Estimations of the best constant involving the <span class="mathjax-formula">$L^ \infty $</span> norm in Wente’s inequality

A NNALES DE LA FACULTÉ DES SCIENCES DE T OULOUSE

S ^AMI B ^ARAKET

Estimations of the best constant involving the L

^∞

norm in Wente’s inequality

Annales de la faculté des sciences de Toulouse 6

^e

série, tome 5, n

^o

3 (1996), p. 373-385

<http://www.numdam.org/item?id=AFST_1996_6_5_3_373_0>

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- 373 -

Estimations of the best constant involving

the L~

norm

in Wente’s inequality(*)

SAMI

BARAKET(1)

Annales de la Faculte des Sciences de Toulouse Vol. V, n° 3, ¹⁹⁹⁶

R,ESUME. - Dans ce travail, ^on s’interesse a la meilleure constante dans

une inegalite due a H. Wente. On montre que cette constante est bornée independamment du domaine sur lequel ^on travaille. En particulier, lorsque le domaine est simplement connexe, la meilleure constante associee a la norme Loo est

1/2 7r.

ABSTRACT. - In this work, ^we study ^{the best constant in the so called} Wente’s inequality. Our main result relies on the fact that the constant in Wente’s estimate can be bounded from above independently ^{of the} domain on which the problem ^is posed. ^In particular, if the domain is bounded and simply connected, ^we show that the best constant involving

the Loo norm is

1 ~2~r.

Introduction and statement of the results

Let Q be a smooth and bounded domain in

R2.

Given two functions a

and b on H such that

We denote

by 03C6

^the

unique

^{solution in}

L2 (S2)

^{of the Dirichlet}

problem

where

subscripts

^denote

partial

differentiation with

respect

^to coordinates.

( *) Reçu le 11 avril 1994

(1) C.M.L.A. U.R.A. 1611, ^{61 av. du} President-Wilson, ^{94235 Cachan}

(France)

^et

Faculté des Sciences de Tunis, Departement ^de Mathematiques, Campus ^Univer- sitaire, TN-1060 Tunis

(Tunisie)

In the context of the

equation

^{of the mean}

curvature,

H. Wente in

[5],

also H. Brezis and J.-M. Coron in

[3]

^{have obtained the}

following striking

result.

THEOREM 0

(~3J, ~5J).

^- The solution _~p

of (2)

^{and is a continuous}

function

^{on SZ and its}

gradient belongs

^to

L2(03A9).

Moreover there exists a.

constant which

depends

^on S2 such that

F. Bethuel and J.-M.

Ghidaglia

ⁱⁿ

[1]

showed that under the

hypothesis

of Theorem

0,

^{there exists a constant}

Ci

^{which does not}

depend

of S2 such that

(4)

^{holds true:}

Remarks 1

1 It is clear that

Co(f2)

ⁱⁿ

(4)

^{is invariant under} translation and dilatation of

H,

it could however

depend

^{on its}

shape.

The result in

[1]

^{means it is}

proved

^{that the constant}

Co(f2)

^{is bounded}

independently

^{of f2.}

2 The constant is invariant under a conformal transformation of Q.

We denote

by Coo(Q)

^{the best constant}

involving

^{the L°° norm and}

by

the

L2

^norm. Then we have:

Our _purpose in this _paper is to

study

We first remark that:

Our results are the

following

^theorems.

THEOREM 1. ^- Assume that SZ ⁼

R2,

^then:

THEOREM 2. ^- Let S~ be a. smooth and bounded domain in

II~2,

^then:

Moreover

if

^{we assume} that S2 is a bounded

simply

^{connected domain in}

then:

Proof of

^{Theorem 1. . - In}

~3~

H. Brezis and J.-M. Coron showed that

1/27T,

^{the next}

step

^{is to show that}

1/2~r.

^{VVe let}

and g

^{will be} chosen later. We

obtain,

By (2),

^{we have}

Thus we

compute

and

Hence

Now we

choose ge = e-~~2 ,

^we first observe that

and hence

Then, by (15)

where r denotes the _gamma function. We conclude that

when £ tends _{to zero,} so

Proof of

^{Theorem ~. . - We first assume} that S2 is a smooth and bounded domain

in 2,

^not

necessarily simply

connected. Here we let also

and g

^{will be chosen later.}

Next we introduce the

following

^notations

we obtain

by ( 15)

It was established in the

proof

^{of Theorem 1 that:}

when n tends to

infinity. Let g

^be

fixed, ~

^{E such that:}

First

step

We

put g~, (~) _

^{and we show the}

_following

^Lemma.

LEMMA 1. ^- With the above

notations,

^{zue have}

when a tends to

infinity.

Proof of

^{Lemma 1. . - We} have as 03BB tends to

infinity,

Thus

where

By

^the

inequality

^of

Cauchy-Schwarz,

^{we obtain}

then

Returning

to gn, ^we prove that for n fixed ^-~

L(gn)

^{as a tends to}

infinity. By (24),

^-

then

by (22)

Denote ^{= we} have _supp

g~, C ~ 0 , 2~~, ~.

^Let

= we have _supp

hn

C

~ 0 , 1/2 ~ .

We remark that L is invariant

by

dilatation. Let ⁼

g( au),

^{an easy}

computation

^shows

that

L(ga )

⁼

L(g),

^then

Denote

by

We introduce the solution

03C8n

^{of the}

following problem

We have

Since

then, by (27)

Second

step

Let Q be a

smooth, nonempty

and bounded domain in

I182,

^{then there}

exists ro >

0,

zo such that

B(zo, ro)

^{C S~.}

If we note

an (z)

⁼

an ((z

^-

zo)/ro) bn(z)

⁼

bn ((z

^-

zo)/ro)

^{we have}

since

Let

~n

^{be a solution of the}

following problem

we shall _prove the

following

^result.

LEMMA 2.2014 Under the above

hypothesis,

^{we have :}

Proof of

^{Lemma ~. Let}

SZl

⁼

B(zo, (3/4)ro) , ^S22

⁼

(1/2)ro~,

~1 ~ C~0(R2), supp ~1 C

03A91,

~1 ⁼ 1 on

?-o/2)

^{and ~2 = 1 - ~1, so}

supp x2 C

~2.

For _{r~ E}

since

supp 03C8n

^C

B(z0, r0/2)

^{and ~2 = 0 in}

B(z0, r0/2),

^{we have}

Hence

Returning

^{to the}

proof

of Theorem

2,

^since

we obtain

by (28)

and

finally

which _proves

(8).

V’e now return to the

special

^case where S2 is a

simply

^{connected domain.}

First we prove the

following

^result.

LEMMA 3. ^- Let _~p be the solution

of

the Dirichlet

problem

then we have

where _{y E}

B1, E(x - y)

⁼

yl

^{and v} is the exterior normal vector.

Choosing

y ⁼ 0 in

(35),

^{we have}

It is _{easy to} see from

(2)

^that

so we deduce

Using

we have

then

and

using

^the

Cauchy-Schwarz inequality,

^we deduce that

We consider a function T which is defined from

D2

to

D2 by: T(z) = (zo

⁺

z)/(1 - zoz)

^{where zo E}

^D2.

^{We remark} that ~’ is a conformal transformation and

T(0)

^{= zo.}

Denote

and §

^{= p o}

^T,

^{an easy}

computation gives

Applying (5) for §

^{we obtain}

then

thus

Returning

^{to the}

proof

^{of Theorem}

2,

since f2 is

simply connected,

^there

exists a conformal transformation T such that

T(f2) =

^let

by

^{a similar}

argument

^{as in the}

proof

^of

(37)

^and

^by (38)

^{we find}

Finally

which

together

with the first

step

^{of the}

proof yields (9).

Remark 2. ^{. -} We do not know whether

1/2~r

^for every bounded domain f2 of

R~,

^not

necessarily simply

connected. But the

following suggests

^{that this} may be true.

where the above maximum is taken over map ~p E a, b E

H1 (S2)

such that

(a, ~2) _

PROPOSITION l. ^- the above

hypothesis

^{we have}

Proof of Proposition

^{1. . -}

Using (2)

^and

(3)

^{we deduce that}

then

where

r ~ R2 . Hence,

Now we claim that

Indeed, computation yields

and

Using

^the

inequality of Cauchy-Schwarz,

^{we obtain}

Finally

^{we have}

n

hence we

proved (41).

By

^{a similar}

computation

^{as in the}

proof

^{of Theorem}

2,

^{u~e have}

using (40)

^and

(41),

^{we obtain}

and

finally

- 385 -

Acknowledgements

The author thanks Frederic Helein and Jean-Michel

Ghidaglia

^who

brought

^{him to this}

problem

and for their

support.

References

[1]

^BETHUEL

(F.)

and GHIDAGLIA

(J.-M.) .

²⁰¹⁴ Improved regularity ^of elliptic equa- tions involving jacobians ^and applications, J. Math. Pures Appl.

(to appear).

[2]

^BETHUEL

(F.)

and GHIDAGLIA

(J.-M.) .2014

^Some applications ^{of the coarea} formula to partial differential equations,

(to appear).

[3]

^BREZIS

(H.)

^{and CORON}

(J.-M.) .

²⁰¹⁴ Multiple ^solutions of H-systems ^{and Rellich’s} conjecture, Comm. Pure Appl. ^{Math. 37}

(1984),

pp. 149-187.

[4]

^GILBARG

(D.)

and TRUDINGER

(N. S.) .

²⁰¹⁴ Elliptic partial differential equations of second order, 2nd Edition Springer-Verlag

(1984).

[5]

^WENTE

(H.) .

²⁰¹⁴ An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. ²⁶

(1969),

pp. 318-344.