The best constant for the Sobolev trace embedding from W

Nonlinear Analysis 59 (2004) 1125 – 1145 www.elsevier.com/locate/na

The best constant for the Sobolev trace embedding from W 1,1(
) into L1(*
) F. Andreua , J.M. Mazóna , J.D. Rossib,∗ a Departamento de Análisis Matemático, Universitat de Valencia, Spain b Dept. de Matemáticas, Universidad Católica de Chile, Spain

Received 3 April 2004; accepted 29 July 2004

Abstract In this paper we study the best constant, 1 (
) for the trace map from W 1,1 (
) into L1 (*
). We show that this constant is attained in BV (
) when 1 (
) < 1. Moreover, we prove that this constant can be obtained as limit when p  1 of the best constant of W 1,p (
) → Lp (*
). To perform the proofs we will look at Neumann problems involving the 1-Laplacian, 1 (u) = div(Du/|Du|). © 2004 Published by Elsevier Ltd. MSC: 35J65; 35B33 Keywords: Sobolev trace embedding; Functions of bounded variation

1. Introduction Let
be a bounded set in RN with Lipschitz continuous boundary *
. Of importance in the study of boundary value problems for differential operators in
are the Sobolev trace inequalities. In particular, W 1,1 (
) → L1 (*
) and hence the following inequality holds:

uL1 (*
)
u1,1 ,

∗ Corresponding author.

E-mail address: [email protected] (J.D. Rossi). 0362-546X/$ - see front matter © 2004 Published by Elsevier Ltd. doi:10.1016/j.na.2004.07.051

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F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

for all u ∈ W 1,1 (
). The best constant for this embedding is the largest  such that the above inequality holds, that is,


1 (
) = inf




|u| +




|∇u|: u ∈ W 1,1 (
),

*


 |u| = 1 .

(1)

Our main interest in this paper is to study the dependence of the best constant 1 (
) and extremals (functions where the constant is attained) on the domain. A related problem was studied by Demengel in [8] (see Remark 2). We remark that the existence of extremals is not trivial, due to the lack of compactness of the embedding. For 1 < p
N , let us consider the variational problem
 inf




|u|p +




|∇u|p : u ∈ W 1,p (
),

 *


 |u|p = 1 .

(2)

If we denote by p (
) the above infimum, we have that


p (
) = inf

  +
|∇u|p 1,p : u ≡ / 0 on *
, u ∈ W (
) p *
|u|

p
|u|

(3)

is the best constant for the trace map from W 1,p (
) into Lp (*
). Due to the compactness of the embedding W 1,p (
) → Lp (*
), it is well known (see for instance [10]) that problem (3) has a minimizer in W 1,p (
). These extremals are weak solutions of the following problem: 

p u = |u|p−2 u in
, * u = |u|p−2 u on *
, |∇u|p−2 *

(4)

where p u = div(|∇u|p−2 ∇u) is the p-Laplacian, */* is the outer unit normal derivative and if we use the normalization uLp (*
) = 1, one can check that  = p (
), see [11]. Our first result says that 1 (
) is the limit as p  1 of p (
) and provides a bound for 1 (
). Theorem 1. We have that lim p (
) = 1 (
)

p1

and


 |
| 1 (
)
min ,1 , P (
) where P (
) stands for the perimeter of
.

F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

1127

Therefore, it seems natural to search for an extremal for 1 (
) as the limit of extremals for p (
) when p  1. Formally, if we take limit as p  1 in Eq. (4), we get 

 u Du  = in
,  u := div  1 |Du| |u| (5)   Du ·  = 1 (
) u on *
. |Du| |u| Hence we will look at Neumann problems involving the 1-Laplacian, 1 (u)=div(Du/|Du|) in the context of bounded variation functions (the natural context for this type of problems). To our knowledge the results obtained here have independent interest. We shall say that
has the trace-property if there exists a vector field z
∈ L∞ (
, RN ), with z
∞
1 such that div(z
) ∈ L∞ (
) and [z
, ] = 1 (
)HN−1 -a.e.

on *
.

Our main result states that for any domain having the trace property, the best Sobolev trace constant, 1 (
), is attained by a function in L1 (
) whose derivatives in the sense of distributions are bounded measures on
, that is a function with bounded variation. Theorem 2. Let
be a bounded open set in RN with the trace-property. Then, there exists a nonnegative function of bounded variation which is a minimizer of the variational problem (1) and a solution of problem (5). We will see that every bounded domain
with 1 (
) < 1, has the trace-property. Hence we have proved that, if 1 (
) < 1 then there exists an extremal. Moreover, using results from [12], we can find examples of domains (a ball or an annulus) such that 1 (
) = 1 and verify the trace property (and therefore they have extremals). We also prove that every planar domain
with a point of curvature greater than 2 verifies 1 (
) < 1. Organization of the paper. In Section 2 we collect some preliminary results and prove Theorem 1. In Section 3 we deal with the Neumann problem for the equation div(z) = 1. Finally, in Section 4 we use these results to prove the main theorem, Theorem 2. Throughout this paper C and c denote constants that may change from one line to another.

2. Preliminary results. Proof of Theorem 1 Let us begin with some notation and definitions. Recall that a function u ∈ L1 (
) whose partial derivatives in the sense of distributions are measures with finite total variation in
is called a function of bounded variation. The class of such functions will be denoted by BV (
). Thus u ∈ BV (
) if and only if there are Radon measures 1 , . . . , N defined in
with finite total mass in
and   uD i  = −  di





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F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

for all  ∈ C0∞ (
), i = 1, . . . , N. Thus the gradient of u is a vector valued measure with finite total variation 
 u div :  ∈ C0∞ (
, RN ), |(x)|
1 for x ∈
. |Du|(
) = sup


The set of bounded variation functions, BV (
), is a Banach space endowed with the norm   |u| + |Du|. uBV (
) := u1 + |Du|(
) =





A measurable set E ⊂ RN is said to be of finite perimeter in
if E ∈ BV (
), in this case the perimeter of E in
is defined as P (E,
) := |D E |. We shall use the notation P (E) := P (E, RN ), and Pf (
) shall denote the set of all subset of
of finite perimeter. ∗ For a set of finite perimeter E one can define the essential boundary * E, which is countably N−1 measure, and compute the unit normal E (x) at HN−1 (N −1) rectificable with finite H ∗ almost all points x of * E, where HN−1 is the (N − 1) dimensional Hausdorff measure, ∗ and |D E | coincides with the restriction of HN−1 to * E. It is well known (see, [1,9] or [13]) that for a given function u ∈ BV (
) there exists a sequence un ∈ W 1,1 (
) such that un strict converges to u, i.e.,   |∇un | → |Du|. un → u in L1 (
) and





Moreover, there is a trace operator : BV (
) → L1 (*
) such that (u)L1 (*
)
CuBV (
)

∀u ∈ BV (
)

(6)

for some constant C depending only on
. The trace operator  is continuous between BV (
), endowed with the topology induced by the strict convergence, and L1 (*
). In the sequel we write (u) = u. From the above results, we get 
 
|u| +
|Du| : u≡ / 0 on *
, u ∈ BV (
) . 1 (
) = inf *
|u| Let us recall a generalized Green’s formula given in [5] (see also [4]). For 1
p < ∞, let Xp (
) = {z ∈ L∞ (
, RN ): div(z) ∈ Lp (
)}. 

If z ∈ Xp (
) and w ∈ BV (
) ∩ Lp (
) the functional (z, Dw): C0∞ (
) → R is defined by the formula   wz · ∇ . (z, Dw),  = − w  div(z) −


Then (z, Dw) is a Radon measure in
,   (z, Dw) = z · ∇w








F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

for all w ∈ W 1,1 (
) ∩ L∞ (
) and    (z, Dw)
|(z, Dw)|
z∞ |Dw| B

B

B

1129

(7)

for any Borel set B ⊆
. In [5], a weak trace on *
of the normal component of z ∈ Xp (
) is defined. Concretely, it is proved that there exists a linear operator : Xp (
) → L∞ (*
) such that  (z)∞
z∞ ,

(z)(x) = z(x) · (x) for all x ∈ *
if z ∈ C 1 (
, RN ). We shall denote (z)(x) by [z, ](x). Moreover, the following Green’s formula, relating  the function [z, ] and the measure (z, Dw), for z ∈ Xp (
) and w ∈ BV (
) ∩ Lp (
), is established:    w div(z) dx + (z, Dw) = [z, ]w dHN−1 . (8)





*


Now we can prove Theorem 1. Proof of Theorem 1. We have to prove that lim p (
) = 1 (
)

(9)

p1

and

 |
| ,1 . 1 (
)
min P (
)


(10)

Since p

p

p (
)uLp (*
)
u1,p

∀u ∈ W 1,p (
)

if we set

∗ := lim sup p (
) p1

we have

∗ uL1 (*
)
u1,1

∀u ∈ W 1,1 (
)

from where it follows that ∗
1 (
), that is lim sup p (
)
1 (
).

(11)

p1

Let vp be a minimizer of problem (3). Then, if up := ap vp , with ap satisfying 1/(1−p)

 1/(p−1)  p ap = |vp | |vp | , *


*


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F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

we get 

 *


|up | =

*


|up |p

and consequently 

p (
) =

 +
|∇up |p . *
|up |

p
|up |

(12)

Applying Hölder’s inequality and (12), we have p 1/p

  
|up| + |∇up | 1/p
|up| +
|∇up | 1 (
)

|
| *
|up | *
|up |  1/p

 p p |up | +
|∇up |  
|
|1/p 2(p−1)/p
*
|up | = |
|1/p 2(p−1)/p 

p (
)1/p 1/p . |u | p *




p/p



Hence 21−p p (
) 1 (
)  |
|p/p p

*


|up |

from where it follows that lim inf p (
) 1 (
).

(13)

p1

Now, (9) follows from (11) and (13). Taking u = 
, we obtain that   |
|
|
| +
|D 
| . = 1 (
)
P (
) |  | *

On the other hand, if
ε := {x ∈
: d(x, *
) < ε}, we have   |
ε | + P (
ε ,
)
|
ε | +
|D 
ε | . = 1 (
)
P (
) |  | *

ε Hence, taking ε → 0+ , it follows that 1 (
)
1. Therefore, (10) holds.



It is well known (see for instance [11]) that, for every p > 1, there exist an extremal for the embedding W 1,p (
) → Lp (*
) (this embedding is compact). This is a solution 0
up ∈ W 1,p (
) of the equation    p (
) = |up |p + |∇up |p , |up | = 1, (14)





*


F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

such that up > 0 and satisfies in the weak sense
p up := div(|∇up |p−2 ∇up ) = |up |p−2 up in
, |∇up |p−2 ∇up ·  = p (
)|up |p−2 up on *
.

1131

(15)

Now, by Young’s inequality we have 

    1 2 p p |up | + |∇up |
|up | + |∇up | +  |
| 1 (
)
p p



1 2 = p (
) +  |
|. p p Then, by (9) it follows that   1 (
) = lim |up | + |∇up |. p↓1


(16)




Moreover, by the compact embedding of BV (
) into L1 (
), we can suppose that up → u ∈ BV (
) in the L1 -norm and a.e. in


(17)

weakly∗ as measures.

(18)

and ∇up → Du Then, we have 



∃A := lim

p↓1


|∇up | = 1 (
) −




|u|.

(19)

On the other hand, by the lower-semi-continuity of the total variation respect the L1 -norm, we get    |Du|
lim inf |∇up | = 1 (
) − |u|. p↓1




Hence, 










|Du| +




|u|
1 (
).

(20)

We are interested in the problem: When is u a minimizer of the variational problem (1)? In these cases we would find an extremal for our minimization problem (1). Formally, if we take limit as p  1 in Eq. (15), we get    Du u = |u| 1 u := div |Du| in
, (21) Du u |Du| ·  = 1 (
) |u| on *
. Following [2,6] (see also [4]), we give the following definition of solution of problem (21).

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F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

Definition 1. A function u ∈ BV (
) is said to be a solution of problem (21) if there exists z ∈ X1 (
) with z∞
1,  ∈ L∞ (
) with ∞
1 and ∈ L∞ (*
) with  ∞
1 such that div(z) =  in D (
),

(22)

u = |u| a.e. in
and (z, Du) = |Du| as measures,

(23)

[z, ] = 1 (
)

and

u = |u|HN−1 -a.e. on *
.

(24)

Another problem we are interested in is: In what cases is u a solution of problem  (21)? Note that if v is a solution of problem (21) and *
|v|  = 0, then w := v/ *
|v| is a minimizer of the variational problem (1). Indeed, multiplying (22) by v and integrating by parts, we have      |v| = v = div(z)v = − (z, Dv) + [z, ]v



*
 |v|. = − |Dv| + 1 (
) *





From where it follows that   1 (
) = |w| + |Dw|.





Before we solve the above problems, let us study first the equation div(z) = 1 with Neumann boundary conditions.

3. The equation div(z) = 1 with Neumann boundary conditions Throughout this section we shall denote by
a bounded connected open set in RN , N  2, with Lipschitz continuous boundary *
. Given g ∈ L∞ (*
) with g∞ < 1, consider the functional Eg : L2 (
) →] − ∞, +∞] defined by


Eg (u) :=




|Du| − *
gu

+∞

if u ∈ BV (
) ∩ L2 (
), if u ∈ / BV (
).

In [6] it is proved that

*Eg = Ag , where Ag is the operator in L2 (
) defined by (u, v) ∈ Ag ⇐⇒ u ∈ BV (
) ∩ L2 (
), v ∈ L2 (
) and ∃z ∈ X2 (
), z∞
1

(25)

F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

1133

such that in D (
),

−div(z) = v

(z, Du) = |Du| as measures and [z, ] = g HN−1 -a.e. on *
. Let  : L2 (
) → R the operator defined by  1 (u) := (u(x) + 1)2 dx. 2
We have 0 = argmin(Eg + ) ⇐⇒ 0 ∈ *(Eg + )(0), ⇐⇒ −1 ∈ *Eg (0) ⇐⇒ (0, −1) ∈ Ag . Then, by (25), it follows that 0 = argmin(Eg + ) ⇐⇒ ∃z ∈ X2 (
), z∞
1, such that in D (
),

div(z) = 1

[z, ] = g HN−1 -a.e. on *
. On the other hand, 0 = argmin(Eg + ) if and only if    1 1 |Du| − gu + (u + 1)2  |
| ∀u ∈ BV (
) ∩ L2 (
). 2
2
*


(26)

Replacing u by εu (where ε > 0), expanding the L2 norm, dividing by ε, and letting ε → 0+ we get    gu
|Du| + u ∀u ∈ BV (
) ∩ L2 (
). (27) *








Consequently we have obtained the following result. Lemma 1. Let g ∈ L∞ (*
) with g∞ < 1. Then the following are equivalent: (i) there exists z ∈ X2 (
), z∞
1, such that div(z) = 1

in D (
),

[z, ] = g HN−1 -a.e. on *
. (ii) Eq. (27) holds.

1134

F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

Working as above but using the functional  1 (u) := (|u(x)| + 1)2 dx 2
instead of , we obtain the following result. Lemma 2. Let g ∈ L∞ (*
) with g∞ < 1. Then the following are equivalent: (i) there exist z ∈ X2 (
), z∞
1, and  ∈ L∞ (
), ∞
1 such that div(z) =  in D (
), [z, ] = g HN−1 -a.e. on *
. (ii) the following inequality holds    gu
|Du| + |u| ∀u ∈ BV (
) ∩ L2 (
). *








(28)

We state now the main result of this section. Theorem 3. Let
be a bounded connected open set in RN , N  2, with Lipschitz continuous boundary *
. Assume that |
|/P (
)
 < 1. Then the following are equivalent: (i) there exist z ∈ X2 (
), z∞
1, such that div(z) = 1

in D (
),

[z, ] = HN−1 -a.e. on *
, (ii)



 (iii)

 *


u









|Du| +

 *


u





u

∀u ∈ BV (
) ∩ L2 (
),

(29)




|Du| +




|u| ∀u ∈ BV (
) ∩ L2 (
),

(30)

(iv)


1 (
), (v)

(31)

∗ |E| − HN−1 (* E ∩ *
)
P (E,
)

∀E ∈ Pf (
).

(32)

Proof. By Lemma 1 with g = , (i) and (ii) are equivalent. Obviously, (iii) and (iv) are equivalent, and (ii) implies (iii). Let us see that (iii) implies (v). Taking u = E in (30), with E ⊂
a set of finite perimeter, it follows that ∗

−[|E| − HN−1 (* E ∩ *
)]
P (E,
)

F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

1135

and taking u = 
\E , we get ∗

HN−1 (* (
\E) ∩ *
)
P (
\E,
) + |
\E|. ∗

∗

Now, since P (
\E,
)=P (E,
) and HN−1 (* (
\E)∩*
)=P (
)−HN−1 (* E∩*
), we have ∗

[P (
) − HN−1 (* E ∩ *
)]
P (E,
) + |
| − |E|. Then, since  |
|/P (
), we obtain ∗

|E| − HN−1 (* E ∩ *
)
P (E,
) and (v) holds. Finally, let us see that (v) implies (ii). Given u ∈ BV (
), since for all x ∈
,  0  +∞ {u>t} (x) dt − {u
t} (x) dt, u(x) = −∞

0

using (32) and the coarea formula we get  +∞   0   u(x) dx = {u>t} (x) dx dt − {u
t} (x) dx dt

0 −∞
 +∞  0 = |{u > t}| dt − |{u
t}| dt 0 −∞  +∞ ∗  (HN−1 (* {u > t} ∩ *
) − P ({u > t},
)) dt 0  0 ∗ (HN−1 (* {u
t} ∩ *
) + P ({u
t},
)) dt − −∞   +∞ = u dHN−1 − P ({u > t},
) dt *
−∞ u dHN−1 − |Du| = *


and (29) holds.






Taking  = |
|/P (
) in the above theorem we obtain the following result. Corollary 1. Let
be a bounded connected open set in RN , N  2, with Lipschitz continuous boundary *
. If |
|/P (
) < 1, then the following are equivalent: (i) ∃z ∈ X2 (
), z∞
1, such that div(z) = 1 in D (
), |
| [z, ] = HN−1 -a.e. on *
, P (
)

(33)

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F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

(ii)

1 (
) =

|
| , P (
)

(iii) |E| − |
| HN−1 (*∗ E ∩ *
)
P (E,
) P (
)

∀E ∈ Pf (
).

We do not know if the assumption
connected in Corollary 1 is necessary. So, a natural question is the following: Is there a nonconnected open bounded set
such that |
|/P (
) < 1, verifying (33) and 1 (
) < |
|/P (
)? There are open sets
for which (33) holds and |
|/P (
) = 1, as the following examples show. Example 1. Let
= BR (0) ⊂ RN the ball in RN centered in 0 of radius R. Then, if z(x) := x/N, we have div(z) = 1

in D (
)

and [z, ] =

R |
| HN−1 -a.e. on *
. = P (
) N

Moreover, z∞ =

R |
|
1 ⇐⇒
1. N P (
)

Example 2. Let
= BR (0)\Br (0) ⊂ RN the annulus in RN centered in 0 of radius R and r. Then, it is easy to see that if   x r N−1 R N−1 z(x) := (R N−1 + r N−1 ) − (R + r) N N−1 x N (R + r N−1 ) we have div(z) = 1

in D (
)

and [z, ] =

|
| HN−1 -a.e. on *
. P (
)

Moreover, z∞
1 ⇐⇒

|
|
1. P (
)

F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

1137

Remark 1. Motron in [12] proves that if
= BR (0) is the ball in RN centered in 0 of radius R or
= BR (0)\Br (0) the annulus in RN centered in 0 of radius R and r, then    P (
) |u|
|u| + |∇u| ∀u ∈ W 1,1 (
) (34) |
|

*
and equality holds in (34) if and only if u is constant. From (10) and (34), it follows that if
= BR (0) ⊂ RN or
= BR (0)\Br (0) ⊂ RN , then  |
| |
| 
1, if  P (
) P (
) 1 (
) = |
|  1 if  1. P (
) Moreover, if |
|/P (
)
1, then u = (1/P (
))
is a minimizer of the variational problem (1), being the only minimizer in the case |
|/P (
)=1, and if |
|/P (
) > 1, the variational problem (1) does not have minimizer. In the following example we show that there exists bounded connected open sets
, with |
|/P (
) < 1, for which 1 (
) < |
|/P (
). Example 3. For > 0 and 0 < 
/2, let


, := B1 (0) ∪ (B2+ (0)\B2 (0))  y   ∪ (x, y) ∈ R+ × R+ : x 2 + y 2
2, arctg 2 +



6 − .

Now, if we take u := ]0,1[×]0, [ , we have   +
|u| 2
|Du|  = 1 (
)
< 1 ⇐⇒ 0 < < 2. 2 + |u| *
Therefore, for instance, if = 1 and k = 5, we have |
|/P (
) > 1 and 1 (
) < 1.

F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

1141

4. Proof of Theorem 2 Let
be a bounded connected open set in RN , N  2, with Lipschitz continuous boundary *
. By Lemma 2, if we assume that 1 (
) < 1, there exist z ∈ X2 (
), z∞
1, and div(z)∞
1 such that [z, ] = 1 (
)HN−1 -a.e. on *
. We recall the following definition. Definition 2. Let
be a bounded open set in RN with Lipschitz continuous boundary *
. We shall say that
has the trace-property if there exists a vector field z
∈ L∞ (
, RN ), with z
∞
1 such that div(z
) ∈ L∞ (
) and [z
, ] = 1 (
)HN−1 -a.e.on *
. By the above, we have that every bounded connected open set
in RN , N  2, with Lipschitz continuous boundary and 1 (
) < 1, has the trace-property. Also, as a consequence of Examples 1, 2, and Remark 1, we have that if
=BR (0) ⊂ RN or
=BR (0)\Br (0) ⊂ RN , and |
|/P (
)
1, then
has the trace-property. Therefore there exists
with 1 (
) = 1 satisfying the trace-property. Let us present some examples of planar domains that verify 1 (
) < 1. Example 7. Assume that
⊂ R2 is a bounded open set such that there exists some point, x0 ∈ *
(we may assume x0 = 0), with curvature of the boundary at that point greater than 2, we will show that in this case 1 (
) < 1. So, let us assume that locally near the origin
can be described as
∩ Br (0) = {(x, y): y > ax 2 }. As we are assuming that the curvature at the origin is greater than 2 we have a > 1. Let us consider the function uε = 
∩{y k}. As above, there exists some gk ∈ L1 (
, RN ) such that weakly in L1 (
, RN ) as p  1.

|∇up |p−2 ∇up Bp,k → gk Now, since





|Bp,k | =

(45)




Bp,k (x) dx


|∇up (x)|p p (
) dx
p p k k


L∞ (
)

with  ∞
1, we have for any ∈   (p−1)/p p
|∇up |p−2 ∇up ·  |∇up | |Bp,k |1/p Bp,k






p (
)(p−1)/p



p (
) kp

1/p

=

p (
) . k

Letting p  1, we get that  1 (
) for every k > 0. |gk |
k


(46)

On the other hand, since we have ||∇up |p−2 ∇up 
\Bp,k |
k p−1

for any p > 1,

letting p  1, we obtain that there exists some fk ∈ L1 (
, RN ) with fk ∞
1 such that |∇up |p−2 ∇up 
\Bp,k → fk

weakly in L1 (
, RN ) as p  1.

(47)

Hence, for any k > 0, we may write z = fk + gk , with fk ∞
1 and gk satisfying (46). From where (44) follows. Since up → u a.e. in
, by (41) it follows that

u = |u| a.e. in
and ∞
1. On the other hand, given a measurable subset E ⊂ *
, by Hölder’s inequality we have

 p−1  p−1 up dHN−1
up dHN−1 HN−1 (E)2−p
HN−1 (E)2−p *


E

p−1

from where it follows that {up : 1 < p
2} is a weakly relatively compact subset of L1 (*
). Hence, we can assume that there exists ∈ L1 (*
), such that p−1

up

→

weakly in L1 (*
).

(48)

1144

F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

Moreover, by (48), (14) and applying Fatou’s Lemma, it is easy to see that  ∞
1.

(49)

On the other hand, by (38) and (48), we get

u = |u|HN−1 -a.e. on *
.

(50)

Now, since up is a weak solution of (15), having in mind (41), (9) and (48), if w ∈ W 1,1 (
)∩ C(
) ∩ L∞ (
), we have     z · ∇w = w + z · ∇w div(z)w + z, w*
:=



 p−1 p−2 = lim up w + |∇up | ∇up · ∇w p1
 
 p−1 p−2 = lim up w − div(|∇up | ∇up )w + |∇up |p−2 ∇up · w p1

*
  p−1 = lim p (
) up w = 1 (
)

w. p1

*


*


Thus, having in mind the definition of the weak trace on *
of the normal component of z given in [5], we get [z, ] = 1 (
) .

(51)

Finally, since    |Du| = 1 (
) − |u| = 1 (
) − u

 
 = 1 (
) − div(z)u = 1 (
) + (z, Du) − [z, ]u *


 = 1 (
) + (z, Du) − 1 (
)

u = (z, Du),


we have (z, Du) = |Du| as measures.

*







Remark 2. Let us remark that as a consequence of Theorem 1 in [8] it is obtained the above theorem in the particular case that
is a bounded open subset of RN , whose boundary *
is at least piecewise C2 and 1 (
) < 1. Remark 3. Note that in the above theorem we have proved that if u is the limit as p  1 of the minimizers up of the variational problem
    p p 1,p inf |u| + |∇u| : u ∈ W (
), |u| = 1, (52) 







*


then, if *
|u| = 1, we have that u is a minimizer of the variational problem (1) and a solution of problem (21).

F. Andreu et al. / Nonlinear Analysis 59 (2004) 1125 – 1145

1145

Acknowledgements The first and second authors have been partially supported EC through the RTN Programme Nonlinear Partial Differential Equations Describing Front propagation and other Singular Phenomena, HPRN-CT-2002-00274, and by PNPGC project, reference BFM200201145. The third author acknowledges partial support by ANPCyT PICT No. 03-05009, CONICET PIP0660/98, UBA X431 and Fundación Antorchas. References [1] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, in: Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. [2] F. Andreu, C. Ballester, V. Caselles, J.M. Mazón, Minimizing total variational flow, Differential Integral Equations 14 (2001) 321–360. [4] F. Andreu, V. Caselles, J.M. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progress in Mathematics, vol. 223, Birkhauser, Basel, 2004. [5] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. IV (135) (1983) 293–318. Du ) = u, preprint. [6] G. Ballettini, V. Caselles, M. Novaga, Explicit solutions of the eigenvalue problem −div( |Du| [8] F. Demengel, On some nonlinear equation involving the 1-Laplacian and trace map inequalities, Nonlinear Anal. TMA 48 (2002) 1151–1163 (1983) 293–318. [9] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, in: Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [10] J. Fernández Bonder, J.D. Rossi, Existence results for the p-Laplacian with nonlinear boundary conditions, J. Math. Anal. Appl. 263 (2001) 195–223. [11] S. Martínez, J. Rossi, Isolation and simplicity for the first eigenvalue of the p-Laplacian with a nonlinear boundary condition, Abstract Appl. Anal. 7 (2002) 287–293. [12] M. Motron, Around the best constants for the Sobolev trace map from W 1,1 (
) into L1 (*
), Asymptotic Anal. 29 (2002) 69–90. [13] W.P. Ziemer, Weakly Differentiable Functions, GTM 120, Springer, Berlin, 1989.