In Hilbert space, the greatest biconvex function in a class of biconvex functions satisfying certain boundary conditions was explicitly found by Burkholder [2] in 1986, see Theorem 3.1 in Section 3. Such a result would imply *ζ*-convexity which is one of the three equivalent conditions giving some geometric characterizations of Banach spaces, see [3]:

$$\zeta\mbox{-convex} \quad \Longleftrightarrow\quad \mbox{UMD}\quad \Longleftrightarrow\quad \mbox{HT-space}. $$

To study the extremal problems further in general Banach spaces, we may investigate the characterizations of such extremal functions. We consider the problem of finding the extremal function in the class of real-valued biconvex functions *u* satisfying a boundary condition *ψ* on a domain \(D\times D\) where *D* is a convex domain in \(\mathbb {R}^{2}\). In particular, we restrict the domain *D* to the unit ball in the \(\ell^{p} \)-norm or the unit disk when \(p=2\). We establish that sufficiently smooth solutions to the convex extremal problems with given boundary values are affine on line segments and the domain *D* is foliated by such segments.

In \(\ell^{p}(\mathbb{R}^{2})\), we denote the \(\ell^{p}\)-norm by \(\vert x\vert _{p}\) where \(x=(x_{1},x_{2})\) in \(\mathbb{R}^{2}\). Thus \(\vert x\vert _{p}^{p}=\vert x_{1}\vert ^{p}+\vert x_{2}\vert ^{p}\). We define the open and closed unit balls in \(\mathbb{R}^{2}\) for the \(\ell^{p}\)-metric by \(B= \{x\in\mathbb{R}^{2}:\vert x\vert _{p}<1 \}\) and \(\overline{B}= \{x\in\mathbb{R}^{2}:\vert x\vert _{p}\leq1 \}\).

Let \(u:\overline{B}\times\overline{B}\rightarrow\mathbb{R}\) be a continuous function that is in \(C^{3}\) in \(B\times B\). Fix \(y=(y_{1},y_{2})\in\overline{B}\). Then we can consider *u* as a function of \(x_{1}\) and \(x_{2}\), where \(x=(x_{1},x_{2})\) in *B̅*. For \(x_{1}=\cos{\theta}\), \(x_{2}=\sigma(\theta)=\pm (1-|\cos(\theta )|^{p} )^{1/p}\) and \(y=(y_{1},y_{2})\in\overline{B}\), we define a boundary function *ψ* by

$$ \psi(\theta)=u\bigl(\cos(\theta),\sigma(\theta)\bigr)= \bigl(\bigl\vert \cos(\theta )+y_{1}\bigr\vert ^{p}+\bigl\vert \sigma(\theta)+y_{2}\bigr\vert ^{p} \bigr)^{1/p} = \vert x+y\vert _{p} . $$

(1)

When \(p=2\), this function *ψ* is the same boundary function as in Burkholder’s result [2].

We denote the partial derivatives of a function \(u=u(x_{1},x_{2})\) by using subscripts, for example,

$$u_{x_{1}x_{2}} = \frac{ \partial^{2} u }{ \partial x_{1}\, \partial x_{2} } . $$

We define a real-valued function *A* on *B̅* by

$$ A(x_{1},x_{2})=\frac{u_{x_{1}x_{1}}}{u_{x_{1}x_{2}}}= \frac{u_{x_{1}x_{2}}}{u_{x_{2}x_{2}}}, $$

(2)

assuming that \(u_{x_{1}x_{1}}\), \(u_{x_{2}x_{2}}\) and \(u_{x_{1}x_{2}}\) do not vanish there (a milder requirement would be that \(u_{x_{1}x_{1}}\), \(u_{x_{2}x_{2}}\) and \(u_{x_{1}x_{2}}\) do not vanish on any open set, but we do not address that here). If \(u\in C^{3}\), then differentiation shows that we have the relation

$$AA_{x_{2}}=A_{x_{1}}, $$

which is a special case of the Hopf differential equation. Hence, the general solution \(A=A(x_{1},x_{2})\) of this equation is given implicitly by

$$ \Phi(A)=x_{2}+Ax_{1}, $$

(3)

where Φ is a suitable real-valued function (see [4]). Different admissible choices for Φ give rise to different solutions *A*.

From this perspective and using these notations, we obtain the following theorems.

### Theorem 2.1

*Let*
\(A=A(x_{1},x_{2})\)
*be a continuous real*-*valued function defined for*
\((x_{1},x_{2})\in D\), *where*
*D*
*is the closure of a bounded convex plane domain*. *If* Φ *is a continuous real*-*valued function on an interval of the real axis that contains the set*
\(A(D)\), *and if*
\(\Phi (A(x_{1},x_{2}))=x_{2}+x_{1}A(x_{1},x_{2})\)
*in*
*D*, *then*
\(A(x_{1},x_{2})\)
*is constant on certain line segments that are maximal in the sense that each segment is the intersection of*
*D*
*with a straight line*. *The union of such line segments is*
*D*. *Moreover*, *if the real*-*valued function*
*u*
*is in*
\(C^{2}\)
*in*
*D*
*and satisfies* (2) *there* (*which*, *in particular*, *means that*
\(u_{x_{2}x_{2}}\)
*and*
\(u_{x_{1}x_{2}}\)
*do not vanish in*
*D*), *then*
*u*
*is affine on each such segment*.

### Remark 2.2

When \(D=\overline{B}\), Theorem 2.1 gives a foliation of the entire domain *B̅*.

### Theorem 2.3

*Let*
*u*
*be a convex*
\(C^{2}\)
*real*-*valued function on a bounded convex plane domain*
*D*. *If*
*u*
*is affine on an open line segment*
*L*
*in*
*D*, *then at each point of*
*L*
*we have*

$$ u_{x_{1}x_{1}}u_{x_{2}x_{2}}-u_{x_{1}x_{2}}^{2}=0. $$

(4)

*Furthermore*, *at each point of*
*L*
*where*
\(u_{x_{1}x_{2}}\ne0\)
*and*
\(u_{x_{2}x_{2}}\ne0\), *we have*

$$\frac{u_{x_{1}x_{1}}}{u_{x_{1}x_{2}}}=\frac{u_{x_{1}x_{2}}}{u_{x_{2}x_{2}}} . $$

Let now *D* be the closure of a bounded convex plane domain, let *ψ* be a continuous real-valued function on *∂D*, and let \({\mathcal{F}}\) be the set of real-valued convex functions *v* on *D* such that \(v\leq\psi\) on *∂D*. Define

$$u(x) =\sup\bigl\{ v(x) : v\in{\mathcal{F}} \bigr\} . $$

Following Burkholder, we see that *u* is convex and \(u\in{\mathcal{F}}\), so that *u* is the maximal function in \({\mathcal{F}}\). Suppose that *u* is in \(C^{3}\) in the interior of *D*. The convexity of *u* implies that at each point we have \(u_{x_{1}x_{1}}\geq0\), \(u_{x_{2}x_{2}}\geq 0\), and \(u_{x_{1}x_{1}}u_{x_{2}x_{2}}-u_{x_{1}x_{2}}^{2}\geq0\). As noted by Burkholder, at each point equality must hold in at least one of these inequalities, for otherwise we may modify *u* slightly in a small neighborhood of the point and get an even larger element of \({\mathcal{F}}\). Now if \(u_{x_{1}x_{1}}= 0\) or \(u_{x_{2}x_{2}}= 0\), then from

$$u_{x_{1}x_{1}}u_{x_{2}x_{2}}-u_{x_{1}x_{2}}^{2} = - u_{x_{1}x_{2}}^{2} \geq0 $$

we deduce that \(u_{x_{1}x_{2}}=0\), so that

$$u_{x_{1}x_{1}}u_{x_{2}x_{2}}-u_{x_{1}x_{2}}^{2} = 0 . $$

Thus (4) holds at each point of the interior of *D*.

If we now assume that \(u_{x_{1}x_{2}}\ne0\) and \(u_{x_{2}x_{2}}\ne0\) in *D*, it follows that we may define the function *A* as in (2), and then *A* is given by (3) for a suitable function Φ. Now from Theorem 2.1 we see that *u* is affine on line segments that foliate *D*. Thus we obtain the following result.

### Theorem 2.4

*Let*
*D*
*be the closure of a bounded convex plane domain*, *let*
*ψ*
*be a continuous real*-*valued function on*
*∂D*, *and let*
*u*
*be the maximal real*-*valued convex function on*
*D*
*such that*
\(u\leq\psi\)
*on*
*∂D*. *Suppose that*
\(u\in C^{3}\). *Then* (4) *holds at each point of the interior of*
*D*. *Furthermore*, *if*
\(u_{x_{1}x_{2}}\ne0\)
*and*
\(u_{x_{2}x_{2}}\ne0\)
*in*
*D*, *then we may define the function*
*A*
*as in* (2), *and then*
*A*
*is given by* (3) *for a suitable function* Φ. *Finally*, *u*
*is affine on line segments that foliate*
*D*.

We conclude with the remark that the property of *u* being affine on line segments is equivalent to (4) in a suitable sense.

### Theorem 2.5

*Let*
*D*
*be a bounded convex plane domain*. *Let*
\(u:D \rightarrow\mathbb{R}\)
*be a convex function in the class*
\(C^{3}\). *Then*, *on certain line segments that are maximal in the sense that each segment is the intersection of*
*D*
*with a straight line*, *we have*

$$ \frac{u_{x_{1}x_{1}}}{u_{x_{1}x_{2}}}=\frac{u_{x_{1}x_{2}}}{u_{x_{2}x_{2}}} \quad \textit{if, and only if,} \quad u \textit{ is affine on the line segment}, $$

(5)

*assuming that*
\(u_{x_{1}x_{2}}\ne0\)
*and*
\(u_{x_{2}x_{2}}\ne0\).

Namely, if *u* is affine on line segments as stated, then it follows from Theorem 2.3 that \(u_{x_{1}x_{1}}/u_{x_{1}x_{2}}=u_{x_{1}x_{2}}/u_{x_{2}x_{2}}\) on these segments if the denominators are assumed to be non-zero. Conversely, if the denominators are non-zero and \(u_{x_{1}x_{1}}/u_{x_{1}x_{2}}=u_{x_{1}x_{2}}/u_{x_{2}x_{2}}\), then we may define *A* as in (2), and it follows as explained above that *u* is affine on line segments that foliate *D*. This proves Theorem 2.5.