The Gradient Flow of Infinity-Harmonic Potentials

The $\infty$-Laplace Equation

$$\Delta_{\infty}u\,\equiv\,\sum_{i,j}\frac{\partial u}{\partial x_{i}}\frac{\partial u}{\partial x_{j}}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\,=\,0$$

was introduced by G. Aronsson in 1967 (cf. [Ar1]) to produce optimal Lipschitz extensions of boundary values. It has been extensively studied. Some of the highlights are

• •

  Viscosity solutions for $\Delta_{\infty}$, [BDM]

• •

  Uniqueness, [J]

• •

  Differentiability, [S], [ESa] and [ES]

• •

  Tug-of-War (connection with stochastic game theory), [PSW]

We are interested in the two-dimensional equation

$$\Bigl{(}\frac{\partial u}{\partial x_{1}}\Bigr{)}^{\!2}\frac{\partial^{2}u}{\partial x_{1}^{2}}\,+\,2\,\frac{\partial u}{\partial x_{1}}\frac{\partial u}{\partial x_{2}}\frac{\partial^{2}u}{\partial x_{1}\partial x_{2}}\,+\,\Bigl{(}\frac{\partial u}{\partial x_{2}}\Bigr{)}^{\!2}\frac{\partial^{2}u}{\partial x_{2}^{2}}\,=\,0$$

in so-called convex ring domains $G=\Omega\setminus K$. Here $\Omega$ is a bounded convex domain in $\mathbb{R}^{2}$ and $K\Subset\Omega$ is a closed convex set. We continue our investigation in [LL] of the $\infty$-potential $u_{\infty}$, which is the unique solution in $C(\overline{G})$ of the boundary value problem

$$\begin{cases}\Delta_{\infty}u\,=\,0\qquad\text{in}\qquad G\\ \phantom{\Delta_{\infty}}u\,=\,0\qquad\text{on}\qquad\partial\Omega\\ \phantom{\Delta_{\infty}}u\,=\,1\qquad\text{on}\qquad\partial K.\end{cases}$$

In [LL] we proved that the ascending streamlines, the solutions $\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2})$ of

$$\frac{d\boldsymbol{\alpha}(t)}{dt}=+\nabla u_{\infty}(\boldsymbol{\alpha}(t)),\quad 0\leq t<T_{\boldsymbol{\alpha}}$$

with given initial point $\boldsymbol{\alpha}(0)\in\overline{\Omega}\setminus K$, are unique and terminate at $\partial K$. (The descending ones are not!) Streamlines may meet and then continue along a common arc. Uniqueness prevents crossing streamlines.

Along a streamline one would expect that the speed $|\nabla u_{\infty}(\boldsymbol{\alpha})|$ is constant. Indeed,

$$\frac{d}{dt}|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|^{2}=2\,\Delta_{\infty}u_{\infty}(\boldsymbol{\alpha}(t))=0,$$

but the calculation requires second derivatives. The main difficulty is the lack of second derivatives. Although, the second derivatives are known to exist almost everywhere with respect to the Lebesgue area, see [KZZ] for this new result, this is of little use since the area of a streamline is zero. In [LL] it was shown that the above calculation fails: for most streamlines the speed is not constant the whole way up to $\partial K$. (We shall see that the speed is constant from the initial point till the streamline meets another streamline.)

We use the approximation with the (unique) solution of the $p$-Laplace equation

$$\Delta_{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0,\qquad p>2.$$

in $G$ with the same boundary values as $u_{\infty}$.

We shall use several facts about these $p$-harmonic functions due to J. Lewis, cf. [L]. It is decisive that the level curves $\{u_{p}(x)=c\}$ are convex and that $\Delta u_{p}\leq 0$. See Section 2 for more details.

We also need the facts that (i) $\nabla u_{p}\to\nabla u_{\infty}$ in $L^{2}_{\text{loc}}$ and (ii) the family $\{|\nabla u_{p}|\}$ is locally equicontinuous. (Notice that we wrote $|\nabla u_{p}|$, not $\nabla u_{p}$.) We extract a proof of this from the recent pathbreaking work by H. Koch, Y. R-Y. Zhang and Y. Zhou in [KZZ], complementing their results by applying a simple device, due to Lebesgue in [Leb], to the norm $|\nabla u_{p}|$ of the quasiregular mapping

$$\frac{\partial u_{p}}{\partial x_{1}}-i\frac{\partial u_{p}}{\partial x_{2}},\quad i^{2}=-1.$$

The quasiregularity was obtained by B. Bojarski and T. Iwaniec in [BI].

We prove the following basic result in Section 3.

Combining this with a result in the opposite direction (cf. Lemma 12 in [LL]), we can control the meeting points so that these lie on a few specific streamlines, here called attracting streamlines.

$\Omega$ is a bounded convex domain in $\mathbb{R}^{2}$ and $K\Subset\Omega$ is a compact and convex set, which may reduce to a point. We study the equation in the convex ring $G=\Omega\setminus K$. We assume the following normalization:

$$\boxed{\operatorname{dist}(\partial\Omega,K)=1.}$$

The boundary value problem

$$\begin{cases}\Delta_{\infty}u\,=\,0\qquad\text{in}\qquad G,\\ \phantom{\Delta_{\infty}}u\,=\,0\qquad\text{on}\qquad\partial\Omega,\\ \phantom{\Delta_{\infty}}u\,=\,1\qquad\text{on}\qquad\partial K,\end{cases}$$

has a unique solution $u_{\infty}\in C(\overline{G})$ in general. By [ESa], $\nabla u_{\infty}$ is locally Hölder continuous in $G$. We will assume that also $\nabla u_{\infty}\in C(\overline{\Omega}\setminus K)$. This is fulfilled if for instance $\partial\Omega$ has a piecewise $C^{2}$ regular boundary. See Lemma 2 and Theorem 2 in [HL], Theorem 7.1 in [MPS] and Theorem 1 in [WY].

In [LL] it was established that, for a given initial point $\xi_{0}\in\partial\Omega$, the gradient flow

$$\begin{cases}\displaystyle\frac{d\boldsymbol{\alpha}(t)}{dt}&=\,+\nabla u_{\infty}(\boldsymbol{\alpha}(t)),\quad 0\leq t<T,\\ \boldsymbol{\alpha}(0)&=\,\xi_{0},\end{cases}$$

has a unique solution $\boldsymbol{\alpha}=\boldsymbol{\alpha}(t)$, which terminates at some point $\boldsymbol{\alpha}(T)$ on $\partial K$. (Some caution is required if $|\nabla u_{\infty}(\xi_{0})|=0$.) We say that $\boldsymbol{\alpha}$ is a streamline. Although unique, two streamlines may meet, join, and continue along a common arc.

We shall employ the $p$-harmonic approximation

$$\begin{cases}\Delta_{p}u_{p}\,=\,0\qquad\text{in}\qquad G,\\ \phantom{\Delta_{p}}u_{p}\,=\,0\qquad\text{on}\qquad\partial\Omega,\\ \phantom{\Delta_{p}}u_{p}\,=\,1\qquad\text{on}\qquad\partial K,\end{cases}$$

for $\mathbf{p>2}$. It is known that $u_{p}\in C(\overline{G})$ and it takes the correct values (in the classical sense) at each boundary point. We shall need the following results from [L] (see also [Ja]):

1. 1.

   The level curves $\{u_{p}=c\}$ are convex, if $0\leq c\leq 1$,

2. 2.

   $u_{p}\nearrow u_{\infty}$ uniformly in $\overline{G}$,

3. 3.

   $|\nabla u_{p}|\neq 0$ in $G$,

4. 4.

   $u_{p}$ is real analytic in $G$,

5. 5.

   $\Delta u_{p}\leq 0$.

The streamlines of $u_{p}$ do not meet in $G$. This is due to the regularity of $u_{p}$ and the Picard-Lindelöf theorem. Properties 1), 3), and 5) are preserved at the limit $p=\infty$. Especially, $\nabla u_{\infty}\neq 0$ in $G$.

We keep the normalization $\operatorname{dist}(\partial\Omega,K)=1$. Then $|\nabla u_{\infty}|\leq 1$, but we also need a uniform bound for $|\nabla u_{p}|$. The bound

$$|\nabla u_{p}|\leq 1\quad\text{on }\partial\Omega.$$

(1)

follows by comparison with the distance function

$$\delta(x)=\operatorname{dist}(x,\partial\Omega).$$

In a convex domain, $\delta$ is a supersolution of the $p$-Laplace equation. Since

$$0\leq u_{p}(x)\leq\delta(x)\quad\text{on }\partial G,$$

the same inequality also holds in $G$. In general, $|\nabla u_{p}|$ is unbounded (but $|\nabla u_{\infty}|\leq 1$), so we have to consider a subdomain, say $\{u_{p}<c\}$.

We shall prove that

$$\lim_{p\to\infty}{|\nabla u_{p}|}=|\nabla u_{\infty}|$$

locally uniformly in $G$. From [KZZ] we can extract the following important properties: If $D\Subset G$, then

$$\iint_{D}|\nabla u_{p}-\nabla u_{\infty}|^{2}\,dx_{1}dx_{2}\to 0,\quad\text{as }p\to\infty,$$

$$\iint_{D}|\nabla(|\nabla u_{p}|^{2})|^{2}\,dx_{1}dx_{2}\leq M_{D}<\infty,$$

(J)

for all (large) $p$.

The constant $M_{D}$ depends on $\|\nabla u_{p}\|_{L^{\infty}(E)}$, where $D\Subset E\Subset G$, and $\operatorname{dist}(D,\partial G)$, but not on $p$.

In [KZZ] the estimates were derived for solutions $u^{\varepsilon}$ of the auxiliary equation

$$\Delta_{\infty}u^{\varepsilon}+\varepsilon\Delta u^{\varepsilon}=0$$

while we use $\Delta_{p}u_{p}=0$ written as

$$\Delta_{\infty}u_{p}+\frac{1}{p-2}\,|\nabla u_{p}|^{2}\Delta u_{p}=0.$$

The advantage of our approach is that the inequality $\Delta u_{p}\leq 0$ is available in convex domains for $p\geq 2$.

The conversion from $u^{\varepsilon}$ to $u_{p}$ requires only obvious changes. Formally, the factor $\varepsilon$ in front of an integral in [KZZ] should be moved in under the integral sign and then replaced by $|\nabla u_{p}|^{2}/(p-2)$, upon which every $u^{\varepsilon}$ be replaced by $u_{p}$. This procedure is explained in our Appendix.

In order to prove that the family $\{|\nabla u_{p}|\}$ is locally equicontinuous, we shall use a device due to Lebesgue in [Leb]. A function $f\in C(\overline{B}_{R})\cap W^{1,2}(B_{R})$ is monotone (in the sense of Lebesgue) if

$$\operatorname*{osc}_{\partial B_{r}}f=\operatorname*{osc}_{\overline{B}_{r}}f,\quad 0<r<R,$$

where $B_{r}$ are concentric discs. For such a function

$$\Bigl{(}\operatorname*{osc}_{B_{r}}f\Bigr{)}^{2}\ln\frac{R}{r}\leq\pi\iint_{B_{R}}|\nabla f|^{2}\,dx_{1}dx_{2}.$$

(4)

The proof is merely an integration in polar coordinates, cf. [Leb]. We shall apply this oscillation lemma on the function $f=|\nabla u_{p}|^{2}$. It was shown by Bojarski and Iwaniec in [BI] that the mapping

$$\frac{\partial u_{p}}{\partial x_{1}}-i\,\frac{\partial u_{p}}{\partial x_{2}},\quad i^{2}=-1,$$

is quasiregular. That property implies that its norm $|\nabla u_{p}|$ satisfies the maximum principle, and, where $|\nabla u_{p}|\neq 0$, also the minimum principle. Thus $|\nabla u_{p}|$ is monotone. So is $|\nabla u_{p}|^{2}$. From (4) we obtain

$$\Bigl{(}\operatorname*{osc}_{B_{r}}\{|\nabla u_{p}|^{2}\}\Bigr{)}^{2}\ln\frac{R}{r}\leq\pi\iint_{B_{R}}|\nabla(|\nabla u_{p}|^{2})|^{2}\,dx_{1}dx_{2}.$$

The uniform bound in (2) and a standard covering argument for compact sets yields the following result.

Since $\nabla u_{p}\to\nabla u_{\infty}$ in $L^{2}_{\text{loc}}(G)$ we can use Ascoli’s theorem to conclude that

$$\lim_{p\to\infty}|\nabla u_{p}|=|\nabla u_{\infty}|$$

locally uniformly. (More accurately, we have to extract a subsequence in Ascoli’s theorem, but since the limit $|\nabla u_{\infty}|$ is unique, this precaution is not called for here.)

Caution: The more demanding convergence $\nabla u_{p}\to\nabla u_{\infty}$ holds a.e., but perhaps not locally uniformly.

Let us finally mention that the uniform convergence is not global. For example, in the ring $0<|x|<1$ we have

$$u_{p}(x)=1-|x|^{\frac{p-2}{p-1}},\quad u_{\infty}=1-|x|.$$

Now $|\nabla u_{p}|$ is not even bounded near $x=0$. Thus the convergence cannot be uniform in the whole ring.

In this section, we study the convergence of the streamlines and prove Theorem 1. It is plain that the level curves $\{u_{p}=c\}$ converge to the level curves $\{u_{\infty}=c\}$. However, the convergence of the streamlines requires a more sophisticated proof. (The problem is the identification of the limit as an $\infty$-streamline.)

Suppose that we have the streamlines $\boldsymbol{\alpha}_{p}$ and $\boldsymbol{\alpha}_{\infty}$ having the same initial point $\boldsymbol{\alpha}_{p}(0)=\boldsymbol{\alpha}_{\infty}(0)=x_{0}$. Now

$$\frac{d\boldsymbol{\alpha}_{p}(t)}{dt}=\nabla u_{p}(\boldsymbol{\alpha}_{p}(t)),\quad\frac{d\boldsymbol{\alpha}_{\infty}(t)}{dt}=\nabla u_{\infty}(\boldsymbol{\alpha}_{\infty}(t))$$

when $0<t<T_{p}$, where $u_{p}(\boldsymbol{\alpha}_{p}(T_{p}))=1$. Thus

$$\boldsymbol{\alpha}_{p}(t_{2})-\boldsymbol{\alpha}_{p}(t_{1})=\int_{t_{1}}^{t_{2}}\nabla u_{p}(\boldsymbol{\alpha}_{p}(t))dt.$$

Using the bound

$$|\nabla u_{p}|\leq\Bigl{(}\frac{1}{1-c}\Bigr{)}^{\frac{1}{p-2}},\quad\text{when }u_{p}\leq c,$$

in Lemma 6 we see that

$$|\boldsymbol{\alpha}_{p}(t_{2})-\boldsymbol{\alpha}_{p}(t_{1})|\leq\Bigl{(}\frac{1}{1-c}\Bigr{)}^{\frac{1}{p-2}}|t_{2}-t_{1}|$$

(5)

as long as the curves are below the level $u_{p}=c$, i.e., $u_{p}(\boldsymbol{\alpha}(t_{2}))\leq c$. In particular, the bound is valid in the domain $\{u_{\infty}<c\}$, where $c<1$. Thus, the family of curves is locally equicontinuous. By Ascoli’s theorem we can extract a sequence $p_{j}\to\infty$ such that

$$\boldsymbol{\alpha}_{p_{j}}(t)\to\boldsymbol{\alpha}(t)$$

uniformly in every domain $\{u_{\infty}<c\}$. Here $\boldsymbol{\alpha}(t)$ is some curve with initial point $\boldsymbol{\alpha}(0)=x_{0}$.

The endpoint of $\boldsymbol{\alpha}$ is on $\partial K$. Indeed, let $t_{p}=t_{p}(c)$ denote the parameter value at which $u_{p}(\boldsymbol{\alpha}_{p}(t_{p}))=c$. Take any convergent sequence, say $t_{p}\to t^{*}$. Then

$$c=\lim_{p\to\infty}u_{p}(\boldsymbol{\alpha}_{p}(t_{p}))=u_{\infty}(\boldsymbol{\alpha}(t^{*})).$$

Thus $t^{*}=t_{\infty}(c)$. Then $t_{p}(c)\to t_{\infty}(c)$ for all $c$.

By (5)

$$|\boldsymbol{\alpha}(t_{2})-\boldsymbol{\alpha}(t_{1})|\leq|t_{2}-t_{1}|.$$

Rademacher’s theorem for Lipschitz continuous functions implies that $\boldsymbol{\alpha}(t)$ is differentiable at a.e. $t$.

We claim that $\boldsymbol{\alpha}=\boldsymbol{\alpha}_{\infty}$. Since they start at the same point, the uniqueness of $\infty$-streamlines shows that it is enough to verify

$$\frac{d\boldsymbol{\alpha}(t)}{dt}=\nabla u_{\infty}(\boldsymbol{\alpha}(t)).$$

To this end, we shall employ the convex functions $F_{p}(t)=u_{p}(\boldsymbol{\alpha}_{p}(t))$. Indeed,

$$\frac{dF_{p}(t)}{dt}=\Big{\langle}\nabla u_{p}(\boldsymbol{\alpha}_{p}(t)),\frac{d\boldsymbol{\alpha}_{p}(t)}{dt}\Big{\rangle}=|\nabla u_{p}(\boldsymbol{\alpha}_{p}(t))|^{2}$$

and

$$\frac{d^{2}F_{p}(t)}{dt^{2}}=2\,\Delta_{\infty}u_{p}(\boldsymbol{\alpha}_{p}(t))=-\frac{2}{p-2}\,\Delta u_{p}(\boldsymbol{\alpha}_{p}(t))\,|\nabla u_{p}(\boldsymbol{\alpha}_{p}(t))|^{2}.$$

By Lewis’s theorem, $\Delta u_{p}\leq 0$ in convex ring domains, if $p\geq 2$. Thus,

$$\frac{d^{2}F_{p}(t)}{dt^{2}}\geq 0$$

and so the function $F_{p}(t)$ is convex. The convergence

$$F_{p}(t)=u_{p}(\boldsymbol{\alpha}_{p}(t))\to u_{\infty}(\boldsymbol{\alpha}(t))=F(t)$$

is at least locally uniform, when $p$ takes the values $p_{1},p_{2},p_{3},\ldots$ extracted above. Also the limit $F(t)$ is convex, of course.

We have the locally uniform convergence

$$|\nabla u_{p}(\boldsymbol{\alpha}_{p}(t))|^{2}\to|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|^{2},$$

which follows from Theorem 7 by writing

$$|\nabla u_{p}(\boldsymbol{\alpha}_{p}(t))|-|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|=|\nabla u_{p}(\boldsymbol{\alpha}_{p}(t))|-|\nabla u_{p}(\boldsymbol{\alpha}(t))|+|\nabla u_{p}(\boldsymbol{\alpha}(t))|-|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|.$$

Thus,

$$\frac{dF_{p}(t)}{dt}=|\nabla u_{p}(\boldsymbol{\alpha}_{p}(t))|^{2}\,\to\,|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|^{2}.$$

It follows that³³3$\int|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|^{2}\phi(t)dt\leftarrow\int F_{p}^{\prime}(t)\phi(t)dt=-\int F_{p}(t)\phi^{\prime}(t)dt\to-\int F(t)\phi^{\prime}(t)dt$ $F^{\prime}(t)=|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|^{2}$ for a.e. $t$. We also have by the chain rule

$$\frac{dF(t)}{dt}=\Big{\langle}\nabla u_{\infty}(\boldsymbol{\alpha}(t)),\frac{d\boldsymbol{\alpha}}{dt}\Big{\rangle}$$

a.e., since $\frac{d\boldsymbol{\alpha}}{dt}$ exists for a.e. $t$.

We have arrived at the identity

$$|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|^{2}=\Big{\langle}\nabla u_{\infty}(\boldsymbol{\alpha}(t)),\frac{d\boldsymbol{\alpha}}{dt}\Big{\rangle}$$

valid for a.e. $t$. From

$$\boldsymbol{\alpha}_{p}(t_{2})-\boldsymbol{\alpha}_{p}(t_{1})\leq\int_{t_{1}}^{t_{2}}|\nabla u_{p}(\boldsymbol{\alpha}_{p}(t))|dt,$$

we get

$$\boldsymbol{\alpha}(t_{2})-\boldsymbol{\alpha}(t_{1})\leq\int_{t_{1}}^{t_{2}}|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|dt,$$

and, hence for a.e. $t$

$$\Big{|}\frac{d\boldsymbol{\alpha}(t)}{dt}\Big{|}\leq|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|.$$

We conclude that in the Cauchy-Schwarz inequality

$$|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|^{2}=\Big{\langle}\nabla u_{\infty}(\boldsymbol{\alpha}(t)),\frac{d\boldsymbol{\alpha}}{dt}\Big{\rangle}\leq|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|\Big{|}\frac{d\boldsymbol{\alpha}}{dt}\Big{|}\leq|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|^{2}$$

we have equality. It follows that

$$\frac{d\boldsymbol{\alpha}}{dt}=\nabla u_{\infty}(\boldsymbol{\alpha}(t))$$

for a.e. $t$. In fact, it holds everywhere because now the identity

$$\boldsymbol{\alpha}(t_{2})-\boldsymbol{\alpha}(t_{1})=\int_{t_{1}}^{t_{2}}\nabla u_{\infty}(\boldsymbol{\alpha}(t))dt$$

can be differentiated. This concludes our proof of the fact $\boldsymbol{\alpha}=\boldsymbol{\alpha}_{\infty}$.

We see that the tangent $\frac{d\boldsymbol{\alpha}}{dt}$ is continuous. The proof reveals that the convex functions $F_{p}\to F$ uniformly and hence $F$ is convex as well. Therefore, its derivative

$$F^{\prime}(t)=|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|^{2}$$

is non-decreasing. In other words, $|\nabla u_{\infty}|^{2}$ is non-decreasing along the limit streamline.

This proves Theorem 1.

Curved quadrilaterals and triangles, bounded by arcs of streamlines and level curves, are useful building blocks. It is tentatively understood that at least the interior of the figures are comprised in $G$; the level arcs can be on $\partial\Omega$ and, occasionally, on $\partial K$.

Recall that the $\infty$-streamline

$$\boldsymbol{\alpha}(t),\quad 0\leq t\leq T,$$

with initial point $\boldsymbol{\alpha}(0)=a\in\partial\Omega$ is unique and terminates at $\boldsymbol{\alpha}(T)$ on $\partial K$. On its way, it may (and usually does) meet other streamlines and has common parts with them. By Theorem 1, the speed

$$\left|\frac{d\boldsymbol{\alpha}(t)}{dt}\right|=|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|$$

is non-decreasing. Thus we have the bound⁴⁴4 $$|\nabla u_{\infty}(\boldsymbol{\alpha}(T))|=\lim_{t\to T-}|\nabla u_{\infty}(\boldsymbol{\alpha}(t))|$$

$$|\nabla u_{\infty}(\boldsymbol{\alpha}(t_{1}))|\leq|\nabla u_{\infty}(\boldsymbol{\alpha}(t_{2}))|,\quad 0\leq t_{1}\leq t_{2}\leq T.$$

Sometimes the result below (cf. Lemma 12 in [LL]), valid for curved quadrilaterals and triangles, provides us with the reverse inequality, so that we may even conclude that the speed is constant along suitable arcs of streamlines.

Suppose now that $\xi\in\overline{ab}$ is a point on the lower level curve $\boldsymbol{\sigma}$ at which

$$|\nabla u_{\infty}(\xi)|=\max_{\overline{ab}}|\nabla u_{\infty}(\boldsymbol{\sigma})|=M.$$

Let $\boldsymbol{\mu}$ be the streamline that passes through $\xi$. It intersects $\boldsymbol{\omega}$ at some point $\eta\in\overline{a^{\prime}b^{\prime}}$ (it may have joined $\boldsymbol{\alpha}$ or $\boldsymbol{\beta}$ before reaching $\eta$). See Figure 3. The following result holds:

We can also formulate a similar result for curved triangles. Suppose that the streamlines $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ together with the level curve $\boldsymbol{\sigma}$ form a curved triangle with vertices $a,b$ and $c$. Assume again that $\xi\in\overline{ab}$ is a point at which

$$|\nabla u_{\infty}(\xi)|=\max_{\overline{ab}}|\nabla u_{\infty}(\boldsymbol{\sigma})|=M.$$

Let $\boldsymbol{\mu}$ be the streamline that passes through $\xi$. It passes through $c$ (but may have joined $\boldsymbol{\alpha}$ or $\boldsymbol{\beta}$ before reaching $c$). The following result holds: