Convexity in Multivalued Harmonic Functions

IMMANUEL BEN PORAT

We investigate variants of a Three Circles type Theorem in the context
of $\mathcal{Q}-$valued functions. We prove some convexity inequalities
related to the $L^{2}$ growth function in the $\mathcal{Q}-$valued
settings. Optimality of these inequalities and comparsion to the case
of real valued harmonic functions is also discussed.

§ INTRODUCTION

§.§ Background

The study of multivalued harmonic functions was originated in the
pioneering work of Almgren [5] on Plateau's problem, which asks
for a surface of minimal area among all surfaces stretched accross
a given closed contour. Almgren's theory was further extended and
simplified in [1]. The profound geometric applications of Almgren's
theory to minimal surfaces are not addressed here. Instead, we shall
connect the theory of $\mathcal{Q}-$valued functions to a classical
results from complex analysis, which has some modern reflections.
Let us begin by providing some background to the material that motivated
this work. Let $0\neq u$ be harmonic on the unit ball $B_{1}(0)$.
Then one can associate to $u$ real valued functions $H_{u},D_{u},\overline{H}_{u},I_{u}:\mathrm{(0,1)\rightarrow\mathbb{R}}$
by letting

\[
H_{u}(r)=\underset{\partial B_{r}(0)}{\int}u^{2}(x)d\sigma,D_{u}(r)=\underset{B_{r}(0)}{\int}|\nabla u|^{2}dx,\overline{H}_{u}(r)=\frac{1}{|\partial B_{r}(0)|}H_{u}(r),I_{u}(r)=\frac{rD_{u}(r)}{H_{\mathit{u}}(r)}
\]

$\overline{H}_{u}(r)$ is called the $L^{2}-$growth function of $u$
and $I_{u}(r)$ is called the frequency function of $u$. The motivation
behind the definition of the function $\overline{H}_{u}$ comes from
a classsical result in complex analysis known as the Three Circles

Given a holomorphic function $f$ on the unit ball $B_{1}$, let $M(r)=\underset{B_{r}(0)}{\max}|f|$.
The Three Circles Theorem, proved by Hadamard, states that the function
$\widetilde{M}:(-\infty,0)\rightarrow\mathbb{R}$ defined by $\widetilde{M}(t)=M(e^{t})$
is $\log$ convex, that is $\log\widetilde{M}(t)$ is convex. It is
therefore natural to seek for a Three Circles type Theorem for real
harmonic functions $u:B_{1}(0)\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$.
It was first observed by Agmon [6] that such a theorem holds
if the function $M$ is replaced by an appropriate $L^{2}$-version
on the sphere. Namely, Agmon proves that the function $t\mapsto\overline{H}_{u}(e^{t})$
is $\log$ convex.

In 2015, Lippner and Mangoubi observed the following stronger result:

. Let
$u:B_{1}(0)\rightarrow\mathbb{R}$ be harmonic. Then $\overline{H}_{u}$
is absolutely monotonic, that is $\overline{H}_{u}^{(k)}\geq0$ for
all $k\in\mathbb{N}$. In particular, $H_{u}^{(k)}\geq0$ for all

Since Theorem <ref> in [3] is carried out in a
discrete setting, for the sake of completeness a proof of the continuous
version (as stated above) is presented in the appendix. The second
statement in Theorem <ref> is an immediate consequence
of the first one. It is an exercise to verify that absolute monotinicity
of $\overline{H}$ implies $\log$ convexity of $t\mapsto\overline{H}(e^{t})$
(see [7], II, Problem 123). Roughly speaking, we are interested
in the question whether a Lippner-Mangoubi type theorem can be obtained
in the more general setting of multivalued harmonic functions. Let
us emphasize this could have fascinating applications in the regularity
theory of these objects, since absolutely monotonic functions are
real analytic (due to a celebrated theorem of Bernstein. See [2]).
In some sense, the nonlinear nature of the problem is the main obstacle
in obtaining elliptic regularity type results for multivalued harmonic
functions. We hope that approaching the problem via Bernstein's theorem
may be useful in overcoming some of the difficulties that are created
by the lack of linearity.

§.§ Main Results

Given some $P\in\mathbb{R}^{n}$ we denote by $[[P]]$ the Dirac mass
in $P\in\mathbb{R}^{n}$ and define

\[
\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n}):=\{\stackrel[i=1]{\mathfrak{\mathcal{Q}}}{\sum}[[P_{i}]]|P_{i}\in\mathbb{R}^{n},1\le i\le\mathcal{Q}\}
\]

The set $\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$
is endowed with a metric $\mathcal{G}$, not specified for the moment,
such that the space $(\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}),\mathcal{G})$
is a complete metric space. We then consider functions $f:\Omega\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$,
where $\Omega$ is some domain in $\mathbb{R}^{m}$. We call such
functions $\mathcal{Q}$-valued functions. One key fact is the existence
of a notion of a harmonic $\mathcal{Q}$-valued function.

We adapt the terminology of [1] and call such functions Dir-minimizing.
As their name suggests, Dir-minimizing functions are defined as functions
minimizing a certain class of integrals, by analogy with the classical
Dirichlet principle. For each $f:B_{1}(0)\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n})$
Dir-minimizing we associate a real valued function $\overline{H}_{f}:(0,1)\rightarrow\mathbb{R}$
by letting: $\overline{H}_{f}(r)=\frac{1}{|\partial B_{r}(0)|}\underset{\partial B_{r}(0)}{\int}|f|{}^{2}d\sigma$.
The function $\overline{H}_{f}$ is a generalization of the function
introduced in the beginning. Our first aim would be to generlize Agmon's
Theorem to the multivalued case. We will prove

Let $f:B_{1}(0)\subset\mathbb{R^{\mathit{m}}}\rightarrow\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$
be Dir minimizing such that $H(r)>0$. Define $a:(-\infty,0)\rightarrow\mathbb{R}$
by $a(t)=\log\overline{H}(e^{t})$. Then

$a'(t)\geq0$ for all $t\in(-\infty,0)$

for a.e.

Furthermore, $a$ is convex.

Since (ii) holds merely up to a null set, the convexity of $a$ does
not follow directly. This requires an additional consideration. The
following theorem, is the main result of this work

Let $f:B_{1}(0)\subset\mathbb{R}^{2}\rightarrow\mathcal{A_{Q}}(\mathbb{R}^{n})$
be a Dir minimizing function. Suppose $f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r}(0),\mathcal{A_{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$
for a.e. $0<r<1$. For each $N>0$ define
$\overline{h}_{N,f}:(0,1)\rightarrow\mathbb{R}$ by $\overline{h}_{N,f}(r)=\overline{H}_{f}(r^{N}).$

$\overline{h}'_{\frac{\mathcal{Q}}{2},f}(r)\geq0$ for
all $r\in(0,1)$

for a.e. $r\in(0,1)$

Furthermore, $\overline{h}_{\frac{\mathcal{Q}}{2}}$is convex.

The proof of Theorem <ref> will be followed by a higher
dimensional version, that is, when the domain is the $m-$dimensional
unit baḷl for arbitrary $m>2$. In the higher dimensional version,
the constant $\frac{\mathcal{Q}}{2}$ will be replaced by some constant
depending on $m$ which does not have a simple closed formula. It
should be remarked that unlike in the scenario of Theorem <ref>,
the fact that $\overline{H}_{f}$ (and hence $a$ and $\overline{h}_{N,f}$
) is a.e. twice differentiable (and moreover $C^{1}$) is nontrivial.

A naive version of Theorem <ref> for $\mathcal{Q}$-valued
functions is not valid, as wittnesed by the example $f(z)=\underset{w^{3}=z}{\sum}[[w]]$,
for which the associated $\overline{H}$ function has a negative second
derivative for all $0<r<1$. In addition, the following proposition
demonstrates that we do not have an obvious third derivative version
of Theorem <ref>:

Define $f:B_{1}(0)\subset\mathbb{R}^{2}\mathbb{\rightarrow\mathcal{A}_{\mathrm{2}}\mathrm{(\mathbb{R^{\mathrm{2}}\mathrm{)}}}}$
by $f(z)=\underset{w^{2}=2z-1}{\sum}[[w]]$. Then $f$ is Dir minimizing,$f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r},\mathcal{A_{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$
for all $r\neq\frac{1}{2}$ and $\overline{h}'''_{1,f}(r)=\mathcal{\mathit{\overline{H}_{f}'''}}(r)<0$
for all $\frac{1}{2}<r<1$.

Both Proposition <ref> and the abovementioned example
will be proved and explained in section <ref>. In Proposition
<ref>, our main conribution is performing the formal
computation which shows that $\overline{H}_{f}'''<0$ for all $\frac{1}{2}<r<1$
and showing that the boundary condition $f|_{\text{\ensuremath{\partial}}B_{r}}\text{\ensuremath{\in}}W^{1,2}(\text{\ensuremath{\partial}}B_{r},\mathcal{A_{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$
is indeed satisfied for all $r\neq\frac{1}{2}$. Proving that $f$
is Dir minimizing is a difficult task, and relies on some rather heavy
machinery from geometric measure theory. It should be emphasized that
the domain of $f$ in both counterexamples is the planar unit disk.
Thus, we did not rule out the possibillity that in higher dimensions
the $L^{2}$- growth function of $f$ is more well behaved.

Organization of the paper. In Section <ref>, we
fix some notation and briefly review the frequency function and its
relatives. A detailed exposition may be found in [1]. Section
<ref> is devoted mainly to the proof of Proposition <ref>,
Theorem <ref> and other related convexity inequalities.
In Section <ref> we preform the calculations required to
establish the counterexample given in Proposition <ref>.
In the same context we will prove that the boundary condition $f|_{\text{\ensuremath{\partial}}B_{r}}\text{\ensuremath{\in}}W^{1,2}(\text{\ensuremath{\partial}}B_{r}(0),\mathcal{A_{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$
in Theorem <ref> is in fact verified for a certain class
of Dir-minimizing functions on the unit disk.

§ ACKNOWLEDGEMENT

This work is part of the author's MA thesis. I wish to express my
most deep gratitude towards my supervisor Dan Mangoubi for suggesting
and guiding me in this problem, for many fruitful and stimulating
disscusions and for carefully reading previous versions of this manuscript
and providing insightful comments. In particular, for explaining the
proof of Theorem <ref>. I would also like to thank Camillo
De Lellis for his interest in this work. In particular, for suggesting
the counterexample that appears in Proposition <ref>
and for explaining the proof of Theorem <ref>.

§  PRELIMINARIES

§.§ Notations

$d\sigma=$The surface measure

$\Delta u=$The Laplacian of $u$

$\cdot=$The standard scalar product on $\mathbb{R}^{m}$

$\nu=$The unit normal to the sphere

$C_{m}=$Surface area of the $m-$dimensional unit sphere

$B_{R}(x)=$The ball of radius $R$ centered at $x$

We assume that the reader is familiar with the basic theory of $\mathcal{Q}$-valued
functions. Following [1], we recall some basic notions and terminology.
Given some $P\in\mathbb{R}^{n}$ we denote by $[[P]]$ the Dirac mass
in $P\in\mathbb{R}^{n}$ and define $\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n}):=\{\stackrel[i=1]{\mathcal{Q}}{\sum}[[P_{i}]]|P_{i}\in\mathbb{R}^{n},1\le i\le\mathcal{Q}\}$.
We endow $\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$
with a metric $\mathcal{G}$, defined as follows: For each $T_{1},T_{2}\in\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$
with $T_{1}=\stackrel[\mathit{i=\mathrm{1}}]{\mathcal{Q}}{\sum}[[P_{i}]],T_{2}=\stackrel[\mathit{i=\mathrm{1}}]{\mathcal{Q}}{\sum}[[S_{i}]]$
we define $\mathcal{G\mathrm{(\mathit{T_{\mathrm{1}},T_{\mathrm{2}}})=\underset{\sigma\in\mathscr{P_{\mathcal{Q}}}}{min}\sqrt{\stackrel[\mathit{i=\mathrm{1}}]{\mathcal{Q}}{\sum}|\mathit{P_{i}-S_{\sigma\mathrm{(}i\mathrm{)}}}|^{2}}}}$,
where $\mathscr{P_{\mathcal{Q}}}$ is the permutation group on $\{1,...,\mathcal{Q}\}$.
The space $(\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}),\mathcal{G})$
is a complete metric space. A $\mathcal{Q}$- valued function is a
function $f:\Omega\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$.
Of course, this formalism was designed to capture the notion of a
function attaining multiple values at each point. A regularity theory
can be developed for $\mathcal{Q}$-valued functions. In particular,
the notion of a Sobolev space $W^{1,p}(\Omega,\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$
and the notion of an approximate differential denoted by $Df$.

Suppose $\Omega\subset\mathbb{R}^{m}$ is a bounded domain with smooth
boundary. By analogy with the Dirichlet principle we say that a function
is Dir-minimizing if $f\in W^{1,2}(\Omega,\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$

\[
\underset{\Omega}{\int}|Df|^{2}\leq\underset{\Omega}{\int}|Dg|^{2}
\]

for all $g\in W^{1,2}(\Omega,\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$
whose trace on $\partial\Omega$ agrees with that of $f$. We shall
allways assume $m\geq2$.

3.2. Frequency Function.

We recall the frequency function and its relatives in the context
of $\mathcal{Q}-$valued functions. We have the following Holder regularity
type theorem for Dir-minimizing functions:

There are constants $\alpha=\alpha(m,\mathcal{Q})\in(0,1)$ and $C=C(m,n,\mathcal{Q\mathrm{,}\delta\mathit{\mathrm{)}}}$
with the following property. If $f:B_{1}(0)\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}})}$
is Dir-minimizing then: $\underset{x\neq y\in\overline{B_{\delta}(0)}}{\sup}\frac{\mathcal{G\mathrm{(\mathit{f\mathrm{(\mathit{x\mathrm{),\mathit{f\mathrm{(\mathit{y}))}}}}}}}}}{|x-y|^{\alpha}}$$\leq CDir(f)^{\frac{1}{2}}$
for all $0<\delta<1$.

§  PROOF OF MAIN RESULTS

§.§ Variants of the Three Circles Theorem

In this section we give a proof of Proposition <ref>
and Theorem <ref>. The following identities will play
a crucial role in the study of convexity of the frequency function
and its relatives

Let $f:B_{R}(x)\subset\mathbb{R^{\mathit{m}}\rightarrow\mathcal{A_{Q}\mathrm{(\mathbb{R}^{\mathit{n}})}}}$
be Dir-minimizing.

Then for a.e. $0<r<R$ we have: (i) $(m-2)$$\underset{B_{r}(x)}{\int}|Df|^{2}dx=r\underset{\partial B_{r}(x)}{\int}|Df|^{2}d\sigma-2r\underset{\partial B_{r}(x)}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}|\partial_{\nu}f_{i}|^{2}d\sigma$

(ii) $\underset{B_{r}(x)}{\int}|Df|^{2}dx=\underset{\partial B_{r}(x)}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}(\partial_{\nu}f_{i})\cdot f_{i}d\sigma$

Our starting point is the following theorem

be Dir minimizing. Then:

(a) $H\in C^{1}(0,1)$ and the following identity holds for
all $r\in(0,1):$

\begin{equation}
\end{equation}

(b) If $H(r)>0$ then $I(r)$ is absolutely continuous and
nondecreasing. In particular, $I'(r)\geq0$ for a.e. $r$.

Statement (b) is known as Almgren's monotonicity formula. Since $D$
is absolutely continuous, it is a.e. differentiable. Therefore, in
view of equation <ref>, we see that $H'$ is a.e. differentiable.
Otherwise put, the second derivative of $H$ exists a.e. The regularity
properties of $H$ clearly apply for $\overline{H}$ as well. With
the aid of Almgren's monotonicity formula we are able to extend Agmon's
convexity result ([6]) in the context of $\mathcal{Q}-$valued
functions. This method of proof differs from Agmon's original approach
for real valued harmonic functions, which involves ODEs on Banach

Proof of Proposition 1.2. In light of the above discussion
it is clear that $a$ is $C^{1}(-\infty,0)$ and that $a''$ exists
almost everywhere. For (i), note that

\[
\]

\[
\frac{1}{r^{m-1}}(\frac{m-1}{r}H(r)+2D(r))-(m-1)\frac{1}{r^{m}}H(r)=\frac{2D(r)}{r^{m-1}}\geq0
\]

Where the second equality is due to equation <ref>. So

\begin{equation}
\overline{H}'(r)=\frac{2D(r)}{C_{m}r^{m-1}}\geq0\label{eq:7}
\end{equation}

Therefore, for all $t\in(-\infty,0)$

\begin{equation}
\end{equation}

For (ii), start by noting that $a(t)=\log(\overline{H}(e^{t}))=\log(H(e^{t}))-\log(C_{m}e^{t(m-1)})=\log(H(e^{t}))-\log(C_{m})-(m-1)t$.
Therefore, to prove $a''(t)\geq0$ it suffices to prove $(\log(H(e^{t})))''\geq0$.
By Theorem <ref>, we have that $I'(r)\geq0$ for a.e.
$0<r<1.$ To spare some space, all equalities and inequalities from
now on should be interpreted up to a null set. By virtue of equation

\[
0\leq I'(r)=(\frac{rD(r)}{H(r)})'=(\frac{D(r)}{r^{m-2}\overline{H}(r)})'=\frac{1}{2}(\frac{r(H'(r)-(\frac{m-1}{r})H(r))}{H(r)})'=\frac{1}{2}(\frac{rH'(r)-(m-1)H(r)}{H(r)})'=
\]

\[
\frac{1}{2}(\frac{rH'(r)}{H(r)})'=\frac{1}{2}(\frac{(H'(r)+rH''(r))H(r)-r(H'(r))^{2}}{H^{2}(r)})
\]

Thus, we get

\begin{equation}
\end{equation}

On the other hand by a straightforward calculation:

\begin{equation}
\end{equation}

Combining inequality <ref> with equation <ref> we arrive
at $e^{-t}(\log(H(e^{t}))''\geq0$ which is the same as $(\log(H(e^{t}))''\geq0$.
We are left to explain why $a$ is convex. It is classical that a
continuously differentiable function is convex iff its derivative
is nondecreasing, and so our task reduces to showing that $a'$ is
nondecreasing. By equations <ref> and <ref> we get $a'(t)=\frac{\overline{H}'(e^{t})e^{t}}{\overline{H}(e^{t})}=\frac{2D(e^{t})}{C_{m}e^{t(m-2)}\overline{H}(e^{t})}$.
Since $D(e^{t})$ is a composition of an absolutely continuous function
with a nondecreasing smooth function, it is absolutely continuous.
In addition, $\frac{1}{C_{m}e^{t(m-2)}\overline{H}(e^{t})}$ is differentiable.
So $a'(t)$ is absolutely continuous function on any closed subinterval
of $(-\infty,0)$, as a product of such function. Therefore, the fundamental
theorem of calculus is applicable: if $t_{1},t_{2}\in(-\infty,0),t_{1}<t_{2}$
then $a'(t_{2})-a'(t_{1})=\stackrel[t_{1}]{t_{2}}{\int}a''(t)dt\geq0$.

We draw the reader's attention to a somewhat delicate point, which
will be also relevant in what will come next . The implication “nonnegative
derivative a.e.$\Rightarrow$nondecreasing” is not true in general.
In Proposition <ref> we employed the fact that the
first derivative of $a$ is absolutely continuous in order to deduce
that it is convex. The absolute continuity of the derivatives is a
consequence of equation <ref>. Hence, we implicitly relied
here on the Dir-minimization property. In the case of 1 valued harmonic
functions, this technicality is not created because all functions
involved are smooth. To the best of our knowledge, improved regularity
for the frequency function and its relatives in the multivalued settings
is still an open problem.

The inequality “$\overline{H}''(r)\geq0$ a.e.” is not true in
general, as witnessed by the counterexample in section <ref>.
Nevertheless, we are still able to obtain a convexity result by reducing
the power of the normalization of $H$. More precisely we observe
the weaker

Let $f:B_{1}(0)\subset\mathbb{R^{\mathit{m}}}\rightarrow\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$
be Dir minimizing. Then

(i) $(r\overline{H}(r))'\geq0$ for all $r\in(0,1)$ (ii) $(r\overline{H}(r))''\geq0$
for a.e. $r\in(0,1)$. Furthermore $r\mapsto r\overline{H}(r)$ is

Proof. As for (i) we can in fact derive a stronger result,
namely $\overline{H}'(r)\geq0$ for all $r\in(0,1)$. We compute:

\begin{equation}
\overline{H}'(r)=\frac{1}{C_{m}}[-(m-1)r^{-m}H(r)+r^{-(m-1)}H'(r)]=\frac{r^{-(m-1)}}{C_{m}}[H'(r)-\frac{m-1}{r}H(r)]=\frac{2D(r)r^{-(m-1)}}{C_{m}}\geq0\label{eq:3.4}
\end{equation}

where the last equality is by equation <ref>. For (ii), start
by noting that equation <ref> implies the following equality
for a.e. $r$

\begin{equation}
\overline{H}''(r)=\frac{2}{C_{m}r^{m-1}}[D'(r)-\frac{(m-1)D(r)}{r}]\label{eq:3.4'}
\end{equation}

Gathering our calculations one readily checks that $(r\overline{H}(r))''\geq0\iff rD'(r)-(m-3)D(r)\geq0$.

\[
rD'(r)-(m-3)D(r)\geq rD'(r)-(m-2)D(r)=
\]

\[
r\underset{\partial B_{r}(0)}{\int}|Df|^{2}-(m-2)\underset{B_{r}(0)}{\int}|Df|^{2}=2r\underset{\partial B_{r}(0)}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}|\partial_{\nu}f_{i}|^{2}\geq0
\]

Where the last equality is thanks to (i), <ref>.
The convexity of $r\mapsto r\overline{H}(r)$ follows by a similar
argument to the one demonstrated in Proposition <ref>.

It is clear that Proposition <ref> implies in particular
that the same conclusion holds true for $H$ (and this can also be
derived directly from iterating equation <ref>).

We present now a proof of Theorem <ref>, including a
higher dimensional analog. The main ingredients of the proof are the
variational formulas provided by Proposition <ref>
and the following estimates

Let $f:B_{1}(0)\subset\mathbb{R^{\mathit{m}}}\rightarrow\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}})}$
be Dir-minimizing and suppose that $g_{r}:=f|_{\partial B_{r}(0)}\in W^{1,2}(\partial B_{r}(0),\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}}))}$
for a.e $0<r<1$. Then for a.e $r$: (i) If $m=2$,
$\mathrm{Dir}(f,B_{r}(0))\leq\mathcal{Q}r\mathrm{Dir}(g_{r},\partial B_{r}(0))$
(ii) If $m>2$, $\mathrm{Dir}(f,B_{r}(0))\leq c(m)r\mathrm{Dir}(g_{r},\partial B_{r}(0))$,
where $c(m)<\frac{1}{m-2}$.

Before going into the proof of Theorem <ref>, let us
heuristically explain why it is reasonable the expect the validity
of such a result through the following example.

We recall that a function $f:B_{1}(0)\rightarrow\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}})}$
is $\alpha-$homogeneous ($\alpha>0)$ if $\forall y\in B_{1},y\neq0:f(y)=|y|^{\alpha}f(\frac{y}{|y|})$.
Denote by $\eta:\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}})}\rightarrow\mathbb{R}^{n}$
the center of mass map defined by $\eta(\stackrel[i=1]{\mathcal{Q}}{\sum}[[P_{i}]])=\frac{\stackrel[i=1]{\mathcal{Q}}{\sum}P_{i}}{\mathcal{Q}}$.
We can derive an explicit formula for the $L^{2}-$growth function
of a continuous $\alpha-$homogeneous map:

\[
H(r)=\underset{\partial B_{r}}{\int}|f|^{2}d\sigma=r^{m-1}\underset{\partial B_{1}}{\int}|f|^{2}(ry)d\sigma=
\]

\[
r^{m-1}\underset{\partial B_{1}}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}|f_{i}|^{2}(ry)d\sigma=r^{2\alpha+m-1}\underset{\partial B_{1}}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}|f_{i}|^{2}(y)d\sigma=\kappa r^{2\alpha+m-1}
\]

For some constant $\kappa\geq0$. Hence $\overline{H}$ takes the
form $\overline{H}(r)=\kappa r^{2\alpha}$. Assume now that $m=2$,
is a Dir-minimizing, nontrivial, $\alpha-$homogeneous map with $\eta\circ f=0$
(the simplest example of such a map the 2-valued function $f:B_{1}(0)\rightarrow\mathcal{A}_{2}(\mathbb{R}^{2})$
defined by $f(z)=\underset{w^{2}=z}{\sum}[[w]]$. See Theorem <ref>).
It is proved in [1], Proposition 8.2 that in this case necessarily
$\alpha=\frac{p}{q}\in\mathbb{Q}$ for some $q\leq\mathcal{Q}$. So
$\overline{H}(r)=\kappa r^{\frac{2p}{q}}$, which implies $\overline{H}(r^{\frac{q}{2}})=\kappa r^{p}$,
and the latter function is obviously absolutely monotonic. We are
thus lead to spectulate that more generally, composing $\overline{H}$
with some suitable $\frac{1}{2}[\mathcal{Q}]$-power produces a function
which is more well behaved. Theorem <ref> partially confirms
this speculation.

Proof of Theorem <ref>. That $\overline{h}'_{\frac{\mathcal{Q}}{2}}(r)\geq0$
follows immediately from equation <ref>. Henceforth all equalities
and inequlities should be interpeted up to a null set. A direct calculation

Writing $\xi=r^{N}$, we see that $\overline{h}_{N}''(\xi)\geq0\Leftrightarrow(\frac{N-1}{N})\overline{H}'(\xi)+\overline{H}''(\xi)\xi\geq0$.
Taking $N=\frac{\mathcal{Q}}{2}$ we get

\[
\overline{h}_{\frac{\mathcal{Q}}{2}}''(\xi)\geq0\Leftrightarrow(\mathcal{Q}-2)\overline{H}'(\xi)+\mathcal{Q}\overline{H}''(\xi)\xi\geq0
\]

Owing to equation <ref> we can express $\overline{H}',\overline{H}''$

\[
\overline{H}'(\xi)=\frac{2}{C\xi}\underset{B_{\xi}(0)}{\int}|Df|^{2}
\]

\[
\overline{H}''(\xi)=\frac{2}{C\xi}[\underset{\partial B_{\xi}(0)}{\int}|Df|^{2}-\frac{1}{\xi}\underset{B_{\xi}(0)}{\int}|Df|^{2}]
\]

We now have

\begin{equation}
C\xi[(\mathcal{Q}-2)\overline{H}'(\xi)+\mathcal{Q}\overline{H}''(\xi)\xi]=2(\mathcal{Q}-2)\underset{B_{\xi}(0)}{\int}|Df|^{2}+2\mathcal{Q}\xi[\underset{\partial B_{\xi}(0)}{\int}|Df|^{2}-\frac{1}{\xi}\underset{B_{\xi}(0)}{\int}|Df|^{2}]=2\mathcal{Q}\xi\underset{\partial B_{\xi}(0)}{\int}|Df|^{2}-4\underset{B_{\xi}(0)}{\int}|Df|^{2}\label{eq:3.7}
\end{equation}

Proposition <ref>, (i) combined with Proposition
<ref> yield the following estimate:

\[
\underset{B_{\xi}(0)}{\int}|Df|^{2}dx\leq\xi\mathcal{Q}\mathrm{Dir}(g_{\xi},\partial B_{\xi}(0))
\]

\[
2\underset{B_{\xi}(0)}{\int}|Df|^{2}\leq2\xi\mathcal{Q}\mathrm{Dir}(g_{\xi},\partial B_{\xi}(0))=2\xi\mathcal{Q}[\underset{\partial B_{\xi}(0)}{\int}|Df|^{2}-\stackrel[i=1]{\mathcal{Q}}{\sum}|\partial_{\nu}f_{i}|^{2}d\sigma]=\mathcal{Q}\xi\underset{\partial B_{\xi}(0)}{\int}|Df|^{2}
\]

Where the last equality is due to <ref>, (i).

Combining the last inequality with equation <ref> gives $(\mathcal{Q}-2)\overline{H}'(\xi)+\mathcal{Q}\overline{H}''(\xi)\xi\geq0$,
as wanted. Finally, note that $\overline{h}'_{\frac{\mathcal{Q}}{2}}(r)=\frac{2D(r^{\frac{\mathcal{Q}}{2}})r^{-\frac{\mathcal{Q}}{2}(m-1)}}{C_{m}}r^{\frac{\mathcal{Q}}{2}-1}=\frac{2D(r^{\frac{\mathcal{Q}}{2}})r^{\frac{\mathcal{Q}}{2}(2-m)-1}}{C_{m}}$.
Since $D(r^{\frac{\mathcal{Q}}{2}})$ is a composition of absolutely
continuous nondecreasing functions, it is absolutely continuous. In
view of the previous equation we see that $\overline{h}'_{\frac{\mathcal{Q}}{2}}$
is absolutely continuous. Thus, proceeding as in Proposition <ref>,
it follows that $\overline{h}{}_{\frac{\mathcal{Q}}{2}}$ is convex.

We finish this section by stating an higher dimensional analog of

Let $m>2$ and $f:B_{1}(0)\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{\mathit{n}})$
be a Dir minimizing function. Suppose $f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r},\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$
for a.e. $0<r<1$. Let $c(m)=\frac{1}{m-2}-\epsilon_{m}$
, $0<\epsilon_{m}<\frac{1}{m-2}$ be the constant obtained via proposition
<ref>. Let $\alpha_{m}=\frac{1-\epsilon_{m}(m-2)}{\epsilon_{m}(m-2)^{2}}$.

for all $r\in(0,1)$

for a.e. $r\in(0,1)$

The proof is identical to that of Theorem <ref>, using
estiamte (ii) in Proposition <ref> instead of (i).

We can now conclude that nontrivial Dir minimizing, $\alpha-$homogeneous
function have exponents $\alpha$ far away from $0$.

There are constants $\beta_{m}>0$ with the
following property. Let $f:B_{1}(0)\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{\mathit{n}})$
be a nontrivial Dir minimizing, $\alpha-$homogeneous function.
Suppose $f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r},\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$
for a.e. $0<r<1$. Then $\alpha\geq\beta_{m}$.

Put $\beta_{2}=\frac{1}{\mathcal{Q}}$ and $\beta_{m}=\frac{1}{\alpha_{m}},m>2$.
According to the computation performed in Example <ref>,
$\overline{H}(r)=\kappa r^{2\alpha}$ for some $\kappa>0$. According
to Theorem <ref> and Theorem <ref> we see
that $0\leq\kappa\frac{\alpha}{\beta_{m}}(\frac{\alpha}{\beta_{m}}-1)r^{\frac{\alpha}{\beta_{m}}-2}$
for a.e. $r$, which in particular gives $\alpha\geq\beta_{m}$.

Note that in the specific case that $m=2$ and $\eta\circ f=0$, Corollary
<ref> recovers a weaker form of Proposition 8.2, [1]
which was already mentioned in Example <ref>.

§  COUNTEREXAMPLES

The following theorem allows us to produce nontrivial examples of
Dir-minimizing functions:

$a\neq0,b\in\mathbb{R}$. Define $u:B_{1}(0)\subset\mathbb{R}^{2}\mathbb{\rightarrow}\mathcal{A_{\mathcal{Q}}}(\mathbb{R}^{2})$
by $u(z)=\underset{w^{\mathcal{Q}}=az+b}{\sum}[[w]]$. Then $u$ is

Define $f:B_{1}(0)\subset\mathbb{R}^{2}\mathbb{\rightarrow\mathcal{A}_{\mathrm{3}}\mathrm{(\mathbb{R^{\mathrm{2}}\mathrm{)}}}}$
by $f(z)=\underset{w^{3}=z}{\sum}[[w]]$. That $f$ is Dir minimizing
follows from Theorem <ref>. We compute:

\[
\underset{B_{\rho}(0)}{\int}|f|^{2}dx=\stackrel[0]{2\pi}{\int}\stackrel[0]{\rho}{\int}3r^{\frac{5}{3}}drd\theta=3\stackrel[0]{2\pi}{\int}\stackrel[0]{\rho}{\int}r^{\frac{5}{3}}drd\theta=\frac{9\pi\rho^{\frac{8}{3}}}{4}=C\rho^{\frac{8}{3}}
\]

where $C>0$. Therefore, $H(\rho)=C\rho^{\frac{5}{3}}$ for some
constant $C>0$, and so $\overline{H}(\rho)=\frac{C\rho^{\frac{5}{3}}}{2\pi\rho}=C\rho^{\frac{2}{3}}$,
hence $\overline{H}''(\rho)<0$.

Our next aim is to show that the boundary regularity condition appearing
in Proposition <ref> is indeed verified for a certain
Dir-minimizing functions on the planar unit disk. This will be the
content of Lemma <ref>, which is of interest by its own
right. Any $z\in\mathbb{C}-\{0\}$ admits a representation of the
form $z=Re^{i\omega}$ for some $R>0,\omega\in[0,2\pi)$. We shall
use the convention $\sqrt{z}=\sqrt{R}e^{\frac{i\omega}{2}}$.

Fix $r\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$
and define $h:(0,2\pi)\rightarrow\mathbb{C}$ by $h(\theta)=\sqrt{2re^{i\theta}-1}$.
Then $h\in W^{1,2}((0,2\pi),\mathbb{C})$.

Proof. $\square$

We recall that $f\in W^{1,2}(\Omega,\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n}))$
if there are $\{\varphi_{j}\}_{j=1}^{m}\subset L^{2}(\Omega,\mathbb{R}_{\geq0})$
such that 1. $x\mapsto\mathcal{G}(f(x),T)\in W^{1,2}(\Omega)$ for
all $T\in\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n})$ and 2. $|\partial_{j}\mathcal{G}(f(x),T)|\leq\varphi_{j}$
a.e. for all $T\in\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n})$ and
$1\leq j\leq m$.

Fix $r\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$ and
define $f:(0,2\pi)\rightarrow\mathcal{A}_{2}(\mathbb{R}^{2})$ by
$f(\theta)=\underset{z^{2}=2re^{i\theta}-1}{\sum}[[z]]$. Then $f\in W^{1,2}((0,2\pi),\mathcal{A}_{2}(\mathbb{R}^{2}))$.

Proof. Let $T=\stackrel[i=1]{2}{\sum}[[T_{i}]]\in\mathcal{A}_{2}(\mathbb{R}^{2})$.
Set $h(\theta)=\sqrt{2re^{i\theta}-1}$ and for $P=(P_{1},P_{2})\in\mathbb{R}^{2}\times\mathbb{R}^{2}$
denote by $d_{P}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ the function
$d_{P}(x)=\sqrt{|x-P_{1}|^{2}+|x-P_{2}|^{2}}$. It is not difficult
to see that for any fixed $P$, $d_{P}$ is Lipschitz.

Put $\alpha_{T}(\theta)=(d_{(-T_{1},T_{2})}\circ h)(\theta),\beta_{T}(\theta)=(d_{(T_{1},-T_{2})}\circ h)(\theta)$.
Note that

\begin{equation}
\mathcal{G}(f(\theta),T)=\frac{\alpha_{T}(\theta)+\beta_{T}(\theta)-|\alpha_{T}(\theta)-\beta_{T}(\theta)|}{2}\label{eq:}
\end{equation}

By Lemma <ref>, $\alpha_{T},\beta_{T}$ are a composition
of a Lipschitz function on a $W^{1,2}(0,2\pi)$ function. Therefore
$\alpha_{T},\beta_{T}$ are also $W^{1,2}(0,2\pi)$. In view of Equation
<ref> it is now apparent that $\theta\mapsto\mathcal{G}(f(\theta),T)\in W^{1,2}(0,2\pi)$.
In addition, there is at most one $\theta_{0}\in(0,2\pi)$ for which
$\alpha_{T}(\theta_{0})=0$. An elementary calculation shows that
the following estimate is obeyed for $\theta\in(0,2\pi)-\{\theta_{0}\}$:
and $\partial_{\theta}h_{i}\in L^{2}(0,2\pi)$, $i=1,2$ by Lemma

The same is true for $\beta_{T}$. Combining these estimates with
equation <ref> we see that

\[
|\partial_{\theta}\mathcal{G}(f(\theta),T)|\leq8(|(\partial_{\theta}h_{1})(\theta)|+|(\partial_{\theta}h_{2})(\theta)|)\in L^{2}(0,2\pi)
\]

for all but finitely many $\theta\in(0,2\pi)$. So strictly by definition,
$f\in W^{1,2}((0,2\pi),\mathcal{A}_{2}(\mathbb{R}^{2}))$. $\square$

Proof of Proposition <ref>. That
$f$ is Dir minimizing follows from Theorem <ref>. Furthermore,
by Lemma <ref> $f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r},\mathcal{A}_{\mathrm{2}}\mathrm{(\mathbb{R^{\mathrm{2}}\mathrm{)}}})$
for all $r\neq\frac{1}{2}$.

We compute:

\[
\underset{B_{\rho}(0)}{\int}|f|^{2}dx=2\stackrel[0]{\rho}{\int}\stackrel[0]{2\pi}{\int}r\sqrt{(2r\cos\theta-1)^{2}+4r^{2}\sin^{2}(\theta)}d\theta dr
\]

\[
\]

which implies

\[
\overline{H}(\rho)=\frac{1}{\pi}\stackrel[0]{2\pi}{\int}\sqrt{(2\rho\cos\theta-1)^{2}+4\rho^{2}\sin^{2}(\theta)}d\theta=\frac{1}{\pi}\stackrel[0]{2\pi}{\int}\sqrt{1+4\rho^{2}-4\rho\cos\theta}d\theta\coloneqq\frac{1}{\pi}A(\rho)
\]

We compute $A'''(\rho)$ for all $\frac{1}{2}<\rho<1$:

\[
\]

\[
\]

\[
\stackrel[0]{2\pi}{\int}\frac{4(1+4\rho^{2}-4\rho\cos\theta)-(4\rho-2\cos\theta)^{2}}{(1+4\rho^{2}-4\rho\cos\theta)^{\frac{3}{2}}}d\theta=\stackrel[0]{2\pi}{\int}\frac{4\sin^{2}\theta}{(1+4\rho^{2}-4\rho\cos\theta)^{\frac{3}{2}}}d\theta
\]

\begin{equation}
\end{equation}

\[
\]

We note that as long as $\frac{1}{2}<\rho<1$, the RHS of equation
<ref> is strictly negative, hence the claim. $\square$

§ APPENDIX

This section was explained to me by Dan Mangoubi. We present here
a proof of Theorem <ref>.

The converse statement of Theorem <ref> is clearly not
true. As an example we can take $m=1$ and $u(x)=x^{2}$. Clearly
$u$ is not harmonic. However,

which is obviously absolutely monotonic.

Let $u$ be harmonic on $B_{1}(0)$.
Let $\phi:[0,1]\rightarrow\mathbb{R}$ be smooth and nondecreasing.
Define $d(r)=\underset{B_{1}(0)}{\int}u^{2}(rx)\phi'(||x||\text{\texttwosuperior})dx$.
Then $d$ is absolutely monotonic.

We denote by $u_{i}$ the derivative of $u$ with respect to the $i-$th
variable. Define $\psi(\xi)=\phi(\frac{||\xi||^{2}}{r^{2}})$.

\[
d'(r)=\underset{\text{\ensuremath{B_{1}(0)}}}{{\normalcolor \int}}[2\stackrel[\text{\ensuremath{i}=1}]{m}{\sum}u(rx)u_{i}(rx)x_{i}]\phi'(||x||\text{\texttwosuperior})dx=\underset{B_{r}(0)}{\int}[2\stackrel[\text{\ensuremath{i}=1}]{m}{\sum}u(\xi)u_{i}(\xi)\frac{\xi_{i}}{r^{m+1}}]\phi'(\frac{||\xi||^{2}}{r^{2}})d\xi=
\]

\[
=\frac{1}{r^{m-1}}\underset{B_{r}(0)}{\int}[2\stackrel[\text{\ensuremath{i}=1}]{m}{\sum}u(\xi)u_{i}(\xi)\frac{\xi_{i}}{r^{2}}]\phi'(\frac{||\xi||^{2}}{r^{2}})d\xi=\frac{1}{2r^{m-1}}\underset{B_{r}(0)}{\int}\nabla u^{2}\cdot\nabla\psi d\xi\overset{\ast}{=}
\]

\[
\frac{1}{2r^{m-1}}[\underset{\partial B_{r}(0)}{\int}\psi\nabla_{\nu}u^{2}d\sigma-\underset{B_{r}(0)}{\int}\psi\Delta u^{2}d\xi]=\frac{1}{2r^{m-1}}[\underset{\partial B_{r}(0)}{\int}\phi(1)\nabla_{\nu}u^{2}d\sigma-\underset{B_{r}(0)}{\int}\psi\Delta u^{2}d\xi]\overset{\ast\ast}{=}
\]

\[
=\frac{1}{2r^{m-1}}[\underset{B_{r}(0)}{\int}\phi(1)\text{\textgreek{D}}u^{2}d\xi-\underset{B_{r}(0)}{\int}\psi\Delta u^{2}d\xi]=\frac{1}{2r^{m-1}}\underset{B_{r}(0)}{\int}(\phi(1)-\psi)\Delta u^{2}d\xi
\]

The equality $\ast$ is by Green's identity and the equality $\ast\ast$
is by the divergence theorem. Taking advantage of the identity $\Delta u^{2}=|\nabla u|^{2}$
for $u$ harmonic we obtain

\[
d'(r)=\frac{1}{2r^{m-1}}\underset{B_{r}(0)}{\int}(\phi(1)-\psi)|\nabla u|^{2}d\xi=\frac{1}{2}r\underset{B_{1}(0)}{\int}(\phi(1)-\phi(||x||^{2}))|\nabla u|^{2}(rx)dx
\]

Let $\Phi$ be some anti derivative of $\phi$ and define $\varphi:[0,1]\rightarrow$
by $\varphi(\rho)=\phi(1)\rho-\Phi(\rho)$. Evidently $\varphi$ is
nondecreasing and according to the computation we preformed

\[
\]

Since each $u_{i}$ is harmonic we can iterate the the same argument
in order to obtain $d^{(k)}(r)\ge0$ for all $k$.

As a corollary we obtain a proof of theorem <ref>:

Let $u$ be harmonic on $B_{1}(0)$. Then $\overline{H}$
is absolutely monotonic.

\[
\overline{H}(r)=\frac{1}{C_{m}r^{m-1}}\underset{\partial B_{r}(0)}{\int}u^{2}(x)d\sigma=\frac{1}{C_{m}r^{m-1}}\frac{d}{dr}[\underset{B_{r}(0)}{\int}u^{2}(x)dx]=\frac{1}{C_{m}r^{m-1}}\frac{d}{dr}[\underset{B_{r}(0)}{\int}u^{2}(x)dx]
\]

\[
\]

\[
\frac{1}{C_{m}}[m\underset{B_{1}(0)}{\int}u^{2}(rx)dx+r\frac{d}{dr}[\underset{B_{1}(0)}{\int}u^{2}(rx)dx]]
\]

Taking $\phi(t)=t$ in Proposition <ref>, we see
that last expression is a sum of absolutely monotonic functions, and
hence absolutely monotonic. $\square$

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[4] Camillo De Lellis. Private

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Immanuel Ben Porat

Einstein Institute of Mathematics, The Hebrew University of Jerusalem

Edmond J. Safra Campus; Givat Ram, Jerusalem, Israel; 9190401