ) is a priori bounded. Similar to the above proof for the interior maximum of W it is possible to obtain a bound for the second derivatives, if W̃ attains its maximum at an interior point xW̃ ∈ Ω.

6. Longtime existence and convergence

So far we have obtained uniform u̇, Du and D^{2}u-estimates as long as our solution exists. Furthermore, we know that there is a positive constant c such that

−c + v^{∞} · t ≤ u ≤ +c + v^{∞} · t,  (6.1)

where v^{∞} is the velocity of a translating solution. This is obtained by enclosing our initial values by translating solutions. These translating solutions will then enclose u as long as our solution u exists due to the maximum principle. We can apply Hölder estimates for the second derivatives due to Evans, Krylov and Safonov as well as Schauder estimates, see [3]. As our problem is invariant in “z-direction”, we get longtime existence with uniform bounds for all derivatives of u.

To show convergence to a translating solution we consider w := u − u^{∞}, similar to [1, 5]. Using the mean value theorem we see that w satisfies a parabolic flow equation of the form

\{ \begin{matrix}ẇ = & a^{ij}w_{ij} + b^{i}w_{i} & in Ω, \\ w_{β} = & 0 & on ∂Ω. \end{matrix}

Thus the strong maximum principle implies that w is constant or its oscillation is strictly decreasing in time. In the first case u is already a translating solution. In the second case we wish to exclude that the oscillation is strictly decreasing but does not tend to zero. In the case when the oscillation of w(·, t) tends to ε > 0 as t → ∞, we consider for t_{n} → ∞

u^{n}(x, t) := u(x, t + t_{n}) − v^{∞} · t_{n}.

Due to the uniform estimates in any C^{k}-norm for the derivatives of u^{n}, and the locally (in time) uniform bounds for the C^{0}-norm, see (6.1), a subsequence of the functions u^{n} converges locally (in time) uniformly in any C^{k}-norm to a solution u^{∗} of our flow equation (1.1) that exists for all time. Due to the monotonicity of the oscillation we see that the oscillation of u^{∗} −u^{∞} is equal to ε, independent of t. This is a contradiction as the strong maximum principle implies that a positive oscillation is strictly decreasing. As the oscillation of w tends to zero, we see that u tends to a translating solution as t → ∞ in C^{0}. Adding a constant to the translating solution u^{∞} we may assume that u → u^{∞} as t → ∞ in the C^{0}-norm. Smooth convergence is obtained by using interpolation inequalities of the form

∥Dw∥_{C^{0}(\overline{Ω})}^{2} ≤ c(Ω) · ∥w∥_{C^{0}(\overline{Ω})} · (∥D^{2}w∥_{C^{0}(\overline{Ω})} + ∥Dw∥_{C^{0}(\overline{Ω})})

for w := u − u^{∞}. This completes the proof of Theorem 1.1.

Appendix A. Prescribing Gauß curvature for entire graphs

Here we present an application of our previous existence result on bounded domains to construct unbounded hypersurfaces with prescribed Gauß curvature. Assuming that the hypersurface is given as an entire graph the problem is then to find a solution to

\frac{det D^{2}u}{(1 + |Du|^{2})^{\frac{n+2}{2}}} = g(x).  (A.1)

Observe that this equation fits in the context of the present paper, cf. (4.1)

v^{∞} = log det D^{2}u − log f(x, Du),

just by defining

f(x, p) = g(x)/h(p) with h(p) = h(|p|) = (1 + |p|^{2})^{− \frac{n+2}{2}}

and looking for translating solutions with speed v^{∞} = 0.

Following an argument of Altschuler, Wu [1] we will construct entire rotationally symmetric translating solutions from solutions on growing disk-type domains. Using the graph of the lower half sphere for various radii it is a direct consequence of the strong maximum principle that there cannot exist an entire strictly convex translating solution for any constant value of the Gauß curvature. In the rotationally symmetric case we have the following result.

Theorem A.1. Let g(x) = g(|x|) be positive, smooth and integrable. There is an entire strictly convex solution u to (A.1) if and only if

v := log \frac{∫_{ℝ^{n}} h(|p|)dp}{∫_{ℝ^{n}} g(|x|)dx} ≥ 0.

The solution constructed here has a uniformly bounded gradient if and only if v > 0.

We remark that the rotationally symmetric setting does also allow for a proof by reducing the problem to an ordinary differential equation.

Proof. We will use that for given constants R > 0, ρ > 0 there is a unique, strictly convex solution (v^{∞}, u^{∞}) to the following problem

\{ \begin{matrix}e^{v^{∞}} = & det D^{2}u^{∞} \frac{h(|Du^{∞}|)}{g(|x|)} & in B_{R}(0), \\ u_{ν}^{∞} = & −ρ & on ∂B_{R}(0). \\ u^{∞}(0) = & 0 \end{matrix}  (∗_{R,ρ})

This is a direct application of our elliptic result in Section 4. Furthermore, the uniqueness of the solution also implies its rotational symmetry. The solution could also be found by imposing a second boundary value problem.

From the strict convexity we deduce that Du^{∞} is an isomorphism from B_{R}(0) onto B_{ρ}(0). Integrating (∗_{R,ρ}) we obtain

e^{v^{∞}} ∫_{B_{R}} g(|x|)dx = ∫_{B_{R}} det D^{2}u^{∞}h(|Du^{∞}|)dx = ∫_{B_{ρ}} h(|p|)dp,

which determines the speed

v^{∞} = v^{∞}(R, ρ) := log \frac{∫_{B_{ρ}} h(|p|)dp}{∫_{B_{R}} g(|x|)dx}

uniquely as a function of the parameters R, ρ. Note that v^{∞}(R, ρ) is strictly decreasing in R and strictly increasing in ρ with v^{∞}(R, ρ) → −∞ for ρ → 0.

1. Nonexistence for v < 0: We argue similarly as in the afore mentioned case of constant Gauß curvature. But here we replace lower half spheres by suitably constructed solutions on finite domains with arbitrarily large gradients at the boundary. Since v < 0 there exists a unique R̂ such that ∫_{BR̂} g(|x|)dx = ∫_{ℝ^{n}} h(|p|)dp. Assuming that we have an entire solution u of (A.1), there is a ρ > 0 such that |Du(x)| < ρ/2 for all |x| < R̂. Now take the unique R < R̂ satisfying

∫_{B_{R}} g(|x|)dx = ∫_{B_{ρ}} h(|p|)dp

and consider the solution (v^{∞}, u^{∞}) of (∗_{R,ρ}). By definition we know that v^{∞} = v^{∞}(R, ρ) = 0, hence, u^{∞} solves equation (A.1) in B_{R}(0). Furthermore, the Neumann boundary condition and our choice of ρ yield |Du^{∞}| > |Du| in a neighborhood of ∂B_{R}(0). Thus, there is a translate u^{∞} +m, m ∈ ℝ, which is strictly greater than u. Now we shift back until the graphs touch first at a point x ∈ B_{R}(0). By the strong maximum principle this is impossible as u and u^{∞} solve the same elliptic equation in B_{R}(0).

2. Existence for v ≥ 0: We construct our solution by choosing a sequence of increasing radii R_{k} tending to ∞. By the monotonicity properties of the function v^{∞}(R, ρ) we can find for each R > 0 a unique ρ_{R} such that v^{∞}(R, ρ_{R}) = 0. We remark that ρ_{R} is an increasing sequence. Again the result from Section 4 gives us a unique, smooth, rotationally symmetric solution u_{R} to (∗_{R,ρ_{R}}), which is defined on B_{R}(0) and satisfies v^{∞} = 0. Note that for fixed R the speed v^{∞} and the normal derivative at the boundary −ρ_{R} are uniquely related. Hence the solution u_{R} must coincide on smaller balls B_{R′}(0), R′ < R, with the previous solutions u_{R′} to (∗_{R′,ρ_{R′}}). Therefore, as R tends to ∞, the sequence {u_{R}} will converge uniformly on compact sets to a limit u defined on all of ℝ^{n}. Clearly, u is a rotationally symmetric solution to (A.1). Observe that the sequence ρ_{R} will diverge in the case v = 0, whereas it stays bounded for v > 0. This proves the boundedness of |Du| in the latter case. □

Proceeding as in the existence part of the proof we get easily that nonintegrable g(x) also allow for solutions provided that h(p) is non-integrable too.

This observation can be extended to the function h(p) arising in the equation of prescribed Gauß curvature in Minkowski space

\frac{det D^{2}u}{(1 − |Du|^{2})^{\frac{n+2}{2}}} = g(x).  (A.2)

Hence, h(p) = h(|p|) = (1 − |p|^{2})^{− \frac{n+2}{2}} and h is not integrable on B_{1}(0).

Theorem A.2. For all positive and smooth functions g(x) = g(|x|) there is an entire strictly convex solution u to (A.2). The gradient of a solution satisfies |Du| < 1. Moreover, for a solution constructed here, |Du| ≤ 1 − ε, ε > 0, if and only if ∫_{ℝ^{n}} g < ∞.

Proof. We proceed similarly as in the proof of Theorem A.1 and use notations introduced there. Here, the speed function v^{∞}(R, ρ) is only defined for ρ < 1. But still v^{∞}(R, ρ) is strictly decreasing in R and strictly increasing in ρ and in addition v^{∞}(R, ρ) → −∞ for ρ → 0 and v^{∞}(R, ρ) → ∞ for ρ → 1. As in part 2 of the proof of Theorem A.1 we can find for any R > 0 a unique ρ_{R} (with 0 < ρ_{R} < 1) satisfying v^{∞}(R, ρ_{R}) = 0. Since ρ_{R} < 1 we can choose a smooth function h̃(p) defined on ℝ^{n} such that h(p) = h̃(p) for all |p| ≤ ρ_{R}. Again the result from Section 4 gives us a unique, smooth, convex rotationally symmetric solution u_{R} to (∗_{R,ρ_{R}}) with h replaced by h̃, which is defined on B_{R}(0) and satisfies v^{∞} = 0. We remark that the convexity of u implies that |Du| ≤ ρ_{R} on B_{R}(0). Thus, u is also a solution for the original function h instead of h̃ in the elliptic equation. Again, for R > R′, u_{R} will coincide with solutions u_{R′} obtained on smaller balls B_{R′}(0). Hence, for R tending to infinity, u_{R} converges to an entire solution u of (A.2). In the present case the sequence ρ_{R} stays uniformly bounded away from 1 if ∫_{ℝ^{n}} g(|x|)dx < ∞, whereas ρ_{R} converges to 1 if g is non-integrable. □

In the general case without rotational symmetry, a theorem corresponding to Theorem 1.1 in Minkowski space can be obtained easily from the techniques of this paper, provided that there holds a uniform a priori bound of the form |Du| < 1 − ε, ε > 0.

Appendix B. Illustrations

The pictures included here show the velocity of a solution to a discrete version to the flow equation

u̇ = log det D^{2}u in Ω × [0, ∞),
⟨Du, ν⟩ = ⟨Du_{0}, ν⟩  on Ω × [0, ∞),
u(·, 0) = u_{0}  on Ω,

where Ω = {(x, y) ∈ ℝ^{2} : 1.1 · (x^{2} + (2y)^{2}) < 1} and u_{0}(x, y) = 1.5x^{2} + y^{2} − 0.1y^{4}. The numerical integration has been carried out on a 200 × 100 grid corresponding to [−1, 1] × [−0.5, 0.5] ∈ ℝ^{2}. Let Ω_{0} consist of those points on the grid that belong to Ω and ∂Ω_{0} be those points on the grid such that the distance to Ω_{0} (with respect to the sup-norm) is exactly the minimal distance of two points on the grid. For simplicity, we do not introduce new notations for the discretized quantities.

In each time step we compute D^{2}u in Ω_{0}. Thus, we obtain u̇ and set

u(x, t + Δt) = u(x, t) + u̇(x, t) · Δt.

Then we make the corrections due to the boundary condition: For all x_{0} ∈ ∂Ω let y_{0} := x_{0}+ν(x_{0})·τ_{0}, where ν(x_{0}) is the normalized negative gradient of x^{2} + 2y^{2} and τ_{0} = inf{τ : x_{0} + ν(x_{0}) · τ ∈ convex hull(Ω_{0})}. We set u(x_{0}) := u_{0}(x_{0}) − u_{0}(y_{0}) − u(y_{0}). Here u(y_{0}) is obtained by linear interpolation.

(a) t = 0.0  (b) t = 0.1  (c) t = 0.5

Figure 2. Time evolution on an elliptic domain

Figure 2 shows a grayscale plot of the velocity at different times. It can be seen that the velocity tends to a constant, which reflects the property that u tends to a translating solution.

(a) δ(t)  (b) \frac{−1}{t} log δ(t)

Figure 3. Convergence to constant velocity

To illustrate the rate of convergence we show in Figure 3(a) the decay of δ(t) = ||v(t) − v̄(t)||_{L^{2}(Ω)}^{2}, where v̄(t) = \frac{1}{|Ω|} ∫ v(x, t) dx is the mean velocity. The expected exponential convergence can be seen from Figure 3(b). Here we plot \frac{−1}{t} log δ(t), which saturates nicely for larger times.

References

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Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany

E-mail address: Oliver.Schnuerer@mis.mpg.de

Current address: Harvard University, Department of Mathematics, One Oxford Street, Cambridge, MA 02138, USA

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