数理解析研究所講究録
第 770 巻 1991 年 1-20

1Keio university, 1991

A Free Boundary Problem for Minimal
Surface Equation
Yoshihiko YAMAURA(山浦義彦)

1. Introduction

In this note we introduced a free boundary problem for minimal surface equation.
The following variational problem had been treated by H.W.Alt, L.A.Caffarelli and A.Freedman $[1],[2]$ :(1.1)$\{\begin{array}{l}\int_{\Omega}(F(|\nabla u|^{2})+Q^{2}\chi_{u>0})dL^{n}arrow\min u\in IC\equiv\{w\in L_{1oc}^{2}(\Omega)|\nabla w\in L^{2}(\Omega),w=u^{0}on S\}\end{array}$where
is a connected Lipshitz domain contained in n-dimensional Euclidean space “. $F=F(t)$ is afunction belonging to.
) with $F(0)=0,0<c\leq\partial_{t}F\leq C<\infty$ andis a given measurable function with $0<Q_{\min}\leq Q(x)\leq Q_{m}\ldots<\infty$ for all $x\in\Omega$ , anddenotes$R$$\Omega$$C^{2,1}[0, \infty$$0 \leq\frac{1}{1+t}\partial_{t}^{2}F\leq C<\infty$$Q$$\chi_{u>0}$

the characteristic function of domain $\Omega(u>0)def\equiv\{x\in\Omega|u(x)>0\}$ .
denotes the integration by nLebesgue
belongingmeasure.
is a given non-negative function
towithdimenisonalandis a subset ofwith positive $H^{n-1}$ -measure, where $H”-1$ is the $(n-1)$ -dimensional Hausdorff
measure. They obtained the next three main results:(1) The existence of a minimum:
minimum attained in function space $K$ .
(2) The global regularity of the minimum:
Let be the $\min$ imum for (1.1). $Tll$ en $u\in C^{0,1}(\Omega)$ .
(3) The regularity of a free boundary for the minimum:, free bound ary of :Let be the $\min$ imum for (1.1). $Th$ en if$dL^{n}$$u^{0}$$S$$\nabla u^{0}\in L^{2}(\Omega)$$L_{1oc}^{2}(\Omega)$$\partial\Omega$

$A$

$is$

$u$

$Q\in C^{\alpha}(\Omega)(^{\exists}\alpha\in(0,1))$$u$is an $(n-1)- dim$ en sional-surface$su\sqrt{l}ciently$
here the free bound ary isilat in some sense (Pricisely ee ],$\partial\Omega(u>0)def\equiv\Omega\cap\partial[\Omega(u>0)]$near the point

$w$

$u$

$C^{1,\beta}$$(^{\exists}\beta\in(0,1))$$s$$\int l$[2].)
On the other hand, S.Omata [10] and S.Omata&Y.Yamaura [11] proved the same results for the nonlinear version of (1.1) when $n=2$ :(1.2)$\{\begin{array}{l}\int_{\Omega}(a^{|j}(u)D.uD_{j}u+Q^{9}\sim\chi_{u>0})dL^{\iota}arrow lninu\in K\equiv\{w\in L_{1oc}^{2}(\Omega)|\nabla w\in L^{2}(\Omega),w=u^{0}on S\}\end{array}$where $a^{ij}(z)$ is a smooth function with the following property: there exist positive numbers
and A independent of such that, and matrix
for all[a $ij(z)$ ] is positive defnite:
for all.
Now in this note we would like to treat the following variational problem:$\backslash$$z$$0<\lambda|\xi|^{2}\leq a^{\mathfrak{i}j}(z)\xi_{i}\xi_{j}\leq\Lambda|\xi|^{2}<\infty(^{\forall}z\in R^{1})$$0\leq\dot{a}^{ij}(z)\xi_{i}\xi_{j}(^{\forall}z\in R^{1})$$\{\begin{array}{l}\int_{\Omega}(\sqrt{l+|\nabla u|^{2}}+Q^{2}\chi_{u>0})dL^{n}arrow\min u\in\overline{l\zeta}\equiv\{w\in TW^{I,1}(\Omega)|w=u^{0}onS\}\end{array}$*Post Graduate of KEIO university$\xi\in R^{n}\backslash \{0\}$$\xi\in R^{n}$

(13)

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Y.YAMAURA

The first term of energy in (1.3) denotes the area of the graph of in and then we naturally assume is, andbounded in addition to the conditions above mentioned.
is a given function belonging tothe other notations are as above.
, the positive part of the graph of the minimum for problem
In the special casein and(1.3): $graph^{+}u=\{(x, u(x))|x\in\Omega(u>0)\}$ describes the soap film, which is constructed by the following
physical experiment: Prepare a connected framework and a sufficiently large container flled with soap liquid.
We lift up the framework, which has been completely flooded under soap liquid at the beginning. When thewhole of the framework rises above the surface of the soap liquid, we get a soap flm with the edge consistingof both the given framework and a free boundary on the surface of the soap liquid.
Now unlike (1.1) or (1.2) the minimum in (1.3) is not necessarily attained in , because the function
spacedoes not have -compactness. Thus we must extend admissible function spaceto$BV(\Omega)$ and generalize the problem itself in the same way as [7]:$\Omega$$u$$\Omega$$u^{0}$$Q^{2}\equiv 1$$W^{1,1}(\Omega)$$S=\partial\Omega$$\Omega$$I^{\sim_{\zeta}}$$L^{1}$$W^{1,1}(\Omega)$$W^{1,1}(\Omega)$$arrow\min$

$(P)$

.

$\{u\in\Omega J(u)=\int_{BV(\Omega)}\sqrt{1+|\nabla u|^{2}}+\int_{\Omega}Q^{2}\chi_{u>0}dL^{n}+\int_{S}|u-u^{0}|dH^{n-1}$

is the space of functions, whose distributional derivatives are Radon measures of locally total variation, and the first term of $J(u)$ is well-defined as Radon measure:$BV(\Omega)$$\int_{\Omega}\sqrt{1+|\nabla u|^{2}}$$def\equiv 9\in C_{0}^{1}(\Omega,R_{1}\sup_{|g|\leq}n+1)\int_{\Omega}(g^{n+1}+udiv^{\wedge}g)dL$“.

reover it is well-known that BV-function has a -trace on the Lipshitz bondary, then for a given nonnegative BV-function , the third term of $J(u)$ is also well-defined. Other notations, which will also be
used throughout this note are as follows:$Mo$$L^{1}$$u^{0}$$\Omega$$Q^{2}$: bounded, connected Lipshitz subdomain of: given positive$\chi_{u>0}$$S$$L^{1}$-function in$\Omega$$R^{n}$,,: the characteristic function of $\Omega(u>0)$ ,: non-empty connected open subset of$\partial\Omega$.

It is problem $(P)$ that we will treat in this note. We now remark that even if a minimum for problem $(P)$exists, the trace of the minimum on does not necessarily coninside with . Taking it and the results forproblem (1.1) into account, the following four questions arise(1’) Is a minimum of attained in function space $BV(\Omega)$ ?
(2’) If the minimum exists, how global regularity does it have?
(3’) If the minimum exists, does it have boundary regularity: $u=u^{0}$ on ?
have?
(4’) If the minimum exists, how regularity does a free boundary
study(1’) and (2’) in this note. We will see the affirmative answer for (1’) in section 2. Moreover inWeection 2 we show the maximum principle for the minimum. In section 3 we obtain the first variation formula,which tells us the information about the gradient on the free boundary. For question (2’), We expect the, thenfollowing results: If$S$$u^{0}$$J$$S$$\partial\Omega(u>0)$$s$$\partial\Omega(u>0)\neq\phi$$\{(a)u\in C^{0,}(\Omega)\exists(c)B^{0,\alpha}(0,1))$$whenwhenwhenQ^{2}(x)\leq 1in\Omega^{<1in\Omega}Q_{2}(x)\leq Q_{\max_{2}}1<Q_{\min}^{2}\leq^{2}Q(x)\leq Q_{\max}^{2}<\infty$in$\Omega$.

(1.4)

To study the behavior of the graph , in section 4,5 we deal with the parametric argument, which is createdby De Giorgi. Using the result there, we construct a solution for radially symmetric free boundary problem$u$3
A Free Boundary Problem for Minimal Surface Equationin $n=2$ (Section 6). In particular when $Q^{2}>1$ , we will obtain the example of minimum, which does nothave even-regularity in the whole of the domain .
$W^{1,1}$$\Omega$

2. Existence Theorem

THEOREM 2.1 (Existence)Th ere exists a function $u\in BV(\Omega)sucl\iota tl\iota$ at$J(u)=$.

$\inf$$J$$BV(\Omega)$

Proof. There exists a bounded Lipshitz domain
we set function
defined in extended domainsuch thatas follows:$V$$\Omega\cup V$$\hat{Q}^{2}$$V\cap\partial\Omega=S$. For given function$Q^{2}\in L^{1}(\Omega)$$\hat{Q}^{2}=\{\begin{array}{l}Q^{2}in\Omega 0inV\backslash \overline{\Omega}\end{array}$Moreover we define a function$w$

belonging to$BV(V\backslash \overline{\Omega})$,$\{w=0^{=(u^{0})^{tr+}}w^{tr-}$($inonV\backslash \overline{\Omega_{c}}S$

such that$\Omega_{c}=${$x\in R^{n}|$dis $(x,$$\Omega)<\epsilon$}),

where $tr+$ , tr-denote the inner, outer trace operator on respectively, and is a sufficiently small positive
number such that. To prove the assertion of theorem it is sufficient to show the existence of aminimum for the next variational problem:$S$$\epsilon$$V\backslash \overline{\Omega_{\zeta}}\neq\phi$$(\hat{P})$$\{\begin{array}{l}\hat{J}(u)=\int_{\Omega\cup V}\sqrt{l+|\nabla u|^{2}}+\int_{\Omega\cup V}\hat{Q}^{2}\chi_{u>0}dL^{n}arrow\min u\in X(\Omega\cup V)\equiv\{v\in BV(\Omega\cup V)|v=winV\backslash \overline{\Omega}\}\end{array}$

In fact when we denote the extension of $u\in BV(\Omega)$ to the domain
gives the byjection from $BV(\Omega)$ to $X(\Omega\cup V)$ , and$\Omega\cup V$

by

$w$

as

$\wedge u\in X(\Omega\cup V)$, mapping$uarrow\hat{u}$for $v_{u}\in BV(\Omega)$ .

$\hat{J}(u\wedge)=J(u)+\int_{V\backslash \overline{\Omega}}\sqrt{1+|\nabla w|^{2}}$(2.1)

The second term of the right hand side of (2.1) is a constant independent of $u\in BV(\Omega)$ , and thus theminimum for problem $(P)$ is obtained by restricting the minimum for problem
to , if the latter exists.
We now show the existence of a minimum for problem. We take a minimizing sequence, then obviously$(\hat{P})$$(\hat{P})$$\Omega$$\{u_{j}\}_{j1}^{\infty_{=}}\subset$$X( \Omega\cup V):\lim_{jarrow\infty}\hat{J}(u_{j})=\inf_{X}\hat{J}$$\int_{\Omega\cup V}$V$u_{j}|$

for$\leq M_{1}$In order to estimate the -norm of trace of onand subsequently we denote $(n-1)$ -dimensional ball bysuch that$L^{1}$$u_{j}$$\partial(\Omega\cup V)$$\forall_{j}\in \mathcal{N}$

.

(2.2)

, we deform domain
to $B_{R}(0)x(0, R)$ (Here) for some $R>0$ by Lipshitz homeomorphism$\Omega\cup V$$B_{R}.$$\Phi$$\{\begin{array}{l}\Phi(V\backslash \overline{\Omega_{e}})=D_{\delta}^{R}def\equiv[\mathcal{B}_{R}(0)\backslash B_{R-\delta}(0)]x(0,R)\cup \mathcal{B}_{R}(0)x(R-\delta,R)\Phi(\partial(\Omega\cup V)\cap\Omega_{c})=\mathcal{B}_{R- 5}(0)\cross\{0\}\end{array}$for some positive number $\delta<<R$ . Moreover we define$u_{j}\sim(x)def\equiv u_{j}(\Phi^{-1}(x))$$\tilde{u}_{j}$by$x\in B_{R}(0)x(0, R)$.

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Y.YAMAURA
Since $\tilde{u}_{j}\in BV(B_{R}(0)x(0, R))$ , for some $t_{j}\in(R-S, R)$ and for$\int_{B_{R}(0)}|u_{j}\sim$

RecaJling that$\tilde{u}_{j}=0$in$D_{\delta}^{R}$

for all(Zif, )$\epsilon$$j\in \mathcal{N}$$-\tilde{u}_{j}$(ZE,$t_{j}$)$L^{1}$-a.a.

$\epsilon\in(0, R)$$|d \overline{x}\leq\int_{\mathcal{B}_{R}(0)x[c,t_{3}]}|\nabla u\sim_{j}|$, this inequality implies$\int_{B_{R}(0)}|\overline{u}_{j}(\overline{x}, \epsilon)|d\overline{x}\leq\int_{B_{R}(0)x(0,R)}|\nabla u_{j}\sim|\leq\overline{M_{1}}$Letting$\epsilon\downarrow 0$.

.

, by the definition of the trace,for$\int_{B_{R}(0)}|(u\sim_{j})^{tr}|dH^{n-1}\leq\overline{M_{1}}$$\forall_{j}\in \mathcal{N}$.

In this way we get$\int_{\partial(\Omega\cup V)}|u_{j}|dH^{n-1}\leq M_{2}$for$\forall_{j}\in \mathcal{N}$

.

(2.3)

From (2.2) and (2.3)$\int_{R^{n}}|\nabla\overline{u}_{j}|=\int_{\Omega\cup V}|\nabla u_{j}|+\int_{\partial(\Omega\cup V)}|u_{j}|dH^{n-1}\leq M_{3}$for$\forall_{j}\in\Lambda^{r}$

,

where
$\overline{u}_{j}=t_{0}^{u_{j}}$Using BV-version Sobolev’s imbedding theorem
spt
for all $j\in N$ ,$in\Omega\cup Votherwise$.

$BV_{0}(R^{n})-L^{\frac{n}{r-1}}(R^{n})$, Holder inequality and the fact that$\overline{u}_{j}\subset\overline{S\Omega t\cup V}$$\int_{R^{\mathfrak{n}}}|\overline{u}_{j}|dL^{n}\leq\{\int_{R^{\mathfrak{n}}}|\overline{u}_{j}|\neg_{n-}^{n}\}^{\frac{n-1}{n}}L^{n}(\Omega\neg\cup VnL\leq c(n)\int_{R^{n}}|\nabla\overline{u}_{j}|\leq M_{4}$,and therefore
for$\int_{\Omega\cup V}|u_{j}|dL^{n}\leq M_{4}$Since$\forall_{j}\in \mathcal{N}$

.

(2.4)

is the bounded Lipshitz domain, (2.2) and (2.4) enable us to apply BV-version Rellich’s compactness theorem $BV(\Omega)arrow L^{1}(\Omega)$ : there exists a subsequence
and a function
belonging tosuch that.
in$\Omega\cup V$$\{u_{k}\}\subset\{u_{j}\}$$u_{\infty}$$L^{1}(\Omega\cup V)$$u_{k}arrow u_{\infty}$$L^{1}(\Omega\cup V)$

By tbe lower-semi-continuity (0.4) we can easily clteck tltat $u_{\infty}\in X(\Omega\cup V)$ .
Now sequence
is uniformly bounded with respect to norm
exists a subsequence
and a bounded function such that$\{\chi_{u_{k}>0}\}_{k1}^{\infty_{=}}\subset L^{\infty}(\Omega\cup V)$$\{u_{l}\}\subset\{u_{k}\}$$||\cdot||_{\infty}$$\gamma$$\chi_{u_{l}>0}arrow\gamma$

in $weakly*L^{\infty}(\Omega\cup V)$ .

We can readily show that$\{_{\gamma(x)=1}^{0\leq\gamma(x)\leq}1$$forL^{n}- a.aforL_{n}- a.a..x\in\Omega\cup Vx\in\{\xi\in\Omega\cup V|u_{\infty}(\xi)>0\}$.

Thus we get$J(u_{\infty}) \wedge\leq\int_{\Omega\cup V}\sqrt{1+|\nabla u_{\infty}|^{2}}+\int_{\Omega\cup V}\hat{Q}^{2}\gamma dL^{n}$.

$\leq\lim inflarrow\infty\int_{\Omega\cup V}\sqrt{1+|\nabla u_{l}|^{2}}+\lim_{larrow\infty}\int_{\Omega\cup V}\hat{Q}^{2}\chi_{u_{1}>0}dL^{n}=\inf_{X}\hat{J}$, then there

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A Free Boundary Problem for Minimal Surface Equation

Q.E.D.

Le $tu$ be the $\min$ imum for $(P)$ . Th en $0 \leq u\leq\sup_{S}u^{0}$ in

THEOREM 2.2 (Maximum principle)$\Omega$.

Proof. We show $0\leq u$ in .
(a) Reduction 2.
It is sufficient to prove that for $u^{-}= \min(0, u)$$\Omega$$\int_{\Omega}|\nabla u^{-}|=0$

.

(2.5)

In fact if (2.5) holds, then by the definition of the variation measure,for$\int_{\Omega}u^{-}divgdL^{n}=0$$\forall_{g}\in C_{0}^{1}(\Omega, R^{n})$.

in for some non-positive
Then we have $\nabla u^{-}=0$ in in the sense of weak derivative, and henceon ,constant . Especially it holds that $C=0$ , because if $C<0$ , then using the as sumption$u^{-}\equiv C$$\Omega$$\Omega$$u^{0}\geq 0$

$C$

$S$

$J(u)=L^{n}( \Omega)+\int_{S}|u^{0}-C|dH^{n-1}$$=L^{n}( \Omega)+\int_{S}|u^{0}|dH^{\mathfrak{n}-1}+\int_{S}|C|dH^{n-1}$,$>L^{\mathfrak{n}}( \Omega)+\int_{S}|u^{0}|dH^{n-1}=J(0)$which contradicts to the minimality of . Thus(b) Reduction 2.
Equality (2.5) follows from the next fact:$u^{-}\equiv 0$

$u$

If

$\int_{\Omega}|\nabla u^{-}|>0$

,

then

in

$\Omega$, and hence we obtain $u\geq 0$ .

,

(2.6)

$\int_{\Omega}\sqrt{1+|\nabla u^{+}|^{2}}<\int_{\Omega}\sqrt{1+|\nabla u|^{2}}$because if (2.5) does not hold, using (2.6) we deduce $J(u^{+})<J(u)$ , which is a contradiction.
The fact we have to show is (2.6), but using the approximation argument it is easy to see that$\geq\int_{\Omega}\sqrt{1+|\nabla u^{-}|^{2}}-L^{n}(\Omega)$,then instead of (2.6) we prove the following:

If(c) Proof of (2.7).
Define a positive number$\int_{\Omega}|\nabla u^{-}|>0$$\delta$,then$\int_{\Omega}\sqrt{1+|\nabla u^{-}|^{2}}>L^{n}(\Omega)$

.

(2.7)

as follows:$S \equiv\min(\frac{1}{2}\int_{\Omega}|\nabla u^{-}|,$

Then there exists vector valued function$g_{0}\in C_{0}^{1}(\Omega, R^{n})$$1_{(b)}^{(a)}$$4L^{\mathfrak{n}}(\Omega))$.

such that$\int_{\Omega}^{0_{u^{\infty_{-}}div^{1}g_{0}}}^{|g}|\leq.dL^{n}>5$

.

(2.8)

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Y.YAMAURA
Now for an arbitrarily fixed number $M\geq 1$ , we choose a function
conditions:$g^{n+1}\in C_{0^{1}}(\Omega)$satisfying the next two(2.9)$\{\begin{array}{l}(a)|g^{n+1}|_{\infty}\leq\frac{\sqrt{M^{2}-1}}{M}(a)\int_{\Omega}g^{n+1}dL^{n}>(2\frac{\sqrt{A\prime I^{2}-l}}{M}-l)L^{n}(\Omega)\end{array}$(2.8-a) and (2.9-a) imply$|( \frac{g_{0}}{M}$,,$g^{n+1})|_{\infty}^{2} \leq\frac{|g_{0}|_{\infty}^{2}}{M^{2}}+|g^{n+1}|_{\infty}^{2}\leq 1$

and hence using (2.8-b) and (2.9-b),$\int_{\Omega}\sqrt{1+|\nabla u^{-}|^{2}}=j\in C_{0}^{1}(\Omega,R^{n+1})\sup\int_{\Omega}(f^{n+1}+u^{-}divf^{\wedge})dL^{n}$$|f|\leq 1$$\geq\int_{\Omega}(g^{n+1}+u^{-}div\frac{g_{0}}{M})dL^{n}$$>$$(2 \frac{\sqrt{M^{2}-1}}{M}-1)L^{n}(\Omega)+\frac{\delta}{M}$.

We now choose$M= \frac{2L^{n}(\Omega)}{\delta}+\frac{\delta}{8L^{n}(\Omega)}$then we get.

$\int_{\Omega}\sqrt{1+|\nabla u^{-}|^{2}}-L^{\mathfrak{n}}(\Omega)\geq\frac{45^{2}L^{n}(\Omega)}{16L^{n}(\Omega)^{2}+\delta^{2}}>0$

In a similar way, we obtain $u \leq\sup_{S}u^{0}$ in$\Omega$.
Q.E.D.

3. The flrst variation formula

THEO’II.EM 3.1 (The first variation formula) Let $u\in BV(\Omega)$ be the minim um for $(P)$ with, then. $Ass$ ume$Q^{2}\in W^{1,1}(\Omega)$$u\in C^{0}(\Omega)$(3.1)$1_{\delta\frac{i}{\in}} m_{L^{0}}\int_{\partial\Omega(u>5)}(Q^{2}-(1-\frac{1}{\sqrt{1+|\nabla u|^{2}}}))\langle\eta, \nu_{t}\rangle dH^{n-1}=0$, an for somefor allwith $L^{1}((0, \sup_{\Omega}u)\backslash L)=0$ whereis the unit ou $ter$$no$ rm
for the boundary of domain $\Omega(u>\delta)=\{x\in\Omega|u(x)>S\}$. Moreoveris uniform for any$\eta\in C_{0}^{1}(\Omega, R^{\mathfrak{n}})$$d$$L \subset(0, \sup_{\Omega}u)$$\nu_{\delta}$$al$$\lim_{5arrow 0}$$\eta\in B_{M}(\Omega)\equiv${$\varphi\in C_{0^{1}}(\Omega,$$R^{n})||\varphi|+|\nabla\varphi|\leq M$

Proof. We first note that by the assumptionin$\Omega$}, where$u\in C^{0}(\Omega)$$div(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}I=0$$M$

is an arbit rarily fixed positive numbe .
$r$

it holds that $u\in C^{\infty}(\Omega(u>0))$ and$u$satisfies

in $\Omega(u>0)$

in the classical sense. Moreover we can easily obtain $u\in W^{1,1}(\Omega)$ .
Now let $\tau_{c}(x)=x+\epsilon\eta(x)$ and $u_{c}(x)=uo\tau_{c^{-1}}(x)$ . Since$u^{tr}=u_{\zeta}^{tr}$

on

(3.2)

$S$

for sufficiently $smaU\epsilon>0$ ,$J(u_{c})= \int_{\Omega}(\sqrt{1+|\nabla u_{c}|^{2}}+Q^{2}\chi_{u_{\iota}>0})dL^{n}+\int_{S}|u-u^{0}|dH^{n-1}$.

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A Free Boundary Problem for Minimal Surface Equation
Calculating the first integration-term of$J(u_{\zeta})$,$\int_{\Omega}(\sqrt{1+|\nabla u_{c}|^{2}}+Q^{2}\chi_{u_{e}>0})dL^{n}$$= \int_{\Omega}(\sqrt{1+|\nabla u(\tau_{c}^{-1}(x))D\tau_{c}^{-1}(x)|^{2}}+Q^{2}(x)\cdot x_{u>0}(\tau_{c^{-1}}(x)))dL^{n}(x)$“.

$= \int_{\Omega}(\sqrt{1+|\nabla u(D\tau_{c})^{-1}|^{2}}+Q^{2}o\tau_{\zeta}\cdot x_{u>0})|\det D\tau_{c}|dL$

Using $(D\tau_{c})^{-1}=I-\epsilon D\eta+O(\epsilon^{2}),$
we omit the-term.).

$|\det Dr_{\zeta}|=1+\epsilon div\eta+O(\epsilon^{2})$. (Here and subsequently in this proof$O(\epsilon^{2})$$\int_{\Omega}(\sqrt{1+|\nabla u_{c}|^{2}}+Q^{2}\chi_{u_{\iota}>0})dL^{n}$$= \int_{\Omega}(\sqrt{1+|\nabla u-\epsilon\nabla uD\eta|^{2}}+Q_{0\mathcal{T}_{C}}^{2}\cdot\chi_{u>0})(1+\epsilon div\eta)dL^{n}$$= \int_{\Omega}(\sqrt{1+|\nabla u|^{2}}+\epsilon div\eta\cdot\sqrt{1+|\nabla u|^{2}}-\epsilon\frac{\nabla u(\nabla u\cdot D\eta)}{\sqrt{1+|\nabla u|^{2}}}$$+Q^{2}o\tau_{\zeta}\cdot\chi_{a>0+\epsilon}Q^{2}o\tau_{\zeta}\cdot\chi_{u>0}div\eta)dL^{n}$.

Then$[J(u_{\zeta})-J(u)]= \epsilon\int_{\Omega(u=0)}div\eta dL^{\mathfrak{n}}$$+ \epsilon\int_{\Omega(w>0)}(div\eta\cdot\sqrt{1+|\nabla u|^{2}}-\frac{\nabla u(\nabla u\cdot D\eta)}{\sqrt{1+|\nabla u|^{2}}}IdL^{n}$$+ \int_{\Omega(u>0)}((Q^{2}o\tau_{c}-Q^{2})+\epsilon(Q^{2}o\tau_{c})div\eta)dL^{n}$$= \epsilon\int_{\Omega(u>0)}\{div\eta\cdot\sqrt{1+|\nabla u|^{2}}-\frac{\nabla u(\nabla u\cdot D\eta)}{\sqrt{1+|\nabla u|^{2}}}+div((Q^{2}-1)\eta)\}dL^{n}$

,

(3.3)

We thus get$0= \lim_{carrow 0}\frac{1}{\epsilon}[J(u_{c})-J(u)]$$= \int_{\Omega(\sim>0)}\{div\eta\cdot\sqrt{1+|\nabla u|^{2}}-\frac{\nabla u(\nabla u\cdot D\eta)}{\sqrt{1+|\nabla u|^{2}}}+div((Q^{2}-1)\eta)\}dL^{n}$.

is smooth $(n-1)$ -dimensional curve forHere we remember the Sard theorem, thenforby
. Moreover the Co-area formula BV-function ([5],[7],[12]):$\partial\Omega(u>\delta)$$\delta\in(0,\sup_{\Omega}u)$,$\int_{0}^{\sup_{\Omega}u}d\delta\int_{\Omega}|\nabla\chi_{\Omega(u>\delta)}|=\int_{\Omega}|\nabla u|<\infty$

for $L^{1}- a.a$ . $5\in(0, \sup_{\Omega}u)$ .
with $L^{1}((0, \sup_{\Omega}u)\backslash L)=0$ such that
Thus there exists a set$H^{n-1}(\partial\Omega(u>\delta))<\infty$$L \subset(0, \sup_{\Omega}u)$$\partial\Omega(u>\delta)$

is smooth and has $H^{n-1}- finite$ measure for all$\delta\in L$.

Now$\lim_{\in L}s\sim 0\int_{\Omega(r>\delta)}\{div\eta\cdot\sqrt{1+|\nabla u|^{2}}-\frac{\nabla u(\nabla u\cdot D\eta)}{\sqrt{1+|\nabla u|^{2}}}+div((Q^{2}-1)\eta)\}dL^{n}=0$,$L^{1}$-a.a.

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Y.YAMAURA
where we can easily check that for any
surface equation (3.2),$\eta\in B_{M}(\Omega)\equiv\{\varphi\in C_{0^{1}}(\Omega, R^{n})||\varphi|+|\nabla\varphi|\leq M\}$

. Using the minimal$0= \lim_{\sim,s\epsilon\iota^{0}}\int_{\Omega\langle u>5)}div(\eta\sqrt{1+|\nabla u|^{2}}-\frac{\nabla u\langle\nabla u,\eta\rangle}{\sqrt{1+|\nabla u|^{2}}}+(Q^{2}-1)\eta)dL^{n}$$\nu_{t}\}dH^{n-1}$$= \lim_{\iota\epsilon\iota}sarrow 0\int_{\partial\Omega(u>\delta)}\langle\eta\sqrt{1+|\nabla u|^{2}}-\frac{\nabla u\langle\nabla u,\eta\rangle}{\sqrt{1+|\nabla u|^{2}}}+(Q^{2}-1)\eta,$,$= \lim_{\sim_{L}}\delta_{\delta\in}0\int_{\partial\Omega(u>S)}(Q^{2}-(1-\frac{1}{\sqrt{1+|\nabla u|^{2}}}))\langle\eta, \nu\rangle dH^{n-1}$

where we use Green’s formula (noting that$\nu_{\delta}=-\nabla u/|\nabla u|$

, on

for

$\partial\Omega(u>5)$$\delta\in L.$).
Q.E.D.

4. A construction of the Radon measure

We would like to get a local estimate for the perimeter of which is the subgraph of minimum for(See Lemma 5.6). To do that we need the parametric representation for $Q^{2}X.>0$ -term. Thus our aim, for general BV-function , andof this section is to construct the Radon measure corresponding toto obtain the parametric representation of $J(w)$ .
Before that we recall some defintions: Borel set $E\subset R^{n+1}$ is called a Caccioppoli set when
$D$

$u$

$(P)$

$Q^{2}\chi_{w>0}$

$w$

(4.1)

$\int_{D}|\nabla\chi_{B}|def\equiv g\in C_{0}^{1}(D,R\sup_{|g|\leq 1}n+1\int_{D})\chi_{B}divgdL^{n+1}$

is finite for each bounded open set $D\subset R^{n+1}$ . We call the value deined by (4.1) a perimeter of $E$ in domain. Moreover Caccioppoli set $E$ is called a minimal set in some bounded domain if
$D$

$D$

(4.2)

$\int_{D}|\nabla^{\chi_{B}|}\leq\int_{D}|\nabla^{\chi_{P}|}$for every Borel set $F$ with $spt(\chi_{B}-\chi_{P})\subset D$ .
In this and later chapters we assume$Q^{2}\in L^{\infty}(\Omega)$

LEMMA 4.1

Define. We first show the following Lemma:

as follows:$Q^{2}\in L^{\infty}(\Omega xR^{1})$$Q^{2}(x, t)def\equiv Q^{2}(x)$Then for arbitra rily frxed $v\in BV(\Omega xR^{1})$ an open set$d$$g \mapsto\int_{D}vQ^{2}\partial_{t}gdL^{n+1}$

Proof. Let
to t-variable,$g$

be a function belonging to, and then$C_{0^{1}}(D)$

with

for

$\forall_{\mathfrak{i}}\in R^{1}$$D\subset\Omega xR^{1}$.

, the following funct ion$(g\in C_{0}^{1}(D))$$|g|_{\infty}\leq 1$.

. Noting that$al$is bounded:(4.3)$Q^{2}$

is constant with respect$Q^{2}\partial_{t}g=\partial_{t}(Q^{2}g)$$\int_{D}vQ^{2}\partial_{t9}dL^{n+1}=\int_{D}v\partial_{t}(Q^{2}g)dL^{n+1}$

Now let$(Q^{2}g)_{e}$be the mollified function of$Q^{2}g$.

, then$\int_{D}v\partial_{t}[(Q^{2}g)_{c}]dL^{n+1}=Q_{\max}^{2}\int_{D}v\partial_{t}[\frac{1}{Q_{\max}^{2}}(Q^{2}g)_{c}]dL^{n+1}$$\leq Q_{\max}^{2}\int_{D}|\nabla v|$

.

(4.4)

9
A Free Boundary Problem for Minimal Surface Equation

Letting$\epsilonarrow 0$, we get$\int_{D}v\partial_{t}(Q^{2}g)dL^{n+1}\leq Q_{\max}^{2}\int_{D}|\nabla v|<\infty$.

Combining (4.4) and this inequality, we obtain. the condusion.
Q.E.D.

DEFINITION 4.2
as follows (See $[12J$);

We den $ote$ th $eR$ adon me as $ure$ , which is uniqu $ely$ determi $ned$ for function $al(4.3)$$\int_{D}Q^{2}|\partial_{\ell}v|$.

We are now in a position to construct the Radon measure. Let $w\in BV(\Omega)$ , and $W$ be the subgraph of$w$:$W=\{(x, t)\in\Omega xR^{1}|w(x)>t\}$.

It is well known that $\chi_{w}\in BV(\Omega xR^{1})$ (See [7]). Then the following Radon measure is well defined:$\int_{D}Q^{2}|\partial_{\ell}\chi_{W}|$for$\forall_{D}\subset\Omega xR$,

We remark that by the construction of Radon measure (See[12]) it holds that
(4.5)
$I_{D}^{Q^{2}|\partial_{1}\chi_{w1}}= 9\in_{|g|\leq 1}\sup_{C_{0}^{1}(D)}\int_{D}Q^{2}\chi_{w}\partial_{t}gdL^{n+1}$. We show that this measure corresponds to the second term of
for all open set
with the smooth case, though that will not be used later.
$D\subset\Omega xR_{+}^{1}$

PROPOSITION 4.3$J$. We begin

Let $w\in C^{2}\cap BV(\Omega)$ , th en.

$\int_{\Omega}Q^{2}\chi_{w>0}dL^{n}=\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{1\chi_{w1}}$

Proof. The next equality is proved by using the Green’s formula in the same way as [7]:$\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{W}|=\int_{\partial W\cap[\Omega xR_{+}^{1}]}Q^{2}|\nu_{t}|dH^{n}$where$\nu_{t}$is a t-compornent of unit outer normal for$\partial W$,. Thus$\int_{\partial W\cap[\Omega xR_{+}^{1}]}Q^{2}|\nu_{t}|dH^{n}=\int_{\partial W\cap[\Omega(u>0)xR^{1}]}Q^{2}|\nu_{t}|dH^{n}$$= \int_{\Omega(w>0)}Q^{2}\frac{1}{\sqrt{1+|\nabla w|^{2}}}\sqrt{1+|\nabla w|^{2}}dL^{n}$$= \int_{\Omega}Q^{2}\chi_{w>0}dL^{\mathfrak{n}}$.
Q.E.D.

Next the general case is proved by taking an analogous testing function to Lemma 14.6 ([7]).

10
Y.YAMAURATHEOREM 4.4

Let $w\in BV(\Omega)$ , then$\int_{\Omega}Q^{2}\chi_{w>0}dL^{n}=\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{w1}$

Proof. We first suppose that$u$.

is bounded. We define a testing function$\eta_{\zeta}\in C_{0}^{1}(R_{+}^{1})$$\eta_{\zeta}(t)$as follows:with$\eta_{c}=\{\begin{array}{l}0in(0,\frac{\epsilon}{2})\cup(\sup_{\Omega}w+l,\infty)lin(\epsilon,\sup w)\Omega\end{array}$

Then for each fixed$g\in C_{0^{1}}(\Omega)$

with$|g|_{\infty}\leq 1$,$\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{w}|\geq\int_{\Omega xR_{+}^{1}}Q^{2}(x, t)^{\chi_{w}}(x, t)\partial_{\ell}[g(x)\eta_{\zeta}(t)]dxdt$$= \int_{\Omega(w>0)}Q^{2}(x)dx\int_{0}^{w(x)}\partial_{t}[g(x)\eta_{c}(t)]dt$$= \int_{\Omega(w>0)}Q^{2}(x)g(x)dx\int_{0}^{w(x)}\eta_{c}’(t)dt$$= \int_{\Omega(w>0)}Q^{2}(x)g(x)\eta_{e}(w(x))dx$

Letting$\epsilonarrow 0,$$\eta_{c}(w(x))arrow 1$

for each$x\in\Omega$.

, and so$\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{w1}\geq\int_{\Omega(w>0)}Q^{2}gdL^{n}$,and hence$\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{\ell}\chi_{w}|\geq\int_{\Omega}Q^{2}\chi_{w>0}dL^{\mathfrak{n}}$On the other hand, for all$g\in C_{0}^{1}(\Omega)$

with$|g|_{\infty}\leq 1$.

,$\int_{\Omega xR_{+}^{1}}Q^{2}\chi_{w}\partial_{2}gdL^{\mathfrak{n}+1}=\int_{\Omega(w>0)}Q^{2}(x)dx\int_{0}^{w(x)}(\partial_{t}g)(x, t)dt$$= \int_{\Omega(w>0)}Q^{2}(x)[g(x, w(x))]dx$$\leq\int_{\Omega}Q^{2}\chi_{w>0}dL^{n}$.

From (4.5) we deduce that$\int_{\Omega xR_{+}^{t}}Q^{2}|\partial_{t}\chi_{w}|\leq\int_{\Omega}Q^{2}\chi_{w>0}dL^{n}$.

Now when is unbounded, we first apply the above argument to $w_{M}= \min(w, M)$ , and then letting, we obtain the conclusion.
$w$$Marrow\infty$

Q.E.D.

Combinig Lemma 14.6 ([7]) and the preceding theorem, we get the parametric representation of $J(w)$ ,which is the aim of this section:

11
A Free Boundary Problem for Minimal Surface Equation

COROLLARY 4.5

Let $w\in BV(\Omega)$ , then.

$\int_{\Omega}\sqrt{1+|\nabla w|^{2}}+\int_{\Omega}Q^{2}\chi_{w>0}dL^{n}=\int_{\Omega x1R^{1}}||\nabla\chi_{w1}+\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{w1}$5. Local estimate for perimeter of the minimum

Our aim of this section is to obtain a local estimate for measureminimum for $(P)$ . To do that we need the next theorem:
THEOREM 5.1 Let. Th ensubdomain of$u$$|\nabla x_{U}|$, where$U$

is the subgraph of thebe the $\min$ imum for $(P)$ , an $dU$ be the subgraph of , and let $D$ be a bounded$u$$\Omega xR^{1}$$\int_{D}|\nabla^{\chi_{U}|+}\int_{D\cap(\Omega xR_{+}^{t})}Q^{2}|\partial_{t}\chi_{U}|\leq\int_{D}|\nabla^{\chi_{r1+}}\int_{D\cap(\Omega xR_{+}^{1})}Q^{2}|\partial_{t}\chi_{r1}$for all meas $urable$ sets $F$ with $spt(\chi_{P}-\chi_{U})\subset D$ .

We shall show some lemmata to prove this theorem.

$n$

LEMMA 5.2 Let $F$ be a measurable set with $\Omega x(-\infty, O)\subset F\subset\Omega x(-\infty, T),$ $wh$ ere
umber. $We$ deline function
as follows:

is a positive$w_{F}$$w_{F}(x) def\equiv\int_{0}^{T}\chi_{P}(x, t)dt$

$Th$

$T$

for

$v_{x}\in\Omega$.

en$\int_{\Omega}Q^{2}\chi_{w_{F}>0}dL^{n}\leq\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{P}|$

.

(5.1)

Proof. For simplicity we assume $T=1$ . Define$\eta(x, t)=$$\{\begin{array}{l}\int_{0}^{p}\frac{\chi_{F}(x,\tau)}{w_{F}(x)}d\tau for(x,t)\in\Omega(w_{F}>0)x(0,l)2-tfor(x,t)\in\Omega(w_{P}>0)x[l,2]0otherwisein\Omega xR_{+}^{1}\end{array}$, and for each
We have $0\leq\eta\leq 1$ in
follows:(a) $x_{0}\in\Omega(w_{F}>0)$$\Omega xR_{+}^{1}$$\partial_{t}\eta(x_{0}, t)=$$\eta$

is absolutely continuous for t-variable, and$\{\begin{array}{l}\frac{\chi_{P}(x_{0},t)}{w_{P}(x_{0})}fort\in(0,l)-1fort\in(l,2)0fort\in(-\infty,0)\cup(2,\infty)\end{array}$(b) otherwise$\partial_{t}\eta(x_{0}, t)\equiv 0$$x\in\Omega,$.

$\partial_{t}\eta$

is as

12
Y.YAMAURA
Moreover$\partial_{\ell}\eta$belongs to$L^{1}(\Omega xR_{+}^{1})$, because$\int_{\Omega xR_{+}^{1}}|\partial_{t}\eta(x, t)|dL^{n+1}=\int_{\Omega(w_{F}>0)xR_{+}^{1}}|\partial_{\ell}\eta|dL^{n+1}$$= \int_{\Omega(w_{F}>0)}dx\int_{0}^{\infty}|\partial_{\ell}\eta(x, t)|dt$$= \int_{\Omega(w_{F}>0)}dx[\int_{0}^{1}\frac{\chi_{P}(x,t)}{w_{F}(x)}dt+\int_{1}^{2}1dt]$$=2L^{n}(\Omega(w_{F}>0))<\infty$

Now let$g\in C_{0^{1}}(\Omega),$$|g|\leq 1$, then from the property of.

stated above we can take $g(x)\eta(x)t)$ as a testing$\eta$

function, and therefore$\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{1\chi_{P}}|\geq\int_{\Omega xR_{+}^{1}}\chi_{P}(x, t)Q^{2}(x)\partial_{t}[g(x)\eta(x, t)]dxdt$$= \int_{\Omega\langle w_{F}>0)x(0,1)}\chi_{P}(x, t)Q^{2}(x)g(x)\partial_{\ell}\eta(x, t)dxdt$$= \int_{\Omega(w_{F}>0)}Q^{2}(x)g(x)dx\int_{0}^{1}(\chi_{B}(x, t)\cdot\partial_{t}\eta(x, t))dt$$= \int_{\Omega(w_{F}>0)}Q^{2}(x)g(x)dx\int_{0}^{1}(\chi_{F}(x, t)^{2}\cdot\frac{1}{w_{F}(x)})dt$$= \int_{\Omega(w_{F}>0)}Q^{2}(x)g(x)dx\cdot\frac{1}{w_{F}(x)}\int_{0}^{1}\chi_{P}(x, t)^{2}dt$.

$= \int_{\Omega}Q^{2}\chi_{w_{F}>0gdL^{n}}$Thus (5.1) follows on taking the supremum over all such .
$g$

Q.E.D.

Combining Lemma 14.7 ([7]) and the last lemma, we obtain the following:

Let

COROLLARY 5.3$F_{W_{F}}$be as defined in Lemma 5.2. Then.

$\int_{\Omega}\sqrt{1+|\nabla wp|^{2}}+\int_{\Omega}Q^{2}\chi_{w_{F}>0}dL^{n}\leq\int_{\Omega xR^{1}}|\nabla\chi_{r1+}\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{F}|$LEMMA 5.4 Let
subdomain $contained$ in$u$be the minimum for $(P),$. Then$U$

be the subgraph of , and let
$u$

$D$

$\Omega xR^{1}$$\int_{\Omega xR^{1}}|\nabla\chi_{v1+}\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{\iota^{\chi_{U}|}}\leq\int_{\Omega xR^{1}}|\nabla^{\chi_{p}|+}\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{c^{\chi_{F}|}}$

for all me asurable set $F$ with$\{\begin{array}{l}F\supset\Omega x(-\infty,0)spt(\chi_{F}-\chi_{U})\subset D\end{array}$

Proof. Using Corollary 5.3,$\int_{\Omega xR^{1}}|\nabla^{\chi_{P}|+}\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}^{\chi_{F}|}\geq\int_{\Omega}\sqrt{1+|\nabla w_{F}|^{2}}+\int_{\Omega}Q^{2}\chi_{w_{F}>0}dL^{n}$.

be a bounded

13
A Free Boundary Problem for Minimal Surface Equation

Since$w_{F}^{tr}=u^{tr}$

on ,
$S$

$\int_{\Omega}\sqrt{1+|\nabla w_{F}|^{2}}+\int_{\Omega}Q^{2}\chi_{w_{P}>0}dL^{n}\geq\int_{\Omega}\sqrt{1+|\nabla u|^{2}}+\int_{\Omega}Q^{2}\chi_{u>0}dL^{n}$$= \int_{\Omega xR^{1}}|\nabla\chi_{U}|+\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{\ell^{\chi}u1)}$

where in the last equality we use Corollary 5.5.
Q.E.D.

LEMMA 5.5 Let
subdomain $con$ tained in$u$be the minimum for $(P),$. Then$U$

be the $su$ bgra $phofu$ , and let $D$ be a bounded$\Omega xR^{1}$$\int_{\Omega xR^{1}}|\nabla\chi_{U}|+\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{U}|\leq\int_{\Omega xR^{1}}|\nabla^{\chi_{r1+}}\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{P}|$for all meas ura $ble$ set $F$ with $spt(\chi_{P}-\chi_{U})\subset D$ .

Proof. Suppose the lemma is not true, then there exists measurable set $F$ with $spt(\chi_{F}-\chi_{U})\subset D$ suchthat$\int_{\Omega xR^{1}}|\nabla^{\chi_{U}}|+\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{\ell^{\chi_{U}|}}>\int_{\Omega xR^{1}}|\nabla^{\chi_{F}|+}\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}^{\chi_{P}|}$.

Now let $H=\{(x, t)\in R^{n+1}|t<0\}$ , obviously$\int_{\Omega xR^{1}}|\nabla^{\chi_{P}}|\geq\int_{\Omega xR^{1}}|\nabla^{\chi_{F\cup H}}|$and$\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{I},|=\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{1\chi_{F\cup H}}|$.

Th erefore.

$\int_{\Omega xR^{1}}|\nabla\chi_{U}|+\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{\ell}\chi_{U}|>\int_{\Omega xR^{1}}|\nabla\chi_{F\cup H}|+\int_{\Omega xR_{+}^{1}}Q^{2}|\partial_{t}\chi_{F\cup H}|$

But $F\cup H$ satisfies $F\cup H\supset\Omega x(-\infty, 0)$ , then the last inequaJity contradicts to Lemma 5.4.
Q.E.D.

Theorem 5.1 is immediately followed by the las lemma.
Now using Theorem 5.1 we can get a local estimate for a perimeter. Before tltat we recall the follow fact:
Let $D$ be a connected domain in $R^{n+1}$ and let $E$ be a minimal set inin the sense of (5.2). Then it iswell-known that (See[7]):
$t$

$D$

for

$\int_{B_{\rho}}|\nabla\chi_{B}|\leq\frac{1}{2}(n+1)\omega_{n+1}\rho^{n}$$\forall_{B_{\rho}}\subset D$

,

(5.2)

is the measure of the unit ball in “.
where
We can $now$ state a local estimate for a perimeter:$R$$\omega_{n}$THEOREM 5.6

Le $tu$ be the $minim$ um for $(P)$ , and$\int_{B_{\rho}}|\nabla\chi_{U}|\leq\frac{1}{2}(1+Q_{\max})(n+1)\omega_{n+1}\rho^{n}$$U$

be the $su$ bgraph $ofu$ . Thenfor$\forall_{B_{\rho}}\subset\subset\Omega xR^{1}$.

14
Y.YAMAURA
Proof. Let $E$ be a minimal set in$B_{\rho}$

with

in

$E=U$

$B_{\rho}^{c}$.

Then from Theorem 5. 1$\int_{B_{\rho}}|\nabla\chi_{U}|+\int_{B_{\rho}\cap(\Omega xR_{+}^{1})}Q^{2}|\partial_{t}\chi_{U}|\leq\int_{B_{\rho}}|\nabla\chi_{B}|+\int_{B_{\rho}\cap(\Omega xR_{+}^{1})}Q^{2}|\partial_{t}\chi_{B}|$,and so$\int_{B_{\rho}}|\nabla\chi_{U}|\leq(1+Q_{\max})\int_{B_{p}}|\nabla^{\chi_{B}}|$.

Using (5.2), we obtain the result.
Q.E.D.
6. Radially symmetric free boundary problem

In this section we treat the free boundary problem in the radially symmetric situation $(n\geq 2)$ , and finally
we will construct a solution in 2-dimensional case. We set$\Omega=B_{R}=B_{R}(0)\subset R$“: n-dimensional ball,.

$Q^{2}=Const$ $>0$ ,$S=\partial\Omega=\partial B_{R}$,$u^{0}\equiv h=Const$. $>0$

in$B_{R}$,

that is, we consider the following problem:$arrow\min$.

$\{u\in BB_{R^{R}}J_{S}(u)=\int_{V(B)}\sqrt{1+|\nabla u|^{2}}+\int_{B_{R}}Q^{2}\chi.>0dL^{n}+\int_{\partial B_{R}}|u^{tr}-h|dH^{n-1}$$(P_{S})$

Define function space $BVS(B_{R})$ as follows:$BVS(B_{R})dei\equiv\{u\in BV(B_{R})|u(x)=\exists_{\phi(|x|)}\}$,

then we first as sart

PROPOSITION 6.1$J_{S}=$$\inf$$BV(B_{R})$$\inf$$J_{S}$$BVS(B_{R})$.

Proof. We shall only prove$\inf_{BV(B_{R})}J_{S}\geq\inf_{BVS(B_{R})}J_{S}$,because the reverse inequality is trivial. Let be a minimum for problem
sufficient to show that there exists function $v\in BVS(B_{R})$ such that$u$

$J_{S}(u)\geq J_{S}(v)$.

We define$\sim u=\{\begin{array}{l}uinB_{R}hinB_{2R}\backslash \overline{B_{R}}\end{array}$(6.1)$(P_{S})$, then to prove (6.1) it is

(6.2)

15
A Free Boundary Problem for Minimal Surface Equation
Then$\tilde{u}$is the minimum for the next problem:$arrow\min$.

$\{_{w\in}\overline{j_{S(w)=\int B_{2R}\sqrt{1+|\nabla w|^{2}}+\int Q^{2}\chi_{w>0}}}B_{2R}\{\zeta\in BV(B_{2R})|(=hinB_{2R}\backslash B_{R}\}.dL^{n}$Let

be the subgraph of $u:\sim\tilde{U}=\{(x, t)\in B_{2R}xR^{1}|u\sim(x)>t\}$ . From Corollary 5.5$\tilde{U}$$\overline{J_{S}}(\sim u)=\int_{B_{2R}xR^{1}}|\nabla\chi_{U}\sim|+\int_{B_{2R}xR_{+}^{1}}\chi_{U}$.

On the other hand, by the maximum principle it holds thatin$0\leq u\sim\leq h$$B_{2R}$.

Therefore$\overline{J_{S}}(u\sim)=\int_{B_{2R}x[0,h]}\chi_{U^{c}}+\int_{B_{2R}x(0,h]}\chi_{U^{c}}$$= \int_{R^{n}x[0,h]}\chi_{U^{c}}+\int_{R\cdot x(0,h]}\chi_{U^{c}}$Let$(\tilde{U}^{c})_{s}$

be

$a$

symmetrized set for$\tilde{U}^{c}$.

(See [8]), that is,$(\tilde{U}^{c})_{s}def\equiv\{(x, t)\in R^{n+I}||x|<\rho(t)\}$

,

where
$\rho(t)=(\frac{1}{\omega_{n}}\int_{R}$Since$Q^{2}$.

$\chi_{\tilde{U}^{c}}(\cdot, t)dL^{n})^{\frac{1}{n}}$is constant, we can apply [8] and so

.

(6.3)

$\overline{J_{S}}(u\sim)\geq\int_{R^{\mathfrak{n}}x[0,h]}|\nabla\chi_{(\tilde{U}^{c})}|+\int_{R^{n}x(0,h]}Q^{2}|\partial_{t}\chi_{(U^{c})}\sim.|$

By the method of symmetrization we readily deduce$B_{2R}x(h, \infty)\subset(\tilde{U}^{c})_{s}\subset B_{2R}x[0, \infty)$,

and so$\int_{R^{n}x[0,h]}$I$\nabla\chi_{(U^{c})}\sim.|+\int_{R^{n}x(0,h]}\chi_{(U^{c})}$$= \int_{B_{2R}xR^{1}}$V$\chi_{(U^{c})}\sim.|+\int_{B_{2R}xR_{+}^{1}}\chi_{(U^{c})}$

.

(6.4)

Moreover by the definition of Radon measures$\int_{B_{2R}xR^{1}}\chi_{(U^{c})}+\int_{B_{2R}xR_{+}^{1}}\chi_{(U^{\epsilon})}$(6.5)$= \int_{B_{2R}xR^{1}}\chi_{U}+\int_{B_{2R}xR_{+}^{1}}\chi_{U}$where we denote$\tilde{U}^{s}=(B_{2R}xR^{1})\backslash (\tilde{U}^{c})_{s}$. Combinig (6.3),(6.4) and (6.5),$\overline{J_{S}}(\sim u)\geq\int_{B_{2R}xR^{1}}|\nabla\chi_{U}\sim.|+\int_{B_{2R}xR_{+}^{1}}Q^{2}|\partial_{\ell}\chi_{U*}\sim|$

.

(6.6)

16
Y.YAMAURA

Now define function$u\sim$as follows:$u^{s} \sim(x)=\int_{0}^{h}xU\sim.(x, t)d\tau$.

is radiaUy symmetric, because so is , and furthermore by Lemma 14.7 ([7]), $u^{s}\sim\in BV(B_{2R})$ . Thus$u^{s}\sim\in BVS(B_{2R})$ . Using Corollary 6.4, we can estimate the right hand side of (6.6),

Then$\tilde{U}^{s}$$u^{s}\sim$.

$\overline{J_{S}}(u\sim)\geq\int_{B_{2R}}\sqrt{1+|\nabla u^{s}\sim|^{2}}+\int_{B_{2R}}Q^{2}\chi_{u>0}\sim dL^{n}=\overline{J_{S}}(u^{s}\sim)$

It holds that$u\sim‘\equiv h$in$B_{2R}\backslash \overline{B_{r}}$by the construction of$u^{g}\sim$, and therefore$J_{S}(u)\geq J_{S}(u^{s})$where$u‘=u^{s}\sim|_{B_{R}}$,. We establish (6.2) taking $u^{s}\in BVS(B_{r})$ as .
$v$

Q.E.D.

COROLLARY 6.2 If$\inf_{BV(B_{R})}J_{S}$$\inf_{BVS(B_{R})}J_{S}$is $att$ ained nuiquely in $BVS(B_{R})$ , then the function, which attains

in $BV(B_{R})$ is also uniqu .
$e$

Proof. If there exists function$u$belonging to $BV(B_{R})\backslash BVS(B_{R})$ such that
$J_{S}(u)= \inf_{BV(B_{R})}J_{S}$,then by [8] we get the following strong inequality:$J_{S}(u)>J_{S}(u^{s})$,where $u^{*}\in BVS(B_{R})$ is constructed as shown in the proof of Proposition 6.1. This is the contradiction.
Q.E.D.

From Proposition 6.1 to construct a solution for$(P_{S})$

it is sufficient to do that for the next problem:$arrow\min$$\{_{u\in B}B_{R}.+\int_{B_{R}}Q^{2}\chi_{u>0}dL^{n}+\int_{\partial B_{R}}|u^{tr}-h|dH^{n-1}$$(P_{S}’)$.

Since the admissible function space is $BVS(B_{R})$ , by a slight variation of the proof of Proposition 4.4 we canobtain the ame result:
The minimum for (P\’{s}) is uniquely determined as any of the following three type functions:$s$(a) $u>0$
(b)

a.e.

in

$B_{R}$,$(^{\exists}\rho\in(0, R))$$u\{_{>0}^{=0}$(c) $u=0$$a.ea.e.\cdot$

,

(6.7)

$inB_{R}^{\rho}\backslash \overline{B_{\rho}}inB$a.e. in$B_{R}$.

Especially in case of (6.7-a) it is trivial that must be identically by the form of energy
in case of (6.7-b) is a minimum for the next area minimizing problem in
:
$u$

$u$

$h$

$J_{S}$. Furthermore$B_{R}\backslash \overline{B_{p}}$$arrow\min$$\{_{v\in^{R}BV(B_{R}\backslash \overline{B_{p}})}\int_{B\backslash \overline{B_{\rho}}}\sqrt{1+|\nabla v|^{2}}+\int_{\partial B_{\rho}}|v|dH^{n-1}+\int_{\partial B_{R}}|v-h|dH^{n-1}$

.

(6.8)

17
A Free Boundary Problem for Minimal Surface Equation

Thus we can rewrite $(6.7- a)\sim(6.7- c)$ as follows:
The minimum for (P\’{s}) is uniquely determined as any oi the following three type functions:

in

,

(a)

$u\equiv h$(b)$u=\{v0inB_{R}inB^{p}\backslash \overline{B_{\rho}},(^{\exists}\rho\in(0,R))$(c)$u\equiv 0$in$B_{R}$$B_{R}$(6.9),

where is the minmum for (6.8).
We now want to express (6.9-b)-type function by a concrete function. To do that we study the minimumfor (6.8). First the interior regularity of minimal surface (See [7]) tells us that is a classical solution of the:minimal surface equation in the interior of
$v$

$v$

$B_{R}\backslash \overline{B_{\rho}}$in$div(\frac{\nabla v}{\sqrt{1+|\nabla v|^{2}}})=0$$B_{R}\backslash \overline{B_{p}}$.

From the uniqueness of minimal surface is radially symmetric function. For simplicity we use the same
notation to represent l-dimensional function:
$v$

$v$

$v$

Then

$v$

:

$r v(x)$$(|x|=r)$.

satisfies the next ordinary differential equation:in$v”(r)+ \frac{n-1}{r}v’(r)+\frac{n-1}{r}(v’(r))^{3}=0$

In particular when $n=2,$$v$$(\rho, R)$.

can be written by the elementary function:($v(r)=c_{1}\log(r+\sqrt{r^{2}-c_{1}^{2}})+c_{2}$We next study the relation between, and it is not known $v^{tr}=0$ on$v(\rho)=0$ . Then we ontainsume
as$\partial B_{R}$$h$$c_{1},$$c_{2}$: constants).

and the boundary regularity of . In [7] it is well-known $v^{tr}=h$ on
generally. However, to study the boundary regularity on
we
$v$

a

$\partial B_{\rho}$$v(r)=c_{1} \log\frac{r+\sqrt{r^{2}-c_{1}^{2}}}{\rho+\sqrt{\rho^{2}-c_{1}^{2}}}$in$(\rho, R)$$B_{p}$.

Now we consider function $[v(R)]$ : $c_{1} v(R)(c_{1}\in[0, \rho])$ . It is easy to see that$[v(R)]’(c_{1})>0$

for$\forall_{C_{1}}\in[0, \rho]$,and for the range of value, we get$0 \leq[v(R)](c_{1})\leq\rho\log\frac{R+\sqrt{R^{2}-\rho^{2}}}{\rho}$

Here the maximum is attained when$c_{1}=\rho$for:$[v(R)]( \rho)=\rho\log\frac{R+\sqrt{R^{2}-\rho^{2}}}{\rho}$

From these facts it is known that there are following two cases:(1) $0 \leq h\leq\rho\log\frac{R+\sqrt{R^{2}-\rho^{2}}}{\rho}$ :$\forall_{C_{1}}\in[0, \rho]$.

(6.10)

18
Y.YAMAURA

Minimum$v$

satisfies $v=h$ on$\partial B_{R}$and $v=0$ on$\partial B_{\rho}$.
(6.11)(2) $\rho\log\frac{R+\sqrt{R^{2}-\rho^{2}}}{\rho}<h$ :Minimum$v$

satisfies $v=h$ on

In the former case (6.11-1) we define function

DEFINITION 6.3$\partial B_{R}$, but cannot satisfy $v=0$ onin$u_{\rho}^{h}\equiv 0$(2)$u_{\rho}^{h}=\{\begin{array}{l}0on\partial B_{\rho}hon\partial B_{R}\end{array}$$B_{\rho}$in$div(\frac{\nabla u_{p}^{h}}{\sqrt{1+|\nabla u_{\rho}^{h}|^{2}}})=0$

on$v^{tr}\equiv c>0$

$c$

. But

$u_{p}^{c}$as the function satisfyin$g$$B_{R}\backslash \overline{B_{p}}$.

is radially symmetric, and therefore$v$.
, and so from (6.10)$B_{R}\backslash \overline{B_{\rho}}$$[h- \rho\log\frac{R+\sqrt{R^{2}-\rho^{2}}}{\rho},$$c\in$Let$\partial B_{p}$is regular in the interior of$v$$u_{\rho}^{h}$,(1)

Let’s consider the latter case (6.11-2) precisely. The minimum

for some positive constant.

as follows:$u_{\rho}^{h}$

When $0 \leq h\leq\rho\log\frac{R+\sqrt{R^{2}-\rho^{2}}}{\rho}$ we define(3)$\partial B_{p}$$h]def\equiv L$.

be the function satisfying

(1)

in

$u_{\rho}^{c}\equiv 0$(2)$B_{\rho}$,$u_{\rho}^{c}=\{\begin{array}{l}con\partial B_{p}hon\partial B_{R}\end{array}$(3)$div(\frac{\nabla u_{p}^{c}}{\sqrt{1+|\nabla u_{p}^{c}|^{2}}})=0$

in.

$B_{R}\backslash \overline{B_{\rho}}$Direct calculation tells us that$\tilde{J}(u_{p}^{c(h)})\leq\tilde{J}(u_{p}^{c})$for all $c\in L$ ,$canmininizeJ_{S}inProb1emng(69- b)- typefunctions:wherec(h)=h-\rho\log\frac{R+\sqrt{R^{2}-\rho^{2}}}{(P_{S}^{\rho})amo}.Th.usforgiven\rho,inthe$

DEFINITION 6.4

When$h> \rho\log\frac{R+\sqrt{R^{2}-\rho^{2}}}{\rho}$inlatter case (6.11-2) the following function

we define$u_{\rho}^{h}$(1)$u_{\rho}^{h}\equiv 0$(2)$u_{p}^{h}=\{\begin{array}{l}h-plog\frac{R+\sqrt{R^{2}-\rho^{2}}}{\rho}on\partial B_{p}hon\partial B_{R}\end{array}$(3)$B_{\rho}$,$div(\frac{\nabla u_{\rho}^{h}}{\sqrt{1+|\nabla u_{\rho}^{h}|^{2}}})=0$

in$B_{R}\backslash \overline{B_{\rho}}$.

as the $fun$ ctio satisfyin
$n$

$g$

19
A Free Boundary Problem for Minimal Surface Equation

In conclusion we need only to consider
in Definition 6.3 and Definition 6.4 as (6.9-b)-typeasfunctions. In this way to construct a solution for (P\’{s}) it is sufficiently to consider
admissible function space instead of $BVS(B_{R})$ .
$u_{\rho}^{h}def\iota ned$$\{u_{\rho}^{h}, 0, h\}_{0<\rho<R}$

REMARK 6.5$\{(l)u_{h}^{h}(2)u_{p}^{\rho}\in\in C^{0}(B)BV(B^{R_{R}})\backslash C^{0}(B_{R})whenwhen0h\leq>h\leq\rho_{\frac{R^{log\frac{R+\sqrt{R^{2}-\rho^{2}}}{\sqrt{}^{\rho}R^{2}-\rho^{2}}}+}{\rho}}\rho\}og$

Now we are in a position to state the result, which is obtained by the direct calculation of energy
Problem (P\’{s}):RESULT 6.6 (The solution for)is $aco$ nstant with $0<Q^{2}\leq 1$ , the $\min$ imum forCase 1 $(0<Q^{2}\leq 1)$ Whenas follows:(6.12)$J_{S}$

in$(P_{S})$$Q^{2}$$(P_{S})$is uniquelydetermined$\{\begin{array}{l}u_{p(h)}^{h}wl\iota enh\leq h(Q^{2})Rhwhenh(Q^{2})R\leq h\end{array}$wh ereis a $solution$ of the next $eq$ uation:$h(Q^{2})$$1+h^{2}+2(2-Q^{2})\log h=0$ ,$\rho(h)$is a larger solution of the next equa tion:$\sqrt{2Q^{2}-Q^{4}}\rho\log\frac{R+\sqrt{R^{2}-(2Q^{2}-Q^{4})\rho^{2}}}{(2-Q^{2})\rho}=h$Case 2 $(1 <Q^{2}<2)$ Whenas follows:$Q^{2}$

is a constant with $1<Q^{2}<2$ , the minim um for$(P_{S})$.

is uniq uely determined$\{\begin{array}{l}0whenh\leq(Q^{2}-l)Ru_{p(h)}^{h}when(Q^{2}-1)R\leq h\leq h(Q^{2})Rhwhenh(Q^{2})R\leq h\end{array}$$where$$h(Q^{2})$is a larger solu tion of the next $equ$ at ion:
$1+ \frac{2}{1+h^{2}}(\log h+(1+Q^{2}))=0$ ,$\rho(h)$is a larger $solu$ tion of the next equation:$\rho\log\frac{R+\sqrt{R^{2}-\rho^{2}}}{\rho}-(1-Q^{2})\rho=h$Case 3 $(2 \leq Q^{2})$follows:$Wl_{1}$en$Q^{2}$

is a constant with $Q^{2}\geq 2$ , the minim um for$\{\begin{array}{l}0w\Lambda enh\leq\frac{1}{2}Q^{2}R_{\prime}hwhenh\geq\frac{1}{2}Q^{2}R\end{array}$.

$(P_{S})$is uniquely determ ined as20
Y.YAMAURA
The former case in Case 1 is $cont$ ained in case (6.12-1), an
REMARK 6.7
Case 2 is contained in case (6.12-2).

$d$the second case in

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[7] GIUSTI, E.: Minimal Surfaces and functions of bounded variation, Basel,1984.
[8] GONZALEZ, E.: Sul Problema della goccia appoggiata, Rend. Sem. Mat. Univ. Padova, 58 (1977).
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[9] MASSARI, U.. Esistenza e regolarita delle ipersurfici di curvatura media assegnata in
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[10] OMATA, S.: A Lipshitz Estimate for a Free Boundary Problem for Non-linear Elliptic Equations, preprint.
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$R^{n}$$\Lambda$