Infinity-Harmonic Potentials and Their Streamlines

We consider certain solutions of the Infinity-Laplace Equation in planar convex rings. Their ascending streamlines are unique while the descending ones may bifurcate. We prove that bifurcation occurs in the generic situation and as a consequence, the solutions cannot have Lipschitz continuous gradients.


Introduction
The solutions of the celebrated ∞-Laplace Equation which is the formal limit of the p-Laplace Equations ∆ p u ≡ ∇·(|∇u| p−2 ∇u) = 0 as p → ∞, have many fascinating properties. The solutions provide the best Lipschitz extension of their boundary values (see [Ar1]) and the equation appears even in Stochastic Game Theory (see [CMS]). A characteristic feature for classical solutions is that the speed |∇u| is constant along a streamline, which is a useful property for applications to image processing, see [PSW]. Indeed, along the streamline x = x(t) with the equation we should have d dt |∇u(x(t))| 2 = 2∆ ∞ u(x(t)) = 0 so that |∇u(x(t))| = constant.
However, the calculation requires second partial derivatives. We shall see that this interpretation of constant speed often fails. The solutions of the ∞-Laplace Equation, the so-called ∞-harmonic functions, are defined in the viscosity sense as in [J], [JLM] and [S]. They are continuous and even differentiable. O. Savin [S] has proved that in the plane their gradient is continuous and even locally Hölder continuous, according to [ESa]. Thus the solutions are of class C 1,α loc in the two dimensional case. In [KZZ] the speed |∇u| is shown to belong to a Sobolev space. In higher dimensions the gradient exists (in the classical sense) at every point by a result of L. Evans and Ch. Smart, cf. [ES]. At the moment of writing, the C 1 loc -property is not known in higher dimensions. This unsettled urgent question is the reason for why we restrict our exposition to two dimensions. In the plane the equation reads ∂u ∂x 1 2 ∂ 2 u ∂x 2 1 + 2 ∂u ∂x 1 ∂u ∂x 2 ∂ 2 u ∂x 1 ∂x 2 + ∂u ∂x 2 2 ∂ 2 u ∂x 2 2 = 0 as in G. Aronsson's work [Ar2] about the streamlines.
Notation. We fix some notation. Suppose that Ω is a convex bounded domain in the plane R 2 containing a compact convex set K with boundary Γ = ∂K. The case when K reduces to a single point is of special interest. The domain G = Ω\K is a "convex ring"; it has the outer boundary ∂Ω and the inner boundary Γ. The object of our work is the Dirichlet boundary value problem (1) The unique solution, say V ∞ , attains the boundary values in the classical sense (this holds for all domains, whether they are convex or not). Hence Some properties. By the Maximum Principle, 0 < V ∞ < 1 in G. (It is convenient to put V ∞ = 1 in K and = 0 outside Ω.) The gradient ∇V ∞ ∈ C α loc (G) for some small α, cf. [ESa]. We use some fundamental properties valid in convex rings, which are due to J. Lewis [L]. See also [Ja]. We need the following We interpret the inequality ∆ p V ∞ ≤ 0 in the viscosity sense. This is equivalent to the usual definition of p-superharmonic functions, cf. [JLM], [JJ]. In particular "∆V ∞ ≤ 0" and so V ∞ is an ordinary superharmonic function.
Streamlines. Let us return to the ascending streamlines x = x(t). They are the trajectories of the gradient flow and intersect the convex level curves orthogonally. (If the initial point x 0 ∈ ∂Ω and ∇V ∞ (x(t 0 )) = 0, some special care is needed.) By Peano's Existence Theorem, there exists at least one solution starting at x 0 . Since ∇V ∞ = 0, the trajectory cannot terminate inside G. In fact, x(t) ∈ G when t 0 ≤ t < T for some finite T and x(T ) ∈ Γ. One of our main results is that the solution is unique.
Theorem 1 (Ascending uniqueness). The solution to the equation (2) of the ascending gradient flow is unique and terminates at Γ.
Despite uniqueness, two trajectories, starting at different points, can meet and join. But the trajectories cannot cross. The first point at which two streamlines meet (after which they become a joint trajectory) is here called a Cl-point. Notice that uniqueness is not valid for the usual descending streamlines coming from the equation with a minus sign! They allow bifurcation. The proof of the uniqueness theorem is delicate, since the Picard-Lindelöf Theorem is not applicable, when ∇V ∞ is not Lipschitz continuous. (Mere Hölder continuity is not sufficient.) We base our reasoning on the expedient inequality valid for any domain D ⊂⊂ G with Lipschitz boundary ∂D. Here n denotes the outer normal. The proof given in Proposition 6 requires several regularizations so that the inequality ∆ p V ∞ ≤ 0 can be used pointwise as in [JJ]. The difficulty is the absence of second derivatives. Our next theorem provides a tricky device for detecting Cl-points.
Theorem 2. Let ξ 0 ∈ ∂Ω and denote If β > α, then there exists a neighborhood of ξ 0 such that every pair of streamlines starting there will meet before reaching Γ.
In general, we have not succeeded in proving that the speed |∇V ∞ (x(t))| is non-decreasing along the streamline. Thus the use of the theorem is somewhat elaborate. Let us mention some immediate consequences. First, the fact that two streamlines meet means that the descending gradient flow does not have unique solutions. By the Picard-Lindelöf Theorem the function −V ∞ cannot therefore belong to the class C 1,1 loc (G) in the presence of Cl-points. By general theory, the descending gradient flow dx dt = −∇u(x) has a unique solution if u is locally semiconvex . It follows that our V ∞ cannot be locally semiconvex 1 . (Neither can ψ(V ∞ ) be for a smooth strictly monotone function ψ, since V ∞ and ψ(V ∞ ) have the same level sets.) To apply the theorem we notice that it is always possible to choose β > 0, see Lemma 9. Thus, if we can find a point ξ 0 ∈ ∂Ω yielding α = 0, we have obtained the inequality β > α. According to a result in [MPS] the following holds in convex domains in the plane: if the boundary has an irregular boundary point which is a corner with interior angle less than π, then |∇V ∞ | = 0 at the corner. This provides an α = 0.
Theorem 3. If ∂Ω has a corner with angle less than π, then there are streamlines that meet in G before reaching Γ. In particular, V ∞ is not of class C 1,1 loc (G). For a special kind of domains the distance function dist(x, ∂Ω) is the ∞potential. A stadium is a domain where the distance function attains its maximum value at all its singular points. These sets have a simple characterization in the plane. Namely, See Theorem 6 in [CF]. The set H is called the High Ridge. The simplest example of a stadium is the unit disk: In a stadium, when Γ is the High Ridge, the solution is smooth and no streamlines meet. We argue that all other convex rings have Cl-points. If Γ is a single point we have the following theorem.
Theorem 4. Assume that Γ is a single point. If Ω is not a disk centered at Γ, then there are streamlines that meet. In fact, all streamlines that are not entirely inside the closed disk with radius dist(Γ, ∂Ω) centered at Γ have Cl-points. In particular, loc (G) has been proved before, see Corollary 1.2 in [SWY]. Also the case when Γ is a subset of the High Ridge (though the domain is not necessarily a stadium) is accessible.
Theorem 5. Suppose Γ is a subset of the High Ridge of Ω. Unless Ω is a stadium and Γ its High Ridge, there are streamlines that meet. In particular, V ∞ is not We also mention that Theorem 4 reveals a queer instability for the ∞-Laplace equation. Indeed, the solution of (1) in the disk 0 < |x| < 1 is smooth, while the corresponding solution in an ellipse exhibits points where the second order derivatives are not bounded. After a coordinate transformation, this implies that in a disk the solution of (1) whith ∆ ∞ replaced by the operator u 2 x u xx + 2(1 + δ)u x u y u xy + (1 + δ) 2 u 2 y u yy exhibits this kind of singularites for any δ > 0, but not for δ = 0. A similar instability occurs if the midpoint of the disk is perturbed. We conclude our work with some remarks about a square. This is a challenging example, indeed. Now the domain Ω is a square and Γ is its midpoint. In this case the gradient ∇V ∞ is continuous also on the sides, but ∇V ∞ = 0 at the four corners (and only there), which gives an α = 0 for free in Theorem 2. By symmetry the diagonals are streamlines, so are the medians. It seems as if all the streamlines, except the four medians, would join a diagonal before reaching the midpoint (see Figure 1). We record three results.
First, we show that there are infinitely many Cl-points near the corners. Second, we show that also near the origin there are are infinitely many Cl-points. Finally, we argue that all the streamlines, except the medians, do have infinitely many Cl-points. (It seems as if all points on the diagonals were Cl-points and that these are the only Cl-points.) It is likely that the ∞-harmonic potential function is related to the ∞-eigenvalue problem, introduced in [JLM]. Indeed, this resemblance was the starting point of our investigation.
The reader is supposed to be familiar with the ∞-Laplacian. For the concept of viscosity solutions we refer to [K] and [CIL]. We use standard notation. We restrict ourselves to the plane, but most of our exposition is valid even in higher dimensions provided that the gradient ∇V ∞ be continuous.

Preliminaries
A fundamental tool is inequality (3) for line integrals. For smooth functions it comes from an integration by parts. We shall use the method in [JJ].
Proposition 6. Let p ≥ 2 and assume that D ⊂⊂ G has a Lipschitz boundary ∂D. Then where n is the outer normal.
Proof. Due to the lack of second derivatives we use two regularizations.
Step 1. Let V ∞,ε be the infimal convolution By standard theory V ∞,ε V ∞ locally uniformly in G and in the viscosity sense, when ε > 0 is small enough. The fact that ∆ p V ∞ ≤ 0 implies this. Furthermore, the function is concave. Therefore it has second derivatives in the sense of Alexandroff. So does V ∞,ε . It follows that inequality (5) holds almost everywhere, when the second derivatives are taken in Alexandroff's sense. At almost every as y → x. Here D 2 V ∞,ε is the Hessian matrix of second Alexandroff derivatives.
Step 2. We claim that It is easy to see that Step 3. To obtain second derivatives we define the convolution where ρ j is a standard mollifier. By the proof of Alexandroff's Theorem in [EG] almost everywhere in D; the second derivatives are in the sense of Alexandroff. The convolution preserves concavity: where I 2 is the identity matrix. It is immediate that Together, these inequalities yield the bound Thus we can use Fatou's Lemma to obtain where inequality (5) was used at the end.
Step 4: By the Divergence Theorem Here the continuity of ∇V ∞,ε was needed.

The function
is often more convenient. It has the same level curves and streamlines as V ∞ . Under the same assumptions as in Proposition 6 we have The proof is similar, since holds for smooth functions u > 0.

Estimates for the Gradient
Lemma 7. We have Proof. That ∇V ∞ = 0 is proved in [L], see also [Ja]. This is a simple consequence of the convexity of the level curves. 2 Since V ∞ is an optimal extension of its boundary values, will do. Now |∇v| = 1/δ almost everywhwere. The upper bound follows.
As a consequence of the above gradient bound we obtain the equicontinuity of streamlines, which is needed later on.
Proposition 8 (Convergence). Suppose that a sequence of streamlines in G is given. Then the family {γ k } is equicontinuous and bounded. Furthermore, if the initial points γ k (0) converge to a point a ∈ G, then the streamlines converge uniformly to the streamline via a.
Proof. Integrating the equation we see that Hence the family is uniformly equicontinuous and bounded. Thus we can apply Ascoli's Theorem to find a uniformly convergent subsequence, say γ k j → γ.
We may take the limit under the integral sign in Differentiating, we get dγ dt = ∇V ∞ (γ(t)), which means that the limit curve is a streamline and γ(0) = a. This was for a subsequence, but using the uniqueness theorem (Theorem 1) one can deduce that also the full sequence γ k converges.
Proof. A simple geometric reasoning provides this. Since the level curves are convex, a level set always lies entirely on one side of the tangent lines. This makes it possible to construct a linear function which lies above V ∞ in that part of Ω which is on the outer side of a tangent and which coincides with V ∞ (ξ) at the tangent point ξ. The slope of the plane can be taken to be ≤ V ∞ (ξ)/diam(Ω) and now V ∞ (ξ) = 1. (The reader may wish to draw a picture.) Then the comparison principle yields the estimate. Proof. By Theorem 1 in [SWY] and c > 0. Let ε > 0. Writing x = r(y + z) where r > 0, |y| < 1, and |z| = 1, we have V ∞ r(y + z) − 1 − cr|y + z| ≤ εr|y + z| < 2εr for 0 < r < r ε (= some number < 1). Keep |z| = 1 fixed. Dividing out r we get sup B(z,1) when 0 < r < r ε . According to Theorem 2 in [SWY], inequality (6) implies that for any δ > 0 we can find an ε δ such that This holds for all 0 < r < r ε where 0 < ε < ε δ and hence it follows as r → 0 that lim x→0 |∇V ∞ (x)| = c > 0, as desired.
Corollary 11. Under the same assumptions as in Proposition 10, Proof. Since the function V ∞ is an optimal Lipschitz extension of its boundary data, it follows from Proposition 10 that .
If Γ is part of the High Ridge, it must be a line segment. Corollary 11 can be extended to this case.
Proposition 12. Let Γ be a segment on the High Ridge of Ω. Then Proof. We normalize the geometry so that Γ is the closed segment joining the points (±a, 0) on the x 1 -axis and dist(ξ, ∂Ω) = 1 whenever ξ ∈ Γ. Construct the largest stadium S with Γ as its High Ridge which is contained in Ω. That is, It follows by comparison that In particular, since the domain is convex, these functions coincide on a rectangle: for −a ≤ x 1 ≤ a and −1 ≤ x 2 ≤ 1. As we shall see, V ∞ is glued together of three pieces (inspired by the example in Section 5 of [JLM2]). Let u L be the solution of (1) with Γ = (−a, 0). Similarly, we define u R with Γ = (a, 0). Now Corollary 11 implies lim x→(−a,0) We claim that in Ω First, it is continuous. Second, it is ∞-harmonic in Ω ∩ {|x 1 | > a} and when |x 1 | ≤ a the function coincides with V ∞ by (7). The desired result follows by comparison.

Proofs of the Theorems
Proof of Theorem 1. Assume that two streamlines x 1 (t) and x 2 (t) for the ascending gradient flow in equation (2) emerge at a point x Cl ∈ G. If they intersect some level curve at the points y 1 and y 2 and y 1 = y 2 , then we apply the fundamental inequality (4) to the domain D bounded by parts of the three curves x 1 (t), x 2 (t), and the level curve. Only the arcs with endpoints: x Cl , y 1 , and y 2 count. (One may think of a curved triangle). By inequality (4) since naturally ∇V ∞ , n = 0 along the streamlines and is the outer normal along the level curve between the points y 1 and y 2 . Since ∇V ∞ is continuous, it must be identically 0 along this level curve. This contradicts the fact that ∇V ∞ = 0 in G. Hence we must have y 1 = y 2 and so the streamlines coincide: For a curved quadrilateral bounded by the arcs of two level curves and of to streamlines we have a convenient comparison for the supremum norm of ∇V ∞ on the level arcs. The result indicates that such quadrilaterals cannot always exist, not if the level difference is too big.
Lemma 13. Assume that • the points x 1 and x 2 are on the same level curve V ∞ = a, • the points y 1 and y 2 both are on the higher level curve V ∞ = b > a, • ascending streamlines join x 1 with y 1 and x 2 with y 2 . Then that is, the lower level curve has the larger maximum norm for the gradient.
Proof. Use inequality (4) on the boundary of the domain D bounded by the four arcs. The streamlines do not contribute to the line integral. Along the level arcs the outer normal has the directions ±∇V ∞ , the minus sign being for the lower arc between x 1 and x 2 . This yields Taking the p−1 th roots and sending p to ∞, we arrive at inequality (8).
Proof of Theorem 2. The theorem follows from the above lemma. Indeed, let ε > 0 be very small. There is a strip near Γ, say dist( This strip contains all sufficiently high level curves. In a neighborhood of ξ 0 we have |∇V ∞ | < α+ε. If two different streamlines, starting at the same level curve in this neighborhood reach the strip without joining, then it follows from inequality (8) that we must have which for a small ε contradicts the assumption β > α. Therefore the streamlines must have joined before reaching the top level.
The following localized version of Theorem 2 is convenient.
Corollary 14. Suppose that a streamline γ joins the points a 0 and b 0 in G, where a 0 is on the lower level, i.e.
then there is a neighborhood of a 0 such that every streamline starting there joins the streamline γ before reaching the level curve of b 0 .
Proof. By continuity, we can find a neighborhood of a 0 and a neighborhood of b 0 such that the strict inequality above holds extended to the neighborhoods. Consider a sequence of points a k on the level curve of a 0 such that a k → a 0 . By Proposition 8 the streamlines γ k starting at a k converge uniformly to γ. This implies that when the index k is big enough, the streamline starting at a k must reach the level of b 0 at a point inside the upper neighborhood. By Theorem 2 this is possible only if the streamline has joined γ already before reaching the upper level. (It means that all these streamlines pass via the point b 0 .) Proof of Theorem 4. We may assume that Γ = {0} and dist(Γ, ∂Ω) = 1 so that lim x→0 |∇V ∞ (x)| = 1 by Theorem 10 and its Corollary. With this normalization B = B(0, 1) is the largest disk centered at 0 which is comprised in Ω. If B = Ω, we can find a point ξ ∈ ∂Ω such that ξ ∈ B. Consider the streamline x = x(t) from ξ to the origin. By Lemma 7 |∇V ∞ | ≤ 1. We have two cases.
If |∇V ∞ (x(t * ))| < 1 at some point x * = x(t * ) then there is a neighborhood U * of x * where |∇V ∞ | ≤ α < 1 for some suitable α. Given a small ε > 0, there is a neighborhood of the top 0 in which |∇V ∞ | > 1 − ε. If ε is so small that α < 1 − ε, the quadrilateral described in Lemma 13 cannot exist, since inequality (8) is violated. This means that any two streamlines passing via the neighborhood U * must join before reaching the top.
We are left with the case |∇V ∞ (x(t)| ≡ 1. Using the arclength as parameter we see that Thus the length of the streamline from ξ to 0 is = 1. But that violates the requirement that |ξ − 0| > 1. Therefore this second case is impossible. The proof reveals that all streamlines starting outside the inscribed disk B have Cl-points.
Proof Theorem 5. The proofs follows the same lines as the proof of Theorem 4. The only difference is that we use Proposition 12 instead of Proposition 11.

The Streamlines in a Square
In this section, Ω is the square defined by −1 < x 1 < 1, −1 < x 2 < 1 and Γ is the origin (0, 0). Thus V ∞ (0, 0) = 1. In this case the ∞-potential V ∞ can be defined in the whole plane by reflection through the sides of the square.
(The principle is the same as the Schwartz reflecion for harmonic functions.) The resulting function is ∞-harmonic except at the isolated points (2m, 2n), m, n = 0, ±1, ±2, ... The gradient ∇V ∞ is now continuous except at the aforementioned points. Moreover, at the corners ∇V ∞ (±1, ±1) = 0 since V ∞ = 0 on the sides of the square.
Comparison yields so that V ∞ is a linear function on the medians (= the coordinate axes). If x p = x p (t) is a streamline for the p-harmonic function V p with the same boundary values as V ∞ so that V p → V ∞ as p → ∞, then since ∆V p ≤ 0 (superharmonic) by Lewis's theorem. Thus the functions are convex. Unfortunately, the streamlines usually move as p → ∞, making the control of the process difficult. However, the diagonals are streamlines for all p.
Thus the limit function on the diagonal from (−1, −1) to (0, 0). Since the limit V ∞ (t, t) has a continuous derivative with respect to t, it follows by Theorem 25.7 in [R], that on the diagonal even the derivatives of V p converge uniformly. 3 It follows that the speed |∇V ∞ | is non-decreasing along the diagonal. We sum up a few properties: 1. From each point on the boundary ∂Ω a unique streamline starts and terminates at the origin. Through each point there passes at least one streamline.

2.
A streamline has a continuous tangent.
3. The diagonals and medians are streamlines.
3 Unfortunately, the uniform convergence ∇V p → ∇V ∞ is not known to us.

4.
No streamline can join the medians.
5. The speed |∇V ∞ | is non-decreasing on the diagonals. 4 6. There are infinitely many Cl-points near the corners.
7. There are infinitely many Cl-points near the origin.
8. There are infinitely many Cl-points along any streamline except the medians.
This can be directly deduced from the previous results except for the three last points, which require some further explanation.
Proof of 6). The gradient is zero at the corners and the gradient is non-zero at all interior points. Therefore there must be infinitely many points a 0 and b 0 near the corners satisfying the assumptions of Corollary 14. This implies that there are infinitly many Cl-points near the corners.
Proof of 7). We prove that in each disk around the origin, there is at least one Cl-point. The result follows from this. We assume towards a contradiction that there is c ∈ (0, 1) such that the set {V ∞ > c} does not contain any such points. We apply Theorem 4 to the restriction of w = (V ∞ − c)/(1 − c) to the set {V ∞ ≥ c} to conclude that the set {V ∞ > c} is a ball B. In particular, |∇V ∞ | = 1 in B.
Denote by y 1 the intersection of B and the lower right diagonal. Let x 1 be the closest point to the midpoint (0, −1) of the lower side, such that the streamline starting at x 1 passes through y 1 . 5 We have two alternatives: 1) x 1 is the corner point (1, −1) and 2) x 1 is not the corner point (it cannot be the midpoint).
In the case of 1), any streamline starting at a point x 2 to the left of the corner, intersects ∂B at a point y 2 = y 1 which is not on the diagonal. Since we may take x 2 as close as we wish to the diagonal, we may assume |∇V ∞ | < 1 2 on the line between x 1 and x 2 . Moreover, on the level set joining y 1 and y 2 (= the circle ∂B), we have |∇V ∞ | = 1. By applying Lemma 13 to the pair of points x 1 , x 2 and y 1 , y 2 , we obtain ∇V ∞ ∞,x 1 x 2 ≥ ∇V ∞ ∞,y 1 y 2 = 1, which is a contradiction. In the case of 2), let x 2 be a point to the left of x 1 and y 2 the corresponding point on ∂B. By definition, y 2 = y 1 . Take z 1 to be a point on the streamline from x 1 to y 1 . Let z 2 be a point on the same level line as z 1 and on the streamline between x 2 to y 2 . By Lemma 13 applied to the pair of points y 1 , y 2 and z 1 , z 2 , we obtain that ∇V ∞ ∞,z 1 z 2 ≥ 1.
Since the pair z 1 , z 2 is arbitrary and since we may choose x 2 arbitrary close to x 1 , this implies that |∇V ∞ | = 1 along the streamline starting at x 1 . Since the distance between x 2 and the origin is strictly larger than 1, this is a contradiction.

Proof of 7)
. Let x be a boundary point which is not a midpoint of a side. Then |x| > 1. Therefore, along any streamline starting at x, there must be a point y where |∇V ∞ | < 1. Since |∇V ∞ | is continuous along the streamline, there must be infinitely many points a 0 and b 0 along this streamline satisfying the assumptions of Corollary 14 and therefore there are infinitly many Cl-points along this streamline.
We conjecture that every streamline except the medians joins a diagonal before reaching the midpoint and that the only Cl-points are the points on the diagonals. This is also suggested by Figure 1.
Epilogue. One may wonder whether |∇ log V ∞ | ≥ 1 in the square. This would show that V ∞ is the same function as the ∞-Ground State described in [JLM]. This is also suggested by numerics. Figure 1: The streamlines of V ∞ when Ω is the square −1 < x 1 < 1, −1 < x 2 < 1.