The behavior of the free boundary for the dam problem

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

H ANS W ILHELM A LT

G IANNI G ILARDI

The behavior of the free boundary for the dam problem

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 9, n

^o

4 (1982), p. 571-626

<http://www.numdam.org/item?id=ASNSP_1982_4_9_4_571_0>

© Scuola Normale Superiore, Pisa, 1982, tous droits réservés.

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http://www.numdam.org/

Boundary

for the Dam Problem (*).

HANS WILHELM ALT - GIANNI GILARDI

1. - Introduction.

In this paper ^we

study

the behavior of the free

boundary

for the dam

problem

^{near the}

given

boundaries to

reservoirs, atmosphere,

^and

impervious layers,

^{where we} restrict ourselves to the two dimensional

homogeneous

^case.

We start with a solution of the

problem

^{as it was obtained in}

[3] by

ap-

proximating

^{the free}

boundary problem by

models for saturated-unsaturated flow

through

porous media. That

is,

^a

pair

u, y is ^a solution of the dam

problem,

^if

Here S~ is an open bounded connected set in R2 with

Lipschitz boundary

^and

denotes the porous medium. On the

boundary

^{of ,S2 three}

relatively

open sets are

given, Sres, Which

^denote

respectively

^the

boundary

^to

reservoirs,

^to

atmosphere,

^{and the}

impervious part

^{of the}

boundary (fig. 1).

It is assumed that these three sets are

disjoint

^{and that the}

boundary

^8Q

is the union of their closures. On

Sres

^and

Sair

the pressure is

given by

^a

function uO E

CO,1(R2),

^{which is}

non-negative

^on

Sr.

^{and zero on}

Sair (which

means that we do not consider the case of a

capillary fringe).

^{The set of}

(*)

This work is

partially supported by

^Deutsche

Forschungsgemeinschaft, Heisenberg-Programm (W. Germany),

^and

by

^the G.N.A.F.A. and the I.A.N. of the C.N.R.

(Italy).

Pervenuto alla Redazione il 23 Novembre 1981.

Fig.l

admissible function in

(1.1)

^and

(1.2)

is defined

by

In

(1.2) e

denotes the vector

(0, 1).

Formally,

^y the variational

inequality

ⁱⁿ

(1.2)

^is

equivalent

^{to the diffe-}

rential

equation

where v ^{= -}

(Vu + ye),

^{and the}

^boundary

^conditions

We are interested in the behavior of the free

boundary 8(u &#x3E;0}

^at

points

of detachment from the fixed

boundary

^{and we will show that the free}

boundary

^{behaves as} indicated in

fig.

^2-4

(see

^also

^[10], 2.2.1).

^{Our proce-}

dure is as follows.

First we summarize the basic

properties

of the solution related to the

Lipschitz continuity

^{up to the}

boundary

^8Q

(section 2).

This allows us to consider linear

blow-ups (2.5),

^{which are the basic} tool in order to obtain the local results in sections 5-8. We will see that u is

Lipschitz

^continuous

in

S2, except

^{at the}

separation points

^between reservoirs and

atmosphere

in the case of

overflow,

^and

except points

^{at which the}

compatibility

^con-

dition is violated.

In section 4 we prove

that y

⁼ which says, that in the ^case of a

homogeneous

^medium the solution consists

only

of saturated

regions

^where

~c &#x3E; 0 and

dry regions where y

^{= 0. For}

^general inhomogeneous

^media

this is not true

(see ^[3] Beispiel 4.6).

^The

comparison

lemma in section 3

together

^{with our} local results about the free

boundary

^{leads to a}

uniqueness proof (9.3,

^see

[9, 13]),

which shows that other different

approaches

gave the same solution

([1, ^{4, 5, 12,} 14],

and for further references

[6]).

The

regularity

^{of the free}

boundary

ⁱⁿ the interior was

proved

ⁱⁿ

[2 ],

and results about the

qualitative

^{behavior were} obtained in

[7, 8, 11]..

In

[7, 11] global arguments

also gave results about the

tangent

of the free

boundary

^at

endpoints

^{in the}

special

^{case of a}

rectangular

^{dam. In}

[8]

^it

was

proved by

^{local methods} that the free

boundary

^is

tangential

^{to the}

fixed

boundary

^{at the}

atmosphere

^{on the}

top.

^{In our paper we use}

only

local methods in order to

study

^{the behavior of the free}

boundary

^{near aS2..}

In section 5 we deal with the free

boundary

^near

points

^on the surface of reservoirs.

Using

^a class of linear _super- and subsolutions we prove that there are three

possibilities: overflow,

^a horizontal free

boundary,

^{or a}

right

angle

between free and fixed

boundary (fig. 2).

F I g . 2 . I Fig. 2. 2 Fi g. 2. 3

In section 6 we

study

^{the free}

boundary

^near

points

^{at the}

atmosphere.

We show that for

points

^{on the lower}

part

of the porous medium

(vertical points

^{of 8Q}

included)

^{the free}

boundary

^{has a vertical}

tangent (Fig. 3).

On the upper

part

of the porous medium the free

boundary

^is

always tangen-

tial to which was

proved

ⁱⁿ

[8].

In section 7 we consider free

boundary points

^{near the}

impervious part

of oil.

Using blow-up techniques

^we prove that the ^free

boundary

^{is tan-}

rig.3.1

~

Fi g. 3 . 2 Fi g. 3. 3

gential

^{to 8Q or}

horizontal,

^{where the case of a non} horizontal

tangent

ap- pears

only

^at

points

^on the upper

part

of the porous medium

(fig. 4). Chang- ing

the porous medium afterwards one can construct

examples

^for

fig.

^3.3

.and

fig. 4.3,

^{where the set}

{u

&#x3E;

0)

^{is an}

^arbitrary

^{closed subset of 8Q.}

Fig.4.1 Fig.4.2 Fig.4.3

Whenever the porous medium lies above its

boundary

^{near the}

point

-of

interest,

^{we need the}

assumption

that there is some

dry neighborhood

above this

point

^{in order to start with our local}

arguments. However,

^under

suitable conditions on the

global

^{data this}

assumption

^can be verified

(see 4.6, 9.2).

^Local

counterexamples

^are

given

ⁱⁿ

5.2.2.),

^{6.5. In all cases we}

assume that the fixed

boundary

^8Q is of class near the detachment

point.

’The

only interesting remaining

^{case are}

separation points

^between

atmosphere

and

impervious parts,

which is treated in section

8,

^{where we allow the fixed}

boundary

^{to have two different}

^tangents

^{at such}

points.

Although

^{we deal with}

homogeneous

^porous

media,

^{it is}

clear,

^{that the}

local results also can be obtained for

inhomogeneous unisotropic

media under certain

regularity assumptions

^{on the}

permeability.

^{In the}

unisotropic

^case

this would

change only

^with the direction of the

possible tangents

^{for the}

free

boundary,

^{which can be}

computed formally. However,

^{if there are}

jumps

in the

permeability,

the situation becomes different. In

general,

^{the free}

boundary

^{is no}

longer

^{a smooth}

graph (see [3], Beispiel 4.6),

^{and it}

^would

be of interest to

study

^{the local behavior} of the solution at least near free

boundary points

in the interior. Another

important generalization

^{is the}

three dimensional case, for which ^we have limit curves instead of limit

points

of the free

boundary

^on

However,

^we think that some of our

techniques

could be used in order _{to prove} that at almost all

points

^{of such a curve the} behavior is as for the two dimensional

problem,

^{if we look at a vertical sec-}

tion to 8Q.

In section 9 we

give global applications

of the local results in the

previous

sections. First ^we show that the solution is

unique

up to

ground

^water

lakes,

where we assume that the

boundary

^sets consist of a finite number of smooth

curves. Then we

study

^the

stability

of the local behavior of the free

boundary

with

respect

^{to small}

perturbations

^{of the}

geometry

of the dam.

Using

^this

we can construct

examples

which show that

actually

^{all cases in}

fig.

^2-4

are

possible.

2. - Basic remarks.

Since our

techniques

ⁱⁿ section 5-8 are of local

nature,

it is convenient to define

2.1 LOCAL SOLUTIONS.

If

^B

is, for

^a

ball,

^{we call U E}

Hl,2 (B)

and y ^E local solution in Q r1

B, if u

^{is a} solutions in this

region

^as

~~

( 1.1 )- ( 1.2 )

^with

respect

^{to its on aB as Dirichlet}

data,

^{that is}

with

boundary

^values

and

for

^test

functions ~

the

inequality

holds

The maximum

regularity

^{for solutions u}

locally

^{in Q is the}

Lipschitz continuity,

^{which was}

proved

ⁱⁿ

[3],

^{Satz 3.6. In the}

^special

^{case of har-}

monic functions this

proof

^{is more}

transparent,

^{and therefore let us}

repeat

it here.

2.2 LEMMA.

I f

u, y is ^a local

solution,

^and

if

^{a ball}

Br(x)

^{c S~ r’1 B r1}

fu

&#x3E;

ol

touches the

free boundary

^{Q n B n} &#x3E;

01,

^then

with a universal constant C.

PROOF. We can assume

B,(x)

^{c S~ r1 B. For small b} &#x3E; 0 let v be the har- monic function in D :=

B~ 1 + a)r( x )~Br~2 ( x )

^with

Then

[3],

^{Satz 3.3}

(see proof

^of

3.4) implies

Since D contains a free

boundary point, u

^{is not} harmonic in

D,

^{hence the}

left side is

positive.

We conclude is ^a monotone function of the dis- tance from

x)

On the other hand

using

^Harnack’s

inequality.

2.3 REMARK. 2.2.

implies (see [3],

^Satz

3.7)

^{that u is}

^Lipschitz

^continuous

locally

^{in S~. If D cc S~} is connected and contains a free

boundary point,

the supremum and the

Lipschitz

^{constant of u in D}

depend only

^{on the}

geometry

^{of .~ and D.}

PROOF. Let x : = dist

(D,

^and ro ^E D n

8(u

&#x3E;

0}.

^Then

we find balls

Bx(x~ ) c S~, j =1, ... , k,

^{with and}

EBx/4(Xj),

^{I where}

k

depends only

^{on x} and the diameter of D. Then

by

^2.2

is ^non

empty .

and

by

^Harnack’s

inequality

We

get

Later we will use the

Lipschitz continuity

^{of u} up ^to the

boundary.

^Therefore

let us prove another version of ^2.2

including

^{the fixed}

boundary.

2.4 BASIC LEMMA. Let u be a local solution in S2 r1

BR,

^{and assume that}

Q n

BR

and aS~ r1

BR

^are

connected,

and that the curvature

of

^r1

BR

^does

not exceed

If

^Q

n BB/4

^{contain,8 a}

free boundary point,

^{then q ‘}

and

Here C is a universal constant.

PROOF.

By homogeneity

^{we can assume R = 4. Let with}

u(x) &#x3E; 0

^and

B,(x)

^the maximum ball contained in

B4B( S~

ⁿ &#x3E;

0~ ).

Then

B2r(x)

^{contains a free}

boundary point and

because of the

assumption

on the curvature of

aS2,

^{w e}

have x f

^{E c SZ for some}

^point

^Xg.

Then,

if

Bar(x)

is contained ⁱⁿ

S~,

^{we conclude as in 2.3 , .}

which,

ⁱⁿ

particular,

proves

1).

^{If is not contained in .~ we have}

(now

let 6 ⁼

1 /4 )

^{’ 1}

for some

point

x1 ^E and as in the first case we can estimate

Now let

P.

be the Poisson kernel of

.~~~~(~1)

^{r1 SZ with}

^pole

^{at ~;.}

^By

^the

assumptions

^{on 8Q we have}

which proves the second

inequality

ⁱⁿ

2).

^Moreover

and

by

^Harnack’s

inequality

the second term can be estimated

by

which is the first estimate in

2).

^For

3)

^use the Green-Neumann function

Gx,

that

is,

- -

Using

^Harnack’s

inequality again

^{we see that}

.I. V ’-’.41"" V1. U1

My lp 1..l.ppVllNVI.L uy

the blow-up

^{limits at this}

point. These blow-ups have

^the

advantage

^that

they

^are

globally defined,

^{and that}

they

^involve

only

^{the data at the}

point

xo, hence

they

^have

analytic boundary

^{data and}

satisfy

^an

analytic

differential

equation,

^{even in the case that we} start with a more

general equation

^{for u.}

Moreover,

many free

boundary problems

^are closed with

respect

^to

blow-up

limits,

therefore these limits are

global

solutions. In

general

^{the definition}

of

this limit has to take into account the

homogeneity properties

^{of the}

special

problem.

^In this paper ^{we are} interested in linear

blow-ups

^{at the}

boundary.

2.5 BLOW-UP LIMITS. Let _u, y ^{be a}

Lipschitz

continuous local solution

in Q n

Bll

^{and assume that ’}

1 )

^{0 E aS2 with =}

0,

2)

^Q has normal v* at

0,

^,

3)

^{near 0 the sets aD r1}

{:l: 0153. ^(iv*)

&#x3E;

01 ^belong

^to

Sair’

^or

Simv

^*

(By

continuation we can assume that ^u and y are

defined

ⁱⁿ

B,,, preserving

the

.Lipschitz

^constant

for u).

Corsider the

blow-up

sequence

Then there are

f unct2ons

~c* ^E and y, ^E such that

for

^{it sub-}

sequence

u* ,

Y * is a global

solution in

S~* : _ {~’~0} (that is, a

^local

Q* for

every

R)

^with

respect

^to

boundary

^data

induced by 3).

PROOF. ur ^are

uniformly Lipschitz

continuous

and yr

^are

uniformly bounded,

^{hence we have} the above convergence

properties.

^{To see that} u*, y* ^{is a}

solution,

^denote

by

^{etc. the}

corresponding boundary

sets of and Q*.

By 2 )

^and

3 )

every

with

compact support and ~

^{= 0 on}

~gy ~0

^{on can be}

approximated strongly

ⁱⁿ

by functions Cr

with these

properties

^{on Then}

2.6 DEFINITION. In

general

^{one considers}

blow-up

sequenoes U’r with

respect

^{to balls}

Br(zr) defined by

for an arbitrary

sequence ^{zr r1}

(u

⁼

0~. If

^the

boundary

^sets

of

^the

blow-up

domains

,S2r :_

converge in ^an

appropriate

^{one has the}

same statements as above.

3. -

Comparison

^lemmas.

The

general

definition for _super- and subsolutions is stated in

3.1,

^but

because of technical details near the fixed

boundary

^{we need some additional}

regularity properties

^{in order to} prove ^the

comparison

^{lemma 3.2.}

However, using

the results of sections 4-8 this lemma

implies

that under certain con-

ditions on the data the solution is

unique (9.3).

^Another

comparison lemma,

which can be used for certain

explicit subsolutions,

^is stated in 3.4.

3.1 DEFINITION. Let with and with

~y~l.

^{We call} u, y a

,

1) supersolution, if

^{on and}

for

every

non-negative’

^E

Hl,2(Q) vanishing

^on

Sres

^U

Sair

ⁱ

2) subsolution, if ^{u c u°}

^{on l~ 7 and}

for

every

non-negative’ c- HI 2(S?) vanishing

^on

Bres.

These definitions are such that u, y is a solution if and

only

if it is a

super- and ^a subsolution.

~,

^3.2 COMPARISON LEMMA FROM ABOVE.

Suppose

u, V is ^a

solution,

^and

v, is a

supersolution positive

^{in a}

neighborhood of Sres

with the addi- tional

property

^that

Q

n

a{v

&#x3E;

0}

^consists

of Lipschitz graphs

in vertical di- rection

locally

^{in Q and a set E c aS~ with}

Then u c v on connected

components of

^{Q r1}

Iv

&#x3E;

01 touching

and y ^{= 0} above these

components.

PROOF.

For e

&#x3E; 0 consider the cut-off function

and for 6 &#x3E; 0 define

and for &#x3E; 0 let

Then

-1 ) - ^v)

^{is an} admissible test function for u, y in

(1.2),

^hence

and

v)

^{is an} admissible test function for v in

3.1.1),

^hence

Adding

^these

integrals

^{we obtain}

therefore

By assumption

^{the last}

integral

converges ^{to zero}

for e

^{- 0}

independent

^{of s}

and 6.

Clearly

^{the first term tends to zero 0. Therefore we have to}

show that the middle term becomes

non-positive

^{in the}

limit ê t

⁰

and 3 j 0,

which is an estimate near

Since

locally Q n {v

&#x3E;

0}

^{is a}

^subgraph

^{in vertical}

direction, de

^increases

with

height

^near Therefore for small s one

part

of the second

integral

above is

-

and

using

^{2.3 we} have for the second

part

Since

Ie

^{is a}

Lipschitz graph by assumption

^and

fu

&#x3E;

01

^a

^subgraph

in vertical direction

(see 4.1 ),

^{and since}

^{d,~ = 0}

^{below this can be esti-}

mated

by

which tends to zero

as ê t

⁰

and 3 §

^0.

We thus

proved

^that

From this the lemma follows

easily.

^{Let D be a connected}

component

^of

S~ r1

{v

&#x3E;

0} containing

^a

given point xo

^{E as}

boundary point.

^{Then v}

is

positive

ⁱⁿ

B,(x,)

^{for some r} &#x3E; 0. If we

replace v

^{in S~ n}

Br(xo) by

the harmonic function with same

boundary values,

^we

get

^{a new} super-

solution,

^{hence we can}

apply

the above estimate for the new v in the fol-

lowing

way. If

have and

The first term

vanishes,

since v is harmonic in Q n

Br(xo)

^{and = 0}

in

(u =0}.

^{The second term is}

This shows that the function w E with ^{w =} 0 in is harmonic in

Br(xo),

^{therefore zero in}

Br(xo). By

continuation the same argu- ment

implies w

^{= 0} in D. Since then u ⁼ 0 and therefore

ahy

^{= 0 above}

S~ n one estimate above shows that for

compact

^{curves 27 =}

graph g c

This

completes

^the

proof

of the lemma.

3.3 REMARK. In the definition of

supersolution

^no overflow condition appears, and the

comparison

lemma says, that the solution is the minimal

supersolution.

This result was

already

obtained in

[1] by

construction.

Here u is called minimal if for _every

supersolution v

^{we bave in Q’}

except

in certain

ground

^{water lakes}

(see [1],

^{4. and the}

uniqueness

^theo-.

rem 9.3 in this

paper).

The

following comparison

^{Lemma is not}

sharp,

but easier to prove.

3.4 LEMMA.

Suppose

u, y is ^a

solution,

and v,

is

^a subsolution with,

{v &#x3E;0}.

^Then

u&#x3E;v in

every connected

component of

^{Q (*)}

{u

&#x3E;

0} touching

^a

segment of Sres.

PROOF

(see [3],

^Satz

3.3).

^{We have}

that

is,

The first term is

non-positive,

since v is a

subsolution,

^and the second

integral

is

non-negative,

^since

by assumption

that

is,

4. - The free

boundary

^{in the interior.}

If u,

y is ^a solution for

inhomogeneous

media with

permeability

a, then-

®y (aue) 0, provided W (ace) 0 ([3],

^Satz

4.4),

^which

immediately gives

^4.1

in our case. Moreover this condition on the

permeability implies

y ⁼ which was

proved

ⁱⁿ

[3],

4.1-4.4. Here we will detail the

proof

^{for two dimen-}

sional

homogeneous

^media.

Knowing this,

^{one can}

apply

^the

regularity

results in

[2],

^that

is,

^{the free}

boundary

^{in the} interior consists of

analytic

curves.

4.1 REMARK. If X

[ho, h,]

^c

S~,

^then

u(yo, hi)

&#x3E; 0

impliesu(yo, ho)

&#x3E; 0.

4.2 LEMMA.

If -

^{then 1}

uniformly for

PROOF. Define 7~:=

X [ho, h,]

^and

-Choose a sequence

(yr, hr)

-~7~ with r ⁼ such that

We can assume that y, &#x3E; yo . Since u is

locally Lipschitz

continuous

(2.3)

l is

finite,

^{and for a}

subsequence

^the

blow-up

sequence u, with

respect

^to

the balls converges ^{to a} limit u*, which satisfies

By

^the

strong

^maximum

principle

^this

implies

Since u*, y* is ^a solution we obtain I ⁼ 0

by

^the

following argument.

^For

non-negative functions ~

^E

0~(R2)

^and

d,,(y, ^{h) :=}

^max

(min ^{(1 + 1),} 0)

we have

The first term tends to zero

for 6 ^0,

and since u* ⁼ 0 on

{y

⁼

0}

^the

second

equals

- -

where the first

integral

^{is of order b2}

by

^the

Lipschitz continuity

of u * and the second

integral

^is

non-negative.

This proves

4.3 SEPARATION LEMMA.

If

^X

]ho, h1[

^{c Q n}

{~

⁼

0~,

^{then there is no}

through

^this

line,

^that

is, from

both sides this line can be considered as

impervious boundary.

PROOF.

Let i

^{e with}

support

^{in a small}

neighborhood

^{of a}

point

on this

line,

^{and define}

Then similar as in the

previous proof

^{we have}

for 3 §

⁰

which tends to zero

by

^4.2.

Using

^{4.2 and 4.3 we can} prove

that y

^{is a} characteristic function.

4.4 THEOREM,

SUPPOSE

^X

[ho, hl[

^{c Q n}

{2

⁼

0}

^with

^{(yo, hl)}

^E

E such that in a

neighborhood of

^this

point

^{aS2 is a}

Lipschitz graph

in vertical direction. Then y ^{= 0} in a

nezghborhood o f X ]ho, hl[.

PROOF. Let

hl.

If u would be

positive

^near

(yo, h2)

^{on one}

side of the line

{y

⁼

^{Hopf’s principle}

would contradict 4.2. Therefore

we find values y- y+ ^near yo with

h2)

^{= 0.}

Then u ⁼ 0 above these

points by 4.1,

^{and 4.3}

implies that u,

y is ^a solu- tion in

with

boundary

^sets

Since u ~ C3 y+ - y_, ^{we can}

apply

^the

comparison

lemma 3.2~

to the

supersolution

and we obtain

Then

also y

^{= 0 in this}

region,

which follows

immediately

^{if we take}

~(y, h)

^{= min}

( C3 - ^(h

^-

h2), 0)

^{as test} function for the

solution u, y

ⁱⁿ

^D.

4.5 COROLLARY Let _xo=

(yo, 8(u

&#x3E;

01

^{and x, =}

^{(yo, hl)}

^the

^first

point

^{on 8Q} above xo .

If

^Xl

satisfies

the conditions in

4.4,

^{then Q r1} &#x3E;

0}

is an

analytic, graph

in vertical direction near Xo and 7, ^{= 0 above} this PROOF. If we

apply

^{4.4 to all}

points

^{of ~~ r1}

0}

^{near xo we}

^get

^that

Y ⁼ in a

neighborhood

^of xo and above this

neighborhood.

^Moreover

y ^{= 0 in a}

neighborhood

^{of xi . For} this situation the

regularity

of the free

boundary

^{near zo was}

proved

ⁱⁿ

[2].

Example

6.5 shows that we need

global assumptions

^{to be sure that}

every free

boundary point

^is

regular.

4.6 THEOREM.

If

^consists

o f

^a

finite

^number

of points,

^and

if

uo &#x3E; 0

on then y ⁼ and

locally

^{Q r1} &#x3E;

0)

^{is an}

anal ytic graph

ⁱⁿ

vertical direction.

PROOF. Let x, ⁼

(yo, ho)

^{be a}

point

^a little bit above a

given

free bound- ary

point.

^{As in the}

proof

^{of 4.4 we find a}

sequence y ,

^{yo with}

^ho)

^{= 0.}

Let

(yk, hk)

^{be the first}

point

^{in 8Q above}

(yk, ho),

^{and choose a}

subsequence

such that

(y,, hk)

^{- x.} Denote the

polygon

^{with corners}

(y,, hk), (y~;, ho)r ho),

^and

by rk

^{and let}

Dk

be the connected

component

^of

containing

^{the lower}

part

of the small

strip

^enclosed

by Fk. By

^the

above choice of the

subsequence

^{and since 3D is}

Lipschitz

^we infer that for

large k

^{the set}

D~

^{is bounded}

by T,

^{and a curve}

Ik

^c

3Dy

^{and that}

and

(y+i , hk+l)

^{are the}

only

^common

points

^of

Ek

^and

Fk.

Moreover for

large k and k

&#x3E; k

-E-

^{1 the} intersection

Ik

^r1

Ik

^is

^empty.

^Since

aSres is

^{finite w e have}

two

possibilities:

1. ca,se :

1:k

^C

Sres

^{for some k. Then u} &#x3E; 0 on

Ek

^{and 4.1}

yields

^u &#x3E; 0 in

D&#x26;y

^which

together

^{with the}

Hopf principle

contradicts 4.2.

2. case : for all

large

^{k. Then we conclude as in 4.4 that}

-u(y, h)

^{= 0 for}

(y, h)

^E

S2,

^with and also y ^{= 0 in} this

region.

Since this is true for all

large k,

^{and since the same}

argument

^holds

from the left

side,

^we

proved that y

^{= 0 in a}

neighborhood

^{of X}

]h., ~o -~- 81

for some E &#x3E; 0. Then the assertion follows as in 4.5.

The

following

^{statement is similar to 4.4 and can be}

interpreted

^{as a}

nondegeneracy property.

^The

proof

^also

applies

^to

higher

dimensions.

4.7 LEMMA. Let

Q

⁼

]-1, 1[X ]0, oa[

and suppose that

Q

^{r~ Q is contained}

in

Q

ⁿ

empty,

^and

Q

^rl

Sair

ⁿ

BifiD

^is

finite. Moreover,

^as-

that

Q

^{n . is a}

Lipschitz graph

in vertical direction. Then

for

⁰

no hl

there is a constant c ⁼ &#x3E;

0,

^{such that S~ n}

aQ implies

y ^{= 0} in ac

neighborhood of (0,

PROOF. Define D : =

Q

ⁿ

Ih g(y)},

^where

^{g : [-1, 1]}

^-

]0, oo[

^{is a} smooth function with

~(± 1 )

⁼

2h1, g(± 1)

⁼

0,

^and

g(o ) ho .

^Consider

the harmonic function v in D with

boundary

^{values v = 0 on}

graph g

^and

-~(y, h)

⁼

2h,

^{- ~ for}

(y, h)

^E

8D%graph

g. There is an 8 &#x3E;

0,

^{such that ev}

is ^a local

supersolution

^{in S~ r1}

Q.

^Since &#x3E; 0 on S~ r1

aQ,

the assertion follows from lemma 3.2

applied

^to the domain

Q

^{r1 S~ and the}

boundary

sets Q n

Q,

¹

Sair

⁼

Sair, 191MD

⁼

SIMD

^{In order to}

verify

the condi- tions in the

comparison

^lemma

choose 9

such that E : = aS2 n

graph g

^con-

sists of ^a finite number of

points belonging

^{to or At such}

points

Vu satisfies ^a

Morrey

^condition

for some a &#x3E; 0.

.5. - The free

b©undary

^near reservoirs.

consider the free

boundary

^{near a}

point

^{on the fixed}

boundary,

^which

lies on the surface of a reservoir.

5.1 AsUmos. _u, V ^{is a} local solution in Q r’1

BR for

^{some .R} &#x3E; 0 and

1)

^{. r1}

.BR

^{is a curve with curvature} less then _"0’ contains the has

tangent

exp

Cido]

^{at the}

origin,

^{where 0 cr,, :rt.}

2)

^{8Q rl}

^BR

^r’1

{h 0~

is contained in with Dirichlet data

is contained in

Sair

with Dirichlet data

5.2 REMARKS

1)

^{Since we are} interested in local

properties

of the free

boundary,

we ^can choose 1~ such that

with a OJ,1 function Y

satisfying

2)

^{We should}

remark,

^{that in}

general

^the

assumptions

^{in 5.]}

give

^no

information about the behavior of the free

boundary

^{at the}

origin.

^For

example,

^if ao &#x3E;

n/2

^{and 0 0’ 0’0 then is a} local solution in

fy

&#x3E;

(cot

^{where ua} is the linear function in 5.5 and

Therefore in the case we need the additional

assumption,

^that

y ⁼ in a

neighborhood

^{of some}

point

^{above the}

origin.

^{But then}

(see 4.6)

^{we can} choose .R small

enough,

such that this is fulfilled r1

3)

^The

special

^{data in}

5.1.2)

^{are not} essential for our

techniques,

^but

since

they

^are the standard ones for the dam

problem,

^for

simplicity

^{we re-}

strict ourselves to this case.

Our

procedure

^{is as follows. In the case thatu is}

positive

^{in a}

neighbor-

hood of the

origin,

^we have overflow in this

neighborhood.

Otherwise the

origin

^{is a limit}

point

of the free

boundary,

^{and we show}

(5.3)

that this is,

equivalent

^{to the}

Lipshitz continuity

^{of u near the}

origin.

^{Therefore we are}

allowed to consider

blow-up functions,

^{which are studied in 5.6.}

Using

these results we can go back ^{to the}

original

function u and obtain the desired result

(5.7).

5.3 OVERFLOW LEMMA.

I f u is

^as in 5.1 with the

properties

ⁱⁿ

5.2.1),

then i

for

^{with a.}

universal constant C.

PROOF. If u is

positive

^{in a}

neighborhood

^{of the}

origin,

^then

for z ^-~

0,

^{that is not}

Lipschitz

continuous. Otherwise 0 is ^a limit

point

of the free

boundary.

Then for x E Q n with

u(x)

&#x3E; 0 let be the maximum ball not

intersecting 8(u

&#x3E;

0}.

^Since

^B,.(x)

^{does not}

contain the

origin,

^{we have to}

distinguish

between the

following

^{three cases::}

1. case : e Q. Then u is harmonic in this ball and ^we

get

^from

2.4.1) (applied

^{to S~ n}

2. case : x E

Br/4(Xl)

^{for some Xl E}

Bres.

^{Then u} is harmonic in Q r1 and aD r1 c The first estimate in

2.4.2) (again ^applied

^to.

S~

n B.5,(X)) ^implies

and from the second

inequality

^{we obtain}

Therefore the harmonic function

w(x) : = u(x) +

^{in S2 ~1}

Br/2(X¡)i

satisfies

and

by

well known estimates ^we obtain

3. for some x, ^E

Sair.

As in the 2. case we conclude-

(now

^{the Dirichlet data on 8Q n}

Br/2(Xl)

^are

zero)

5.4 REMARK. The

length

of overflow can be estimated from below in.

the

following

^{sense. There is a universal constant}

C,

such that the

inequality

~~(~) ~ Clxl

^{for some x c- Q r1}

B RI4 ^{implies u}

&#x3E; 0 in Q r1 ^* This follows from

2.4.2).

In the case of a linear fixed

boundary

^{there is a} class of linear _super- and

subsolutions,

^which

give

^us the essential information about the behavior .of the free

boundary.

5.5 LINEAR SOLUTIONS. For define

jf

positive,

^and

ua(x)

^{= 0 otherwise.}

Fig.5

Fig.

^{5 shows the}

regions

^{of the}

parameters

(10 and or for which ua is ^a

super-and

subsolution in the half space

that

is,

Therefore uu is a

solution,

^{if and}

only

^{if a = 0}

(horizontal

^free

boundary)

or l1 ⁼ (/0

- nj2 (right angle).

uO is the solution at rest.

Therefore it is natural to define

and to state the

following theorem,

^{which we} will prove

using

^the

comparison

lemmas 3.2 and 3.4.

5.6 THFOREM. Assume aS~ is linear near the

origin

and u, y is ^a local solutions ^as in 5.1. Then the

following

statements hold.

1) I f

exp is any

tangent

^vector

of

^the

free boundary

^at

0,

2) If with y = ( relevant only for (10"&#x3E;7&#x26;/2),

then either

u ⁼ or the

free boundary

^{has the}

tangent

exp

[i~_]

^{at 0 and is the}

unique blow-up function.

3) I f u ~ u~+,

then either ^{u =} uG+ or we have

overflow,

^that

is, u

&#x3E; 0 in

a

neighborhood of

^0.

PROOF OF

3).

^{For define}

Let us assume that there is no overflow. Then 5.3 says that u is

Lipschitz

continuous in hence

by

^{2.5 we can choose a} sequence

r ~0

^{such that}

converges ^{to a}

blow-up

limit u*, y*. and _a* do, ^and since

h )

^{= 0 for we have}

Now assume that

u *(xo)

&#x3E; 0 for some xo ^E S~ n

a{u’-

&#x3E;

0}.

^Then

^by

^the

strong

^maximum

principle u* &#x3E; u°~*

ⁱⁿ

{u’- &#x3E;0}y

^{and since u, - u*}

uniformly

^{we must have}

for some 8 &#x3E; 0 and small r. Moreover for the

point

^{x1 E where}

ro = ^we have

by

^the

Hopf principle (v

^{normal of}

aS2)

and since ~r -~ ~ * in C ^{I near} zi, ^we also must have

for some c &#x3E; 0 and small r. On the arc

1:e

^of

joining BE(xo)

^and

we have ~c* &#x3E; therefore also

for some El and small r.

Altogether

^{we see that}

for some E and small r. Since

u’* "

is ^a

subsolution,

^we obtain from the

comparison

^{lemma 3.4}

applied

^{to the}

domain D - D

n

Bro

^{and the}

^boundary

sets

Sres U ^(f2

^{() and}

Sair = Sair

that

is, ~(r) ~ Q* -~ ~,

^a contradiction to the definition of ~* . Therefore ^we must have

and we will that this

implies

In the case

~* C ~c/~

^{we conclude}

(see 4.1) u * = 0

ⁱⁿ

~~°‘* = 0~

^r1

and ⁼ 0 in this

region.

^{Therefore on} &#x3E;

01 (y *

is defined from

above)

(By

^{4.2 the case} ~,~ _ Qo cannot

occur.)

This proves, that on the

&#x3E;

01

The

right

^{side is}

non-positive,

^since

o’~&#x3E;o+ by

^our

assumption

^{in the state-}

ment of the

theorem,

and the left side is

non-negative

^since

~c* ~ u~*.

^There-

fore a*=

~+

^and

o-v(u*-u(1*)

^{= 0 on} the above line. Since the

Cauchy

problem

^{has a}

unique solution,

^we conclude u* ⁼ u~+.

If we now

apply

^{the same}

arguments

^{to u and} 1+ instead of ~c * and ~*

we obtain u ⁼ u’+. _,,

PROOF OF

1).

Define 1* ^as

before,

^where

d*(r) :=

^{if the set}

in the definition is

empty.

We will argue ^as in the

proof

^of

3),

^and

thore-

_{Su ’}

fore let u*, y* be ^a

blow-up limit,

which exists

by 5.3,

^{since there is}

nothing

to prove in the overflow ^case.

i ’°

First we see that the

Lipschitz continuity immediately implies

1* &#x3E; ~o ^-a~.

Then we can

apply

^the

arguments

^{in the}

proof

^of

3)

^{and obtain}

on the line &#x3E;

0},

^which

implies

PROOF oF

2).

^Define

Since we have no

overfiow,

and therefore

by

^{5.3 we can choose a}

blow-up

sequence ur, yr

converging

^to u*, y*.

Then

(we proved

^and

~_ c 6* c ~+.

If ~* _ ~_, the free

boundary

has the desired

tangent.

Therefore let 6* &#x3E; 0’-

(hence

· We will prove u =,a’+ as in the

proof

^of

3),

^{but now we} argue from above.

Assume there is a free

boundary pointzo e

&#x3E;

0}

^r1

~u~- - 0)

^{of u * .}

Then for a

subsequence r j 0

^{there are free}

^{boundary points}

^{xr =}

^{( yr, hr)}

of _Ur

converging

^to xo, since otherwise ~c* would be harmonic in ^a

neigh-

borhood of xo . Then

h)

^{= 0 for}

^{h ~ hr}

^{and similar as in the}

proof

^of

3)

we conclude that for some E &#x3E; 0 and small r in the above

subsequence

where

(In

^{the case} Go &#x3E;

n/2

^cut

^Qr

^{in the}

^region

^where

by assumption

yr ⁼

0.)

Now we

apply

^the

comparison

^lemma from above

(3.2)

^to the domain

S~r

^{= Q m}

Qr

^{and the}

boundary

^sets

Sres = Sres

^{U n} &#x3E;

0~ ) ^and Sair

⁼

’Sair

^or

Sair U

&#x3E;

0~ ) .

^{This can be}

^done,

^{since the sepa-}

ration lemma

(4.3)

and the fact that yr ^{= 0 near} the

top

^of

Qr

^{in the case}

~o &#x3E;

zc/2

^ensure

that ur

^{is a} solution with

respect

^{to the data defined above.}

We obtain in

£5,

^a contradiction to the definition of c~*.

Therefore ~c* is harmonic in &#x3E;

0}

^{and zero on}

a{u’*

&#x3E;

0},

^which

and

ah y * = 0

ⁱⁿ

~ua* - 0~ .

^{But since Yr = = 0 in}

=

0}

^{near 0}

^by

^the

assumption u

^{and 4.5 we see} that y * ⁼

X(u.

&#x3E;O}

and therefore

on the line &#x3E;

01,

^{which as in the}

proof

^of

3) yields

If we

apply

^{the same}

argument

^{to u and} 0’+ instead ^of

u* and 0’*,

^{we obtain}

u =

5.7 THEOREM. Let u, y be ^a local solution ^as in

5.1,

^{such that 0 is a limit}

point of

^the

free boundary,

and in the case assume that y ^{= 0} in a

neighborhood of

^some

point

^{above 0.}

Define

Then the

following

^{is true.}

2 ) If

^{a- -r} cr+, ^{a- and the}

free boundary

^{has the}

tangent

exp

[i~_]

^{at 0.}

3) If t ~ ~+,

^{the 7: =} a~+ ^{and the}

free boundary

^{has the}

tangent

exp at 0.

PROOF OF

1).

^We

proceed

^{as in the}

proof

^of

5.6.1),

^{but since 8Q is not}

a

straight

^{li.ne we have to}

^change

the definition of the functions u(1. For small ^r &#x3E; 0 let ur ^E

]0, n[

be the maximum value for which

and define u’l1 as ul1 in

5.5,

^{but with (To}

replaced by

^{(Jr. Then u" are subsolu-}

tions for and

(Jr t (Jo

^as

r ~ 0. ^Using

^{ura in}

SZ r1

Br

^{instead of u~ in the}

proof

^of

5.6.1)

^we obtain the assertion.

(For ¹⁾

the

assumption

on y ^{is not}

needed.).

PROOF OF

3).

^{Since 0 is a limit}

point

of the free

boundary,

u is

Lipschitz

continuous

(5.3),

^and

blow-up

limits u*, y* exist. We will

show,

^{that every}

blow-up

^limit

equals

^{~a*. For s} &#x3E; 0 consider the domain

By assnmption u

^is harmonic in

B,

^r1

De

^{for small r.} Since

D~

has an

angle

less than n at 0 we conclude _,

which

implies u*&#x3E; u°+.

^{Since ê was}

arbitrary

^{we and}

5.6.3) gives

u* ⁼

11/1+.

This shows that

If

not,

there would be an e &#x3E; 0 and a sequence

r t

⁰ such that u &#x3E; 0 in

[~o~+]),

hence the

corresponding blow-up

^{would be harmonic in}

B,(exp [ZC~+]),

^a contradiction. Therefore we find a sequence of free

boundary

points (y~, hk)

^{-+ 0 with .}

and since u is

Lipschitz

continuous we have

where

Here

(y, g(y))

is the first

point

^{on 8Q above}

(y, 0)

^{or a}

point

^{in the}

neighbor- hood,

^{in which}

by assumption

y ^{= 0. For}

given E

&#x3E; 0 consider the function

Then on for

large k

^and

using

^the

separation

^{lemma 4.3 we can}

apply

^the

comparison

^{lemma 3.2 to}

15~

^with

boundary

^sets

BreB

^_

(Vz &#x3E; 0)

and

8m

⁼ &#x3E;

0}.

^We obtain that the free

boundary

^{of u is contained}

in

{v,

&#x3E;

0}

^{near the}

origin,

^{which proves}

3),

^{since E was}

arbitrary.