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\ellipticalarc axes ratio 2:1 150 degrees from 3.5 -1 center at 5 0
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\put{${\cal S}$} at 3 2.5
\endpicture }
% end of figure 1
\noindent Clearly, this problem is actually
two-dimensional, since the surfaces ${\cal{S}}$ and ${\cal{S}}^*$
are generated by the corresponding arcs
$$
\eqalignno{\Gamma&=\{(x,y):(x,y,0)\in{\cal{S}},y>0\}&(3a)\cr
\Gamma^*&=\{(x,y):(x,y,0)\in{\cal{S}}^*,y>0\}.&(3b)\cr}
$$ % figure 2
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% end of figure 2
\noindent
Notice that the endpoints of $\Gamma$ and $\Gamma^*$ are the
intersections of ${\cal{S}}$ and ${\cal{S}}^*$ respectively with the
$x_1-$axis. Our conclusions regarding the qualitative geometric
properties of ${\cal{S}}$ and ${\cal{S}}^*$ can then be expressed
entirely in terms of the qualitative geometric properties of $\Gamma$ and
$\Gamma^*$.
In order to discuss our results, let us adopt the notation of [2] and [4].
Thus $\Gamma$ and $\Gamma^*$ are oriented curves with initial and
terminal points lying on the $x_1-$axis such that
the $x_1-$coordinate of each initial point is smaller than the
$x_1-$coordinate of the corresponding terminal point.
We define $\vec n(x,y)$ to be the unit normal vector to $\Gamma\cup
\Gamma^*$ at $(x,y)\in\Gamma\cup\Gamma^*$ which points to the
right of the curve (with respect to the direction of the curve).
Further, we have the following:
\smallskip
\noindent {\bf Definition.}
Given a unit vector $\vec\nu$, we call $(x_0,y_0)\in\Gamma$
a $\vec\nu$-minimum ($\vec\nu$-maximum)
of $\Gamma$ if $\vec n(x_0,y_0)=\vec\nu$ and
$(x_0,y_0)$ is a strict local minimum (maximum)
relative to $\Gamma$ of $f(x,y)=\vec\nu\cdot (x,y)$
(see, for example, Figures 2 and 3 in [4]).
\smallskip
\noindent {\bf Definition.}
Given a unit vector $\vec\nu$, we call $(x_0,y_0)\in\Gamma^*$
a $\vec\nu$-minimum ($\vec\nu$-maximum)
of $\Gamma^*$ if $\vec n(x_0,y_0)=\vec\nu$ and either
$(x_0,y_0)$ is a strict local minimum (maximum)
relative to $\Gamma$ of $f(x,y)=\vec\nu\cdot (x,y)$
or there is a closed line segment $\gamma^*\subset\Gamma^*$
such that $(x_0,y_0)\in\gamma^*$ and
$\vec\nu\cdot(x,y)>\ (<)\ \vec\nu\cdot(x_0,y_0)$ for
$(x,y)\in\Gamma^*\backslash\gamma^*$ near $\gamma^*$.
Here $\gamma^*$ is considered
as a single local extremum.
\smallskip
\noindent We may define $\vec\nu-$inflection points of $\Gamma$ and
$\Gamma^*$ similarly (see [2], [16]). Notice that Lemma 2(b.) implies
the definitions of $\vec\nu-$extrema are equivalent.
The following figures illustrate the definition of $\vec\nu-$extrema of
$\Gamma;$ the letters $a,A,b,B,c,C$ represent points at which $\Gamma$
has a $-\vec j-$minimum, $-\vec j-$maximum, $-\vec i-$minimum,
$-\vec i-$maximum, $\vec i-$minimum, and $\vec i-$maximum respectively.
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% end of figure 3
\centerline{\epsffile{fig4.ps}} % figures 4 and 5
Let us assume that $\Gamma^*$ contains a finite number
of maximal line segments (including isolated points) on which
$\vec n(x,y)=\pm \vec i$ or $\vec n(x,y)=-\vec j$.
Let equation (2a) be either Laplace's equation (i.e. (15) ) or the
minimal surface equation (i.e. (18) )
in $\Bbb R^3$, ${\cal{S}}^*$ be a closed surface in
$\Bbb R^3$, and $({\cal{S}},U)$ be a solution of the free boundary
problem for some $\lambda>0$. Suppose ${\cal{O}}={\cal{O}}({\cal{S}}^*,
{\cal{S}})$ is rotationally symmetric with respect to the $x_1-$axis
and set
$$
W=\{(x,y)\in\Bbb R^2:(x,y,0)\in{\cal{O}}\}.
\eqno{(4)}
$$
Let $\partial_iW$ and $\partial_oW$ denote the inner boundary
and outer boundary of $W$ respectively and let $\Gamma$ and
$\Gamma^*$ be given by (3).
Then our main results, which are given in \S 1, include the
following as a special case:
\proclaim Theorem 1.
Suppose there exists $u\in C^2(W\cup\partial_o W\cup\Gamma^*)
\cap C^1({\overline{W}})$ such that
$$
U(x_1,y\cos(\theta),y\sin(\theta))=u(x_1,y)
\eqno{(5)}
$$
for $(x_1,y)\in W$ and $\theta\in\Bbb R$. Suppose also that $\Gamma^*$
and $\partial_oW$ are $C^2$ curves and $W$ satisfies an interior
sphere condition at each point of $\partial_iW$. Then
$\Gamma$ has no more $\vec\nu-$minima (maxima) than does $\Gamma^*$
and each $\vec\nu-$minimum (maximum) of $\Gamma$ can be joined to a
(distinct) $\vec\nu-$minimum
(maximum) of $\Gamma^*$ by a curve along which $\nabla u$ has a
constant direction, for each $\vec\nu=-\vec i,\vec i,-\vec j$.
In particular, if $\Gamma^*$ is a graph over the $x-$axis, then
$\Gamma$ is also a graph over the $x-$axis.
\centerline{\epsffile{fig6.ps}}
The study of the relationship between $\vec\nu-$extrema of the free and
fixed boundaries of solutions of the $N-$dimensional
free boundary problem has previously been restricted to the
case $N=2$. In this case, we have a quasilinear elliptic partial
differential operator $Q$ on $\Bbb R^2$, a constant $\lambda>0$, and
a Jordan curve $\Gamma^*$ in $\Bbb R^2$ and the free boundary problem
consists of finding a Jordan curve $\Gamma$ in $\Bbb R^2$ which surrounds
$\Gamma^*$ and a function $u\in C^2(\Omega)\cap
C^1(\Omega\cup\Gamma) \cap C^0(\overline{\Omega})$
such that
$$
Qu=0 \quad \rm{in} \ \ \Omega,
\eqno{(6a)}
$$
$$
u=1 \quad \rm{on} \ \ \Gamma^*,
\eqno{(6b)}
$$
$$
u=0,\ \vert\nabla u\vert = \lambda \quad \rm{on}\ \ \Gamma,
\eqno{(6c)}
$$
where $\Omega={\cal{O}}(\Gamma^*,\Gamma)$.
The ``geometric study'' of this two-dimensional free boundary
problem began with the consideration of the case in which $Q$ is the
Laplace operator. In this case, the principal model for later work
was established by the first author in [1], [2], and [4], where a
method of curves of constant gradient direction was developed and
applied in an analysis of the number and ordering of the directional
extrema and inflection points of the free boundary.
At approximately the same time, curves of constant gradient direction were
independently used to study ideal fluid flows by Friedman and Vogel ([11]).
The use of curves of constant gradient direction was extended
to solutions of the two-dimensional free boundary problem by Vogel ([18])
and the first author ([3]) when (6a) is Poisson's equation,
by the authors when (6a) is the minimal surface equation ([7])
or the heat equation ([8]), and by the second author ([16]) when $Q$ is
any elliptic partial differential operator of the form
$$
Qu\equiv au_{xx}+2bu_{xy}+cu_{yy},
\eqno{(7)}
$$
where $a,b,c$ depend on $x,y,u_x$, and $u_y$. The conclusion
obtained (in the elliptic cases) is that if $\Gamma$ and $u$
constitute a solution of the
free boundary problem, $\Omega$ is a $C^2$ domain, and
$u\in C^2(\overline{\Omega})$, then each $\vec\nu-$extremum
of the free boundary can be joined to a corresponding (distinct)
$\vec\nu-$extremum of the fixed boundary by a curve $(\gamma)$ along which
$\nabla u$ remained parallel to $\vec\nu$
(i.e. $\nabla u(x,y)=\vert\nabla u(x,y)\vert\ \vec\nu$ for each
$(x,y)$ on $\gamma)$ and, in particular, $\Gamma$ has no more
$\vec\nu-$minima $(\vec\nu-$maxima) than does $\Gamma^*$, for each
$\vec\nu$. In addition, the number of $\vec\nu-$inflection points of
$\Gamma$ cannot exceed the number of $\vec\nu-$inflection points of
$\Gamma^*$.
When the three-dimensional free boundary problem is symmetric with respect
to the $x_1-$axis and $({\cal{S}},U)$ is an axial-symmetric solution,
the function $u(x,y)=U(x,y,0)$ is the solution of a related two-dimensional
free boundary problem. In fact, we obtain immediately the following
\proclaim Proposition.
Suppose $({\cal{S}},U)$ is a solution of the three-dimensional free
boundary problem, $U\in C^1(\overline{{\cal{O}}})$, and there exists
$u\in C^2(W)\cap C^1(\overline{W})$ which satisfies (5)
for $(x_1,y)\in W$ and $\theta\in\Bbb R$, where
$W=\{(x,y): (x,y,0)\in\cal{O}\}$. Let $x=x_1$ and
define $\Omega=\{(x,y)\in W: y>0\}$ and $Q$ to be the quasilinear, elliptic
operator given by
$$
Qu(x,y)=a(x,y,\nabla u)u_{xx}+2b(x,y,\nabla u)u_{xy}
+c(x,y,\nabla u)u_{yy}+d(x,y,\nabla u)
\eqno{(8)}
$$
for $u\in C^2(\Omega)$ and $(x,y)\in\Omega$,
where $a(x,y,p,q)=A_{11}(x,y,0,p,q,0)$,
$b(x,y,p,q)=A_{12}(x,y,0,p,q,0)$,
$c(x,y,p,q)=A_{22}(x,y,0,p,q,0)$, and
$d(x,y,p,q)=B(x,y,0,p,q,0)+{q\over{y}}A_{33}(x,y,0,p,q,0)$.
Then $u$ is a solution of free boundary problem (6) when $Q$ is given by
(8).\par
It is natural to conjecture that the results obtained for the two-dimensional
free boundary problem (6) with $Q$ given by (7) apply to solutions of the
$N-$dimensional free boundary problem for arbitrary $N\ge 3$.
Such a generalization, if true, would be esthetically more satisfactory
than the (3-dimensional) axial-symmetric results we obtain. However,
this conjecture is incorrect, as the first author ([6]) has shown
by means of a counterexample in which $N=3$, $G=\triangle$ is the Laplace
operator, $\lambda>0$, the fixed boundary ${\cal{S}}^*$
has precisely one $\vec\nu-$minimum, and the free boundary
${\cal{S}}={\cal{S}}\sb\lambda$ has two distinct
$\vec\nu-$minima, for some direction $\vec\nu$.
The study of qualitative properties of axial-symmetric solutions in
$\Bbb R^3$ is suggested by the facts that the properties in question seem
to correspond to two-dimensional problems and axial-symmetric
solutions of three-dimensional free boundary problems are of physical
interest (e.g. [15]).
The results in Theorems 1 and 3 about the directional extrema of $\Gamma$
would be more appealing if they applied to arbitrary directions in
$\Bbb R^2$. However, when such a problem is reduced to the two-dimensional
free boundary problem (6), the differential operator (8) may contain a
lower order term (i.e. $d$) which complicates the situation. The
conjecture that the solution of (6) has the same qualitative
properties with regard to arbitrary directions is false.
The first author ([6]) has obtained a counterexample
when $N=3$, $G=\triangle$, and $\lambda>0$
in which the generator $\Gamma^*$ has only one $\vec\nu-$minimum while
the free boundary $\Gamma=\Gamma\sb\lambda$ has two
$\vec\nu-$minima, for some direction $\vec\nu$ (which is not an axial or
radial direction). Thus, while our results seem somewhat restricted, the
most natural and appealing generalizations are false.
The paper is organized as follows. In \S 1, we state our main results.
In \S 2, we present some examples of free boundary problems in
$\Bbb R^3$ to which our results apply. The statements of our
preliminary results, which consist of nine lemmas, are given in \S 3
and these lemmas are proven in \S 4; the statements are separated from
their proofs in the hope of making the paper more readable.
Our main results are proven in \S 5 and we include some concluding
remarks in \S 6.
\bigbreak
\centerline{\bf \S 1. MAIN RESULTS} \medskip\nobreak\noindent
Suppose $({\cal{S}},U)$ is a solution of the free boundary problem,
$U\in C^1(\overline{{\cal{O}}})$, ${\cal{O}}$ is
axial-symmetric, $W$ is given by (4),
and there exists $u\in C^2(W)\cap C^1(\overline{W})$
which satisfies (5). Let us write $x=x_1$. We set
$\Gamma^*=\{(x,y):(x,y,0)\in {\cal{S}}^*,y>0\}$,
$\Gamma=\{(x,y):(x,y,0)\in {\cal{S}},y>0\}$, and
$\Omega=\{(x,y)\in W: y>0\}$.
Define $Q$ to be the quasilinear, elliptic operator given by (8).
Then $u$ is a solution of the free boundary problem (6).
We will assume that linear functions of the form $U(x,y,z)=\alpha x+\beta$
are solutions of (2a); this is equivalent to assuming
$$
B(x,y,0,p,0,0)=0.
\eqno{(9)}
$$
Let us define the ratio of the coefficient $a$ of $u_{xx}$ in $Q$ to
the lower order term $d$ in $Q$ to be
$$
g(x,y,p,q)={d(x,y,p,q)\over{a(x,y,p,q)}}\equiv
{{qA_{33}(x,y,0,p,q,0)+yB(x,y,0,p,q,0)}
\over{yA_{11}(x,y,0,p,q,0)}}.
$$
Notice that $g(x,y,p,0)=0$, and so
${\partial g\over{\partial y}}(x,y,p,0)=0$, for all $x\in\Bbb R,y>0$.
\vskip .2 true in
\proclaim Theorem 2.
Let us assume the three-dimensional free boundary problem
(2) has a solution $({\cal{S}},U)$, $U$ is in
$C^2({\cal{O}}\cup{\cal{S}})\cap C^1({\overline{\cal{O}}})$,
the solution $({\cal{S}},U)$ is axial-symmetric, and
condition (9) holds. Let $\partial_iW$ be the inner portion
of the boundary of $W$ and assume $W$ satisfies an interior
sphere condition at each point of $\partial_iW$.
If we define $\Gamma$ and $\Gamma^*$ as
above and if $\Gamma^*$ is the graph of a $C^1$ function,
then $\Gamma$ is the graph of a $C^2$ function. \par
If we are willing to assume that additional conditions are satisfied,
we can obtain a result which is stronger than that of Theorem 2.
Let us define the function
$$
h(x,y,p,q)={d(x,y,p,q)\over{c(x,y,p,q)}}\equiv
{{qA_{33}(x,y,0,p,q,0)+yB(x,y,0,p,q,0)}
\over{yA_{22}(x,y,0,p,q,0)}}.
\eqno{(10a)}
$$
Let us assume that
$$
{\partial h\over{\partial x}}(x,y,0,q)=0
\eqno{(10b)}
$$
and there is a $C^1$ function $\Phi(y,q)$
satisfying
$$
\Phi(y,q)<0,
\eqno{(10c)}
$$
$$
{\partial\Phi\over{\partial y}}(y,q)<0,
\eqno{(10d)}
$$
$$
{\partial\Phi\over{\partial q}}(y,q)>0,
\eqno{(10e)}
$$
$$
{\partial\Phi\over{\partial y}}(y,q)=h(x,y,0,q)
{\partial\Phi\over{\partial q}}(y,q),
\eqno{(10f)}
$$
$$
{q_1\over{\Phi(y,q_1)}}\le{q_2\over{\Phi(y,q_2)}}
\quad{\rm{when}}\ \ q_2__0$ on $\Gamma^*$. To see
this, let $(x_0,y_0)\in\Gamma^*$ and define $w=1-u$. Then
$Rw=0$ as above, $w\ge 0$ on $\overline{\Omega}$, and
$w(x_0,y_0)=0$. The Hopf boundary point lemma then implies
$\vert\nabla u(x_0,y_0)\vert>0$ and so our observation holds.
Thus $\vert\nabla u\vert>0$ on $\partial W$.
Suppose $\vert\nabla u(x_0,y_0)\vert=0$ for some
$(x_0,y_0)\in W$ and set $z_0=u(x_0,y_0)$.
Then $0 z_0$ on $\omega_2\cup\omega_4\cup
\dots\cup\omega_{2m}$. The Jordan curve theorem and the fact that
$W$ is an annular domain implies that there is a component $\omega$ of
$\{(x,y)\in W:u(x,y)\neq z_0\}$ whose closure does not intersect
$\partial W$ and the maximum principle implies $u\equiv z_0$
in $\omega$, in contradiction to the fact that $u\neq z_0$ in
$\omega$. Thus $\vert\nabla u\vert>0$ in $W$. \hfill Q.E.D.
\bigbreak
\noindent {\bf Proof of Lemma 6.} (a.)
Let $\gamma$ be directed curve in $\overline{\Omega}$ starting
at $(x_0,y_0)\in\Gamma$ along which $u_x=0$ and
$\phi$ is strictly decreasing.
From Lemma 2 (b.) and the fact that $\Gamma$ has a $-\vec j-$minimum
at $(x_0,y_0)$, we see that $\Gamma$ is ``strictly concave
down'' near $(x_0,y_0)$.
Assume the claim is false and $\gamma$ ends at a point
$(x_1,y_1)\in{\overline{\Omega}}$
with $y_1=y_0$ such that
$y__