Konstantinos Gkikas
email: kgkikas@dim.uchile.cl

Laurent Véron
email: veronl@univ-tours.fr

The spherical p-harmonic eigenvalue problem in non-smooth domains

Keywords: 2010 Mathematics Subject Classification. 35J72, 35J92 p-Laplacian operator, polar sets, Harnack inequality, boundary Harnack inequality, p-Martin boundary

come

The spherical p-harmonic eigenvalue problem in non-smooth domains

with β 0 = N -p p-1 and were div and ∇ denote the divergence operator and the covariant gradient on S N -1 endowed with the metric induced by its isometric inbedding into R N . Separable solutions play a key role for describing the boundary behaviour and the singularities of solutions of a large variety of quasilinear equations. When N = 2 the equation is completely integrable and has been solved by Kroll in the regular case β < 0 and Kichenassamy and Véron in the the singular case β > 0. In higher dimension, Tolksdorff [START_REF] Tolksdorf | On the Dirichlet problem for quasilinear equations in domains with conical boundary points[END_REF] proved the following: Theorem A If S is a smooth spherical domain, there exist two couples (β S , ω S ) and (β S , ω S ) where β S > 0 and β S < 0, ω S and ω S are positive C 2 (S)-functions vanishing on ∂S which solve (1.1) with (β, ω) = (β S , ω S ) or (β, ω) = (β S , ω S ). Furthermore β S and β S are unique, and ω S and ω S are unique up to an homothety.

A more general and transparent proof has been obtained by Porretta and Véron [START_REF] Porretta | Separable p-harmonic functions in a cone and related quasilinear equations on manifolds[END_REF], but always in the case of a smooth spherical domain. The aim of this article is to extend Theorem A to a general spherical domain. If we consider an increasing sequence of smooth domains {S k } such that S k ⊂ S k ⊂ S k+1 and ∪ k S k = S we prove the following: Theorem B Assume that S c is not polar. Then the sequence of the β S k > 0 from Theorem A is decreasing and converges to β S > 0. There exists ω S ∈ W 1,p 0 (S) ∩ L ∞ (S) weak solution of (1.1) with β = β S . Furthermore β S > 0 is the largest exponent β such that (1.1) admits a positive solution ω S ∈ W 1,p 0 (S). Under a mild assumption on S it is possible to approximate it by a decreasing sequence of smooth domains S k such that S k ⊂ S k ⊂ S k-1 and ∩ k S k = S Theorem C Assume that S = o S. Then the sequence β S k > 0 is increasing and converges to βS > 0 and there exists ωS ∈ W 1,p 0 (S) ∩ L ∞ (S) weak solution of (1.1) with β = βS . Furthermore βS is the smallest exponent β such that (1.1) admits a positive solution ω S ∈ W 1,p 0 (S). We prove the uniqueness of the exponent β, under a Lipschitz assumption on S. Theorem D Assume that S is a Lipschitz domain, then β S = βS and if ω and ω are two positive solutions of (1.1) in W 1,p 0 (S), there exists a constant c > 0 such that c -1 ω ≤ ω ≤ cω . The proof of Theorem C is based upon a sharp form of boundary Harnack inequality proved in [START_REF] Lewis | Boundary behavior and the Martin boundary problem for p-harmonic functions in Lipschitz domains[END_REF],

ln ω(σ1) ω (σ1) -ln ω(σ2) ω (σ2) ≤ c 1 |σ 1 -σ 2 | α ∀ σ 1 , σ 2 ∈ S, (1.2)
for some c 1 = c 1 (N, p, S) > 0 and α ∈ (0, 1). Actually we have a stronger result, much more delicate to obtain.

Theorem E Let S be a Lipschitz subdomain of S N -1 . Then two positive solutions of (1.1) in W 1,p 0 (S) are proportional.

The proof is based upon a non trivial adaptation of a series of deep results of Lewis and Nyström [START_REF] Lewis | Boundary behavior and the Martin boundary problem for p-harmonic functions in Lipschitz domains[END_REF] concerning the p-Martin boundary of domains.

Acknowledgements This article has been prepared with the support of the collaboration programs ECOS C14E08.

Existence

Estimates

Through this article we assume that S c is not polar, or equivalently that it has positive c S N -1

1,p -capacity. Lemma 2.1. Assume p > 1. Then any solution ω ∈ W 1,p 0 (S) of (1.1) satisfies ω C γ (S) ≤ c 1 ω L p (S) , (2.1
)

if p > N -1 where γ = 1 -N -1 p if p > N -1 and ω L ∞ (S) ≤ c 1 ω L p (S) , (2.2
)

if 1 < p ≤ N -1, where c 1 > 0 depends on p, N , β.
Proof. Multiplying the equation by ω and using Hölder's inequality, we derive

(i) S β 2 ω 2 + |∇ ω| 2 p 2 dS ≤ (β(pβ -(p -1)β 0 )) p 2 S |ω| p dS if p ≥ 2, (ii) S β 2 ω 2 + |∇ ω| 2 p 2 dS ≤ β p-1 (pβ -(p -1)β 0 ) S |ω| p dS if 1 < p < 2.
(2.3)

Notice that these inequalities hold for all p > 1. If p > N -1 (2.1) follows by Morrey'inequality. Here after we assume

1 < p ≤ N -1. Let α ≥ 1 and k > 0. Then ζ = min{|ω| , k} α-1 ω is an admissible test function, hence 1-If p ≥ 2, S β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω.∇ ζ dS = (p -1)β(β -β 0 ) S β 2 ω 2 + |∇ ω| 2 p-2 2 ωζdS ≤ c 2 S |∇ ω| p-2 ω 2 min{|ω| , k} α-1 dS + c 2 β p S |ω| p min{|ω| , k} α-1 dS ≤ c 2 S |ω| p min{|ω| , k} α-1 dS p-2 p S |∇ ω| p min{|ω| , k} α-1 dS 2 p + c 2 β p S |ω| p min{|ω| , k} α-1 dS, (2.4)
where

c 2 = c 2 (N, p, β) > 0. Since S β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω.∇ ζ dS ≥ c 3 (p) S |∇ ω| p min{|ω| , k} α-1 dS, it implies that there exists c 4 = c 4 (N, p, β) such that S |∇ ω| p min{|ω| , k} α-1 dS ≤ c 4 S |ω| p min{|ω| , k} α-1 dS, (2.5)
which yields

S |∇ j(ω)| p dS ≤ c 4 S |j(ω)| p dS, (2.6)
where j(ω) = min{|ω| , k}

α-1 p ω. 2-If 1 < p < 2, then S β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω.∇ ζ dS = S β 2 ω 2 + |∇ ω| 2 p-2 2
|∇ ω| 2 min{|ω| , k} α-1 dS

+ (α -1) S∩{|ω|<k} β 2 ω 2 + |∇ ω| 2 p-2 2 |∇ ω| 2 |ω| α-1 dS. (2.7) Since S β 2 ω 2 + |∇ ω| 2 p-2 2 |∇ ω| 2 min{|ω| , k} α-1 dS = S β 2 ω 2 + |∇ ω| 2 p 2 min{|ω| , k} α-1 dS -β 2 S β 2 ω 2 + |∇ ω| 2 p-2 2 min{|ω| , k} α-1 ω 2 dS ≥ S |∇ ω| p min{|ω| , k} α-1 dS -β 2 S β 2 ω 2 + |∇ ω| 2 p-2 2
min{|ω| , k} α-1 ω 2 dS, we derive (2.9)

S |∇ ω| p min{|ω| , k} α-1 dS ≤ β p-1 (pβ -(p -1)β 0 ) S |ω| p min{|ω| , k} α-1 dS, (2.8
If p < N -1 we derive from Sobolev inequality and putting q = α -1 + p and s

= N -1 N -1-p > 1 S |ω| sq dS 1 s ≤ c 5 S |ω| q dS, (2.10
1 t ≤ c 5 S |ω| q dS, (2.11)
with t = m(N -1) p(N -1-m) = m N -1-m .
The proof follows again by Moser's iterative scheme. Proposition 2.2. Let S 1 and S 2 be two subdomains of S N -1 such that S 1 ⊂ S 1 ⊂ S 2 and S 2 not polar. Let β j > 0, j=1,2, such that there exist positive solutions ω j ∈ W 1,p 0 (S j ) solutions of

-div β 2 j ω 2 j + |∇ ω j | 2 p-2 2 ∇ ω j = (p -1)β j (β j -β 0 ) β 2 j ω 2 j + |∇ ω j | 2 p-2 2 ω j in S j ω j = 0 in ∂S j .
(2.12)

Then β 1 ≥ β 2 .
Proof. Set u j (r, σ) = r -βj ω j (σ), C Sj = (0, ∞) × S J and assume β 1 < β 2 . By Harnack inequality ω 2 ≥ c > 0 on S 1 , thus u 2 (r, σ) ≥ cr -β2 a.e. in C S1 .

For > 0 there exist r > 0 such that

u 2 (x) ≥ u 1 (x) ∀ x ∈ C S1 ∩ B r .
Let δ > 0, there exists R δ > 0 such that

u 1 (x) ≤ δ ∀ x ∈ C S1 ∩ B c R δ . Hence ζ = (u 1 -u 2 -δ) + ∈ W 1,p 0 (Q r ,R δ S1 ), where Q r ,R δ S1 = {x ∈ C S1 : r < |x| < R δ }. This implies 0 = Q r ,R δ S 1 |∇u 1 | p-2 ∇u 1 -|∇( u 1 )| p-2 ∇( u 1 ).∇ζ dx = Q r ,R δ S 1 ∩{u1-u2≥δ} |∇u 1 | p-2 ∇u 1 -|∇( u 1 )| p-2 ∇( u 1 ).∇(u 1 -u 2 ) dx. Therefore ∇(u 1 -u 2 -δ) + = 0 a.e. in Q r ,Rδ S1 , which leads to u 1 -u 2 ≤ δ in the same set. Letting δ → 0 yields R δ → ∞, thus we obtain u 1 ≤ u 2 in C S1 ∩ B c r hence u 1 ≤ 0 in C S1 , contradiction.

Approximations from inside

Proof of Theorem B. Let {S k } be an increasing sequence of smooth domains such that S k ⊂ S k ⊂ S k+1 . We denote by {(β S k , ω k )} the corresponding sequence of solutions of (1.1) with β = β S k and ω = ω k . The sequence {β S k } is uniquely determined by [START_REF] Tolksdorf | On the Dirichlet problem for quasilinear equations in domains with conical boundary points[END_REF], it admits a limit β := β S , and the ω k are the unique positive solutions such that

S k |ω k | dS = 1. If p ≥ 2, we have S k |∇ ω k | p dS ≤ S k β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 |∇ ω k | 2 dS = (p -1)β S k (β S k -β 0 ) S k β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ω 2 k dS ≤ 2 (p-4) + 2 (p -1)β S k (β S k -β 0 ) S k β p-2 S k ω p k + |∇ ω k | p-2 ω 2 k dS ≤ c 7 (N, p, β S k ) S k ω p k dS + 1 2 S k |∇ ω k | p dS. Since β S k ≤ β 1 , we derive S k |∇ ω k | p dS ≤ c 8 , (2.13)
from the normalization assumption with c 8 = 2c 7 (N, p, β 1 ). If 1 < p < 2, we have

S k |∇ ω k | p dS ≤ S k β 2 S k ω 2 k + |∇ ω k | 2 p 2 dS ≤ β S k (pβ S k + (p -1)β 0 ) S k β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ω 2 k dS ≤ β p-1 k (pβ S k + (p -1)β 0 ) S k ω p k dS,
and we obtain (2.13) with c 8 = β p-1

1 (pβ 1 + (p -1)β 0 ). Next we extend ω k by 0 in S c k .
Then there exists ω ∈ W 1,p 0 (S) such that ω k ω weakly in W 1,p 0 (S), up to subsequence that we still denote {ω k }, and

ω k → ω in L p (S). Step 1: We claim that ∇ ω k converges to ∇ ω locally in L p (S). Let a ∈ S and r > 0 such that B 4r (a) ⊂ S. Then for k ≥ k 0 , B 2r (a) ⊂ S k . Let ζ ∈ C ∞ 0 (B 2r (a)) such that 0 ≤ ζ ≤ 1, ζ = 1 in B r (a). For test function we choose η k = ζ(ω -ω k ), then S k β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ η k dS = (p-1)β S k (β S k -β 0 ) S k β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ω k η k dS.
By the above inequality, we have

B2r(a) β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ η k dS = B2r(a) β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω.∇ η k dS -(p -1)β S k (β S k -β 0 ) S k β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ω k η k dS.
Using the weak convergence of the gradient, we have lim k→∞ B2r(a)

β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω.∇ η k dS = 0.
Since ω k is uniformly bounded in W 1,p 0 (S) and ω k → ω in L p (S), we have

lim k→∞ B2r(a) β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ω k η k dS = 0, and
lim k→∞ B2r(a) (ω -ω k ) β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ ζ dS = 0.
Combining the above relations we infer

lim k→∞ B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ (ω -ω k ) dS = 0.
(2.14) Next we write

B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ (ω -ω k ) dS = 1 2 B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 + β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 |∇ (ω -ω k )| 2 dS + 1 2 B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 × |∇ ω| 2 + β 2 ω 2 -β 2 S k ω 2 k -|∇ ω k | 2 dS - 1 2 B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 β 2 ω 2 -β 2 S k ω 2 k dS. (2.15)
If p ≥ 2, we have from (2.4),

B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ (ω -ω k ) dS ≥ 1 2 B2r(a) ζ |∇ ω| p-2 + |∇ ω k | p-2 |∇ (ω -ω k )| 2 dS - 1 2 B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 β 2 ω 2 -β 2 S k ω 2 k dS ≥ min{2 -1 , 2 2-p } B2r(a) ζ |∇ (ω -ω k )| p dS - 1 2 B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 β 2 ω 2 -β 2 S k ω 2 k dS. (2.16) Since ω k → ω in L p (S), β S k → β and ω k , ω are uniformly bounded in W 1,p 0 (S) , we derive B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 β 2 ω 2 -β 2 S k ω 2 k dS → 0 as k → ∞.
|∇ (ω -ω k )| p dS = 0. (2.17) If 1 < p < 2, then B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ (ω -ω k ) dS = B2r(a) ζ β 2 S k ω 2 k + |∇ ω| 2 p-2 2 ∇ ω -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ (ω -ω k ) dS + B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 -β 2 S k ω 2 k + |∇ ω| 2 p-2 2 ∇ ω.∇ (ω -ω k ) dS.
(2.18) Up to extracting a subsequence, we have that ω k → ω a.e. in S and that there exists

Φ ∈ L 1 (S) such that |ω k | p + |ω| p ≤ Φ a.e. in S and ∀ k ≥ 1. (2.19) Since β 2 S k ω 2 k + |∇ ω| 2 p-2 2 |∇ω| ≤ β 2 S k ω 2 k + |∇ ω| 2 p-1 2 ≤ β p-1 S k ω p-1 k + |∇ ω| p-1 , and
β 2 ω 2 + |∇ ω| 2 p-2 2 |∇ω| ≤ β p-1 ω p-1 + |∇ ω| p-1 ,
we derive that

β 2 ω 2 + |∇ ω| 2 p-2 2 -β 2 S k ω 2 k + |∇ ω| 2 p-2 2 |∇ ω| ≤ 2 β p-1 Φ p-1 + |∇ ω| p-1 ,
which implies that

ζ β 2 ω 2 + |∇ ω| 2 p-2 2 -β 2 S k ω 2 k + |∇ ω| 2 p-2 2 ∇ ω → 0 in L p (S)
where p is the conjugate of p, and finally

B2r(a) ζ β 2 ω 2 + |∇ ω| 2 p-2 2 -β 2 S k ω 2 k + |∇ ω| 2 p-2 2 ∇ ω.∇ (ω -ω k ) dS → 0 as k → ∞.
(2.20) For the last term on the right-hand side of (2.18), we have, for

γ ∈ R + and A, B ∈ R N , γ + |B| 2 p-2 2 B -γ + |A| 2 p-2 2 A = 1 0 d dt γ + |tB + (1 -t)A| 2 p-2 2 (tB + (1 -t)A) dt = 1 0 γ + |tB + (1 -t)A| 2 p-2 2
dt (B -A)

+ (p -2) 1 0 γ + |tB + (1 -t)A| 2 p-4 2 tB + (1 -t)A . B -A (tB + (1 -t)A)dt.
This implies

γ + |B| 2 p-2 2 B -γ + |A| 2 p-2 2 A . B -A = 1 0 γ + |tB + (1 -t)A| 2 p-2 2 dt |B -A| 2 + (p -2) 1 0 γ + |tB + (1 -t)A| 2 p-4 2 tB + (1 -t)A . B -A 2 dt.
We observe that

1 0 γ + |tB + (1 -t)A| 2 p-4 2 tB + (1 -t)A . B -A 2 dt ≤ |B -A| 2 1 0 γ + |tB + (1 -t)A| 2 p-2 2
dt, and since 1 < p < 2, we finally obtain

γ + |B| 2 p-2 2 B -γ + |A| 2 p-2 2 A . B -A ≥ (p -1) 1 0 γ + |tB + (1 -t)A| 2 p-2 2 dt |B -A| 2 ≥ (p -1) |B -A| 2 γ + 1 + |B| 2 + |A| 2 p-2 2
.

(2.21)

We plug this estimate into (2.18)

with γ = β 2 k ω 2 k , A = ∇ ω and B = ∇ ω k , then B2r(a) ζ β 2 S k ω 2 k + |∇ ω| 2 p-2 2 ∇ ω -β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ (ω -ω k ) dS ≥ B2r(a) ζ |∇ (ω -ω k )| 2 β 2 k ω 2 k + 1 + |∇ ω k | 2 + |∇ ω| 2 p-2 2
dS.

(2.22)

Set φ(.) = β 2 k ω 2 k + 1 + |∇ ω k | 2 + |∇ ω| 2 , then Br(a) |∇ ω -∇ ω k | p dS = Br(a) |∇ ω -∇ ω k | p φ p(p-2) 4 φ -p(p-2) 4 dS ≤ Br(a) |∇ ω -∇ ω k | 2 φ p-2 2 dS p 2
Br(a)

φ p 2 dS 2-p 2 .
(2.23)

Jointly with (2.14) and (2.22) we conclude that (2.17).

Step 1 follows by a standard covering argument.

Step 2: We claim that ω k converges to ω in W 1,p 0 (S). Up to a subsequence that we denote again by {k}, we can assume that ω k → ω and ∇ ω k → ∇ ω a.e. in S. Let ζ ∈ C ∞ 0 (S), then there exists k ∈ N such that the support

K of ζ is a compact subset of S k for all k ≥ k . If 1 < p < 2, β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 |∇ ω k | ≤ |∇ ω k | p-1 ,
which bounded in L p (K), then uniformly integrable in K and by Vitali's convergence theorem

β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k → β 2 ω 2 + |∇ ω| 2 p-2 2 ∇ ω, in L 1 loc (S). Similarly β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ω k → β 2 ω 2 + |∇ ω| 2 p-2 2 ω, in L 1 loc (S). If p ≥ 2 β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 |∇ ω k | ≤ c |ω k | p-1 + |∇ ω k | p-1 ,
and we conclude again by Vitali's convergence theorem that the previous convergences hold. Since

S k β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ∇ ω k .∇ ζ dS = (p-1)β S k (β S k -β 0 ) S k β 2 S k ω 2 k + |∇ ω k | 2 p-2 2 ω k ζdS
we conclude that ω is a weak solution of (1.1) with β = β S .

Approximations from outside

Proof of Theorem C. Since S c has a non-empty interior, the existence of a sequence {ω k } corresponding to solutions of (1.1) in S k with β = β S k is the consequence of [START_REF] Porretta | Separable p-harmonic functions in a cone and related quasilinear equations on manifolds[END_REF]. The fact that {β S k } is increasing follows from Proposition 2.2. We denote by β := βS its limit, and it is smaller or equal to β S . Estimates (2.4) are valid with S k , ω k and β S k instead of S, ω and β. If we extend ω k by 0 in S c k these estimates are valid with S N -1 instead of S k . Then up to a subsequence the exists ω ∈ W 1,p (S N -1 ) and a subsequence stil denoted by {k} such that ω k ω weakly in W 1,p (S N -1 ), strongly in L p (S N -1 ) and a.e. in S N -1 . Furthermore, as in the proof of Theorem A, for any compact set K ⊂ S, ∇ ω k → ∇ ω in L p (K). This is sufficient to assert that ω is a weak solution of

-div β2 ω 2 + |∇ ω | 2 p-2 2 ∇ ω = (p -1) β( β -β 0 ) β2 ω 2 + |∇ ω | 2 p-2 2 ω in S.
Moreover ω S k belongs to W 3 Uniqueness

Uniqueness of exponent β

Proof of Theorem D. If S is Lipschitz, C S is also Lipschitz. We fix z ∈ S ≈ S N -1 ∩ ∂C S and we apply [START_REF] Lewis | Boundary behavior and the Martin boundary problem for p-harmonic functions in Lipschitz domains[END_REF]Th 2] in G z = C S ∩ B 1 2 (z) to two separable p-harmonic functions u(r, σ) = r -β ω(σ) and u (r, σ) = r -β ω (σ). There exist γ ∈ (0, 1 2 ), c 10 > 0 and α ∈ (0, 1) such that

ln u(y 1 ) u (y 1 ) -ln u(y 2 ) u (y 2 ) ≤ c 10 |y 1 -y 2 | α ∀ y 1 , y 2 ∈ C S ∩ B γ (z). (3.24) Assume |y 1 | = |y 2 | = 1, then ln ω(y 1 ) ω (y 1 ) -ln ω(y 2 ) ω (y 2 ) ≤ c 10 |y 1 -y 2 | α ∀ y 1 , y 2 ∈ S ∩ B γ (z). (3.25)
We denote by (x, y) the geodesic distance on S N -1 and by (x, K) the geodesic distance from a point x ∈ S N -1 to a subset K. Since the set S γ = {σ ∈ S : (σ, ∂S) ≤ γ 2 } can be covered by a finite number of balls with center on ∂S, we infer that Assume now that there exist two exponents β > β > 0 such that r -β ω(.) and r -β ω (.) are pharmonic and positive in the cone C S and vanishes on ∂C S . Put θ = β β , η = ω θ and

ln ω(y 1 ) ω (y 1 ) -ln ω(y 2 ) ω (y 2 ) ≤ c 11 ∀ y 1 , y 2 ∈ S γ . ( 3
T (η) = -div β 2 η 2 + |∇ η| 2 p-2 2 ∇ η -(p -1)β(β -β 0 ) β 2 η 2 + |∇ η| 2 p-2 2 η, then
T (η) = -θ p-2 β 2 ω 2 + |∇ ω | 2 p-2 2 (β -β )ω 2 + (p -1)θ(θ -1) |∇ ω | 2 ≤ 0.
Up to multiplying ω by λ, we can assume that η ≤ ω and that the graphs of η and ω are tangent in S. Since ω ≤ cω, η = o(ω) near ∂S. Hence there exists σ 0 ∈ S such that ω(σ 0 ) = η(σ 0 ) and the coincidence set of η and ω is a compact subset of S. We put w = ω -η, since ∇ω(σ 0 ) = ∇η(σ 0 ) we proceed as in [14, Th 4.1] (see also [START_REF] Friedman | Singular Solutions of Some Quasilinear Elliptic Equations[END_REF] in the flat case) and derive that w satisfies, in a system of local coordinates (σ 1 , ..., σ N -1 ) near σ 0 ,

Lw := - ,j ∂ ∂σ A j, ∂w ∂σ j + j C j ∂w ∂σ + Cw ≥ 0,
where the matrix (A j, ) is smooth, symmetric and positive near σ 0 and the C j and C are bounded. Hence w is locally zero. By a standard argument of connectedness, this implies that the zero set of w must be empty, contradiction. Hence β = β .

Uniqueness of eigenfunction

The proof is based upon a delicate adaptation of the characterisation of the p-Martin boundary obtained in [START_REF] Lewis | Boundary behavior and the Martin boundary problem for p-harmonic functions in Lipschitz domains[END_REF], but we first give a proof in the convex case.

The convex case

Theorem 3.1. Assume S is a convex spherical subdomain. Then two positive solutions of (1.1) are proportional.

Proof. We recall that a domain S is (geodesically) convex if a minimal geodesic joining two points of

S is contained in S. If S ⊂ S N -1 is convex, the cone C S is convex too. Since S is convex,
-∆ p u = 0 in C ,1 S u = -β ω in C S ∩ ∂B u = 0 in (C S ∩ ∂B 1 ) ∪ ∂C S ∩ (B 1 \ B ) . (3.30)
Then (u ω -1) + ≤ u ≤ u ω in C ,1 S . (3.31)
Furthermore → u is increasing. When ↓ 0, u ↑ u 0 where u 0 is positive and p-harmonic in C 1,0 S , vanishes on ∂C 1,0 S \ {0} and satisfies (3.30) with = 0. In particular

lim r→0 r β u 0 (r, σ) = ω(σ) locally uniformly in S. (3.32)
We construct the same approximation u in C ,1 S with ω instead of ω. Mutadis mutandis (3.31) holds and u ↑ u 0 which is positive and p-harmonic in C 1 S , satisfies

(u ω -1) + ≤ u 0 ≤ u ω in C 1,0 S ,
and thus lim

Proof of Theorem E

In what follows we borrow most of our construction from [START_REF] Lewis | Boundary behavior and the Martin boundary problem for p-harmonic functions in Lipschitz domains[END_REF] that we adapt to the case of an infinite cone a make explicit for the sake of completeness. The next nondegeneracy property of positive p-harmonic functions is proved in [START_REF] Lewis | Boundary behavior and the Martin boundary problem for p-harmonic functions in Lipschitz domains[END_REF]Lemma 4.28]. If Ω is replaced by a cone C S , the nondegeneracy property still holds uniformly on ∂C S \ {0}. Let ω, ω ∈ W 1,p 0 (S) ∩ C(S) be positive solutions (1.1). Since ω ω is bounded from above and from below in S by positive constants, we can assume, as in the proof of Theorem D, that ω ≥ ω in S and that the graphs of ω and ω are tangent. hence, if ω = ω , then ω > ω in S and there exists a sequence

{σ n } converging to σ 0 ∈ ∂S as n → ∞ such that lim n→∞ ω (σ n ) ω(σ n ) = 1.
We define δ 1 = sup{δ > 0 : δω < ω }. For t ∈ (δ 1 , 1), we set

φ t = sup {ω , tω} and ψ t = inf t δ 1 ω , ω (3.37)
We also set

v φt (r, σ) = r -β φ t (σ) and v ψt (r, σ) = r -β ψ t (σ) ∀ (r, σ) ∈ (0, ∞) × S. (3.38)
where τ * = (1 -δ 1 )τ + δ 1 and as a consequence

lim τ ↑1 sup |x|≥a |x| β (u ω (x) -v τ,a (x)) = 0 and lim τ ↓0 sup |x|≥a |x| β (v τ,a (x) -u ω (x)) = 0 (3.43) Furthermore 0 ≤ v τ ,a -v τ,a τ -τ ≤ 1 δ 1 -1 v τ ,a ∀ 0 ≤ τ < τ ≤ 1. ( 3
u ω (x) ≤ v φ τ * (x) ≤ v τ,a,b (x) ≤ v ψ τ * (x) ≤ u ω (x) ∀ x ∈ C a,b S , which yields (3.42). Similarly, we have on ∂C a,b S 0 ≤ v τ ,a,b -v τ,a,b τ -τ = u ω -u ω ≤ (δ -1 1 -1)u ω ≤ (δ -1 1 -1)v τ,a,b , (3.45
) equivalently 0 ≤ v τ ,a,b ≤ 1 + (τ -τ )(δ -1 1 -1) v τ,a,b . (3
i,j (x) = |∇v τ,a | p-4 (p -2) ∂v τ,a ∂x j ∂v τ,a ∂x i + δ ij |∇v τ,a | 2 .
L satisfies the following ellipticity condition End of the proof. By the differentiability property of v τ,a with respect to τ , there exists two countable dense sets {(r ν } ⊂ [a, ∞) and {σ µ } ⊂ [a, ∞) such that ∂ τ v τ,a (r ν , σ µ ) exists for almost all τ . We put x ν,µ = (r ν , σ µ ), hence

min{1, p -1} |∇v τ,a | 2 |ξ| 2 ≤ i,j b i,j (x)ξ i ξ j ≤ max{1, p -1} |∇v τ,a | 2 |ξ| 2 ∀ξ ∈ R N . ( 3
ln ω(σ µ ) ω (σ µ ) -ln ω(σ µ ) ω (σ µ ) = ln v 1,a (x ν,µ ) v 0,a (x ν,µ ) -ln v 1,a (x ν,µ ) v 0,a (x ν,µ ) = 1 0 ∂ τ v τ,a (x ν,µ ) v τ,a (x ν,µ ) - ∂ τ v τ,a (x ν,µ ) v τ,

) and c 5 >and c 6 =

 56 0 depends on N , p and β. Iterating this estimate by Moser's method we derive (2.10). If p = N -1 we have for 1 ≤ m < p -1 and m * = m(N -1) c 6 (N, p), hence S |ω| tq dS

  1,p 0 (S k ) for all k. Since ω k = 0 in S c k and converges a.e. to ω, this last function vanishes a.e. in ∪ k S c k = (∩ k S k ) c = S c . Therefore ω vanishes a.e. in S c and since it is quasi continuous, it vanishes, (1 -p)quasi everywhere in S c . From Netrusov's theorem (see [1, Th 10.1.1]-(iii)) there exists a sequence {η n } ⊂ C ∞ 0 (S) which converges to ω in W 1,p (S), thus ω ∈ W 1,p 0 (S).

  it is Lipschitz and by Theorem D, β S = βS := β. Let ω and ω be two positive solutions of (1.1) satisfying sup S ω = sup S ω = 1. We denote by u ω (x) = |x| -β ω(.) and u ω (x) = |x| -β ω (.) the corresponding separable p-harmonic functions defined in C S . If 0 < a < b, we set C a,b S = C S ∩ (B b \ B a ). Then for 0 < < 1 we denote by u the unique function which satisfies

r→0 r β u 0

 0 (r, σ) = ω (σ) locally uniformly in S. (3.33) However, by [10, Th 4] u 0 and u 0 are proportional. Combined with (3.32), (3.33) it implies the claim.

Proposition 3 . 2 . 1 14u

 321 Let Ω ⊂ R N be a bounded Lipschitz domain and 1 < p < ∞. Then there exist constants ρ > 0, c 14 , c 15 > 0 depending respectively on Ω (for ρ), and p, N and the Lipschitz norm M of ∂Ω (for c 14 and c 15 ) with the property that for any w ∈ ∂Ω and any positive p-harmonic function u in Ω, continuous in Ω ∩ B 2ρ (w) and vanishing on ∂Ω ∩ B ρ (w), one can find ξ ∈ S N -1 , independent of u, such thatc -(y) dist (y, ∂Ω) ≤ ∇u(y), ξ ≤ |∇u(y)| ≤ c 14 u(y) dist (y, ∂Ω) , (3.34) for all y ∈ C S ∩ B ρ|w| c 15 (w).

Corollary 3 . 3 . 1 14u 15 ( 1 16u

 331151 Let 1 < p < ∞, S ⊂ S N -1 is a Lipschitz domain and C S the cone generated by S. (i) Then there exist constants ρ < 1 2 , c 14 , c 15 > 0 depending respectively on S (for ρ), and p, N and the Lipschitz norm M of ∂S and diam(S) (for c 14 and c 15 ) with the property that for any w ∈ ∂C S and any positive p-harmonic function u in C S , continuous in C S ∩ B 2ρ|w| (w) and vanishing on ∂C S ∩ B ρ|w| (w) continuous, one can find ξ ∈ S N -1 , independent of u, such that c -(y) dist (y, ∂C S ) ≤ ∇u(y), ξ ≤ |∇u(y)| ≤ c 14 u(y) dist (y, ∂C S ) , (3.35) for all y ∈ B ρ c w) ∩ C S . (ii) Then there exist positive constants κ and c 16 , c 17 depending on S (for κ), and p, N and the Lipschitz norm M of ∂S and diam(S) (for c 16 , c 17 such that for any a > 0 and any positive p-harmonic function u in C a S vanishing on ∂C S ∩ B c a , there holds c -(y) dist (y, ∂C S ) ≤ |∇u(y)| ≤ c 16 u(y) dist (y, ∂C S ) ∀y ∈ C c17a S s.t. dist (y, ∂C S ) ≤ κ |y| . (3.36)

  .44)Proof. The uniqueness and the (strict) monotonicity of (τ, b) → v τ,a,b follow from the monotonicity of τ → τ ω + (1 -τ )ω and the strong maximum principle. The continuity is a consequence of uniqueness and regularity theory for p-harmonic functions. It follows from (3.40) with t = τ * and the fact that v φ τ * and v ψ τ * are respectively a subsolution and a supersolution of -∆ p , that we have

=

  .46) By the maximum principle (3.45) holds in C a,b S . This implies (3.44). As a consequence of (3.44), ∂ τ v τ,a exists for almost all τ ∈ (0, 1) in W 1,p 0 (C a,b S ) for all b > a and it is a solution of Lw = ∇. (p -2) |∇v τ,a | p-4 ∇v τ,a .∇Z ∇v τ,a

Finally

  M (ĉt) -m(t) ≤ ĉ2 (m(ĉt) -m(t)). (3.53) Similarly M (t) -m(ĉt) ≤ ĉ2 (M (t) -M (ĉt)). (3.54) Summing the two inequalities we get (M (t) -m(t)) + (M (ĉt) -m(ĉt)) ≤ ĉ2 ((M (t) -m(t)) -(M (ĉt) -m(ĉt))) , which yields (3.52).

1 ĉ2 +1 < 1 .

 11 a (x ν,µ ) dτ.(3.55) Using the continuity of ω ω and the density of {σ m } we deriveln ω(σ) ω (σ) -ln ω(σ ) ω (σ ) . ≤ M (r ν ) -m(r ν ) ∀(σ, σ ) ∈ S × S. (3.56)We can assume that r ν ≥ ĉνn a for some sequence {ν n } tending to infinity with n, henceln ω(σ) ω (σ) -ln ω(σ ) ω (σ ) ≤ θ n (M (ĉ ν1 ) -m(ĉ ν1 )) ∀(σ, σ ) ∈ S × S ∀n ∈ N * ,(3.57)where θ = ĉ2 -Letting n → ∞ implies the claim.

  .48)It is important to notice that Lv τ,a = (p -1)∆ p v τ,a = 0. The estimate (3.48) combined with (3.36) and the decay of v τ,a and ∂ τ v τ,a implies that they satisfy Harnack inequality and boundary Harnack inequality in C a S . There exists a constant ĉ > c 17 > 1 (see 3.36) such that Lemma 3.6. For t > ĉa there holds Proof. There holds∂ τ v τ,a -m(t)v τ,a ≥ 0 and M (t)v τ,a -∂ τ v τ,a ≥ 0 ∈ C t S .Estimate (3.49) is valid for any couple of positive solutions(h 1 , h 2 ) of Lh = 0 in C a S vanishing on ∂C a S ∩ B c a , in particular for (∂ τ v τ,a -m(t)v τ,a , v τ,a ) and (M (t)v τ,a -∂ τ v τ,a , v τ,a ). Hence 1 ĉ ∂ τ v τ,a (x a ) v τ,a (x a ) -m(t) ≤ ∂ τ v τ,a (x) v τ,a (x) -m(t) ≤ ĉ ∂ τ v τ,a (x a ) v τ,a (x a )

	1 ĉ ∂ τ v τ,a (x a ) v τ,a (x a ) ĉ where x M (ĉt) -m(ĉt) ≤ ≤ ∂ τ v τ,a (x) v τ,a (x) This implies 1 ∂	≤ ĉ2 -1 ĉ ∂ τ v τ,a (x a ) v τ,a (x a ) ĉ2 + 1 (M (t) -m(t)) . ∀x ∈ C ĉa S , -m(t)	∀x ∈ C t S .	(3.49) (3.51) (3.52)

a = (ĉa, σ 0 ) for some σ 0 ∈ S fixed. We set

M (t) = sup x∈C t S ∂ τ v τ,a (x) v τ,a (x) and m(t) = inf x∈C t S ∂ τ v τ,a (x) v τ,a (x) ∀t > a (3.50) τ v τ,a (x a ) v τ,a (x a ) -m(t) ≤ m(ĉt) -m(t),

and

∂ τ v τ,a (x) v τ,a (x) -m(t) ≤ ĉ2 (m(ĉt) -m(t)) ∀x ∈ C t S .

Lemma 3.4. The functions φ t and ψ t are respectively a subsolution and a supersolution of (1.1) in W 1,p 0 (S), v φt and v ψt are respectively a subsolution and a supersolution of -∆ p in C S , and there exists η ∈ W 1,p 0 (S) solution of (1.1) such that

If S t is the subset of η ∈ W 1,p 0 (S) solutions of (1.1) and satisfying (3.39), then ω t = sup{η : η ∈ S t } belongs to S t . It is increasing with respect to t with uniform limits ω when t ↓ δ 1 and ω when t ↑ 1. Finally, if θ t = t-δ1 1-δ1 , there holds

Proof. Clearly φ t and ψ t are respectively a subsolution and a supersolution of the operator T , they belong to W 1,p 0 (S) ∩ L ∞ (S) and they satisfy ω ≤ φ t ≤ ψ t ≤ ω. Furthermore, by Dini convergence theorem lim

uniformly in S. Moreover, in spherical coordinates,

Thus v φt is a subsolution -∆ p in C S and v ψt is a supersolution. Since the operator T is a Leray-Lions operator, it follows by [START_REF] Boccardo | Résultats d'existence pour certains problèmes elliptiques quasilinéaires[END_REF] that there exists η ∈ W 1,p 0 (S) ∩ L ∞ (S) satisfying T (η) = 0 and φ t ≤ η ≤ ψ t in S. We denote by S t the set of η ∈ W 1,p 0 (S) ∩ L ∞ (S) satisfying T (η) = 0 and φ t ≤ η ≤ ψ t in S. Then there exists a sequence {η n } ⊂ S t and ω t ∈ W 1,p 0 (S) ∩ L ∞ (S) such that η n (σ) ↑ ω t (σ) for all σ ∈ Σ, where Σ is a countable dense subset of S. By Lemma 2.1 {η n } is bounded in L p (S), hence in C γ (S) for some γ ∈ (0, 1). By the estimates of the proof of Theorem B-Step 2, {η n } is bounded in W 1,p 0 (S). By standard regularity theory, we can also assume that η n → ω t in the C 1 loc (S)-topology. Hence ω t is a weak solution of (1.1), it belongs to W 1,p 0 (S) ∩ L ∞ (S) and satisfies φ t ≤ ω t ≤ ψ t . Therefore it is the maximal element of S t . The monotonity of ω t is a consequence of the monotonicity of φ t and ψ t and the last statement (3.40) is a straightforward computation.

Next we recall the deformation of p-harmonic functions already used in [START_REF] Lewis | Boundary behavior and the Martin boundary problem for p-harmonic functions in Lipschitz domains[END_REF]. If τ ∈ (0, 1) and 0 < a < b, we denote by v τ,a,b the p-harmonic function defined in C a,b S satisfying


Dr Emmanuelle Jacquet
email: emmanuelle.jacquet@univ-fcomte.fr

Sylvain Joly

Jérôme Chambert

Khansa Rekik

Patrick Sandoz

Ultra-light extensometer for the assessment of the mechanical properties of the human skin in vivo

Keywords: Extensometer, Skin in vivo, Mechanical properties, Standalone instrument, Full field measurement, Digital Image Correlation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Ultra-light extensometer for the assessment of the mechanical properties of the human skin in-vivo

INTRODUCTION

The ability to predict the in vivo response of the human skin to a mechanical loading is crucial to various domains, either clinical (sutures techniques, follow-up of skin pathologies or tissue bio-engineering) or related to body care research (air-bag design or efficiency testing of cosmetic products). For this purpose, surgeons, biologists, doctors and engineers have been trying for a long time to evaluate the mechanical properties of this heterogeneous, viscoelastic, anisotropic, non-linear and adhesive material.

Basically, the mechanical behavior of the skin can be tackled by two categories of tests; i.e. perpendicularly to the skin surface (succion [START_REF] Diridollou | Sex-and site-dependent variations in the thickness and mechanical properties of human skin in vivo[END_REF], contact indentation [START_REF] Jachowicz | Indentometric analysis of in vivo skin and comparison with artificial skin models[END_REF] [3] or contactless indentation [START_REF] Boyer | Non contact method for in vivo assessment of skin mechanical properties for assessing effect of ageing[END_REF]) or in the plane of the skin surface (torsion [START_REF] Rigal | Hardware and Basic Principles of the Dermal Torque Meter in Bioengineering of the Skin, Skin Biomechanics[END_REF] or extension [START_REF] Khatyr | Model of the viscoelastic behaviour of skin in vivo and study of anisotropy[END_REF]). The extension test consists in moving two pads attached to the skin surface and in measuring the force induced by the pad displacements.

Many extensometers for the in vivo skin have been designed in association with a stand. The first of them [START_REF] Thacker | In vivo extensometer for measurement of the biomechanical properties of human skin[END_REF] served as a basis for several more recent ones [START_REF] Manschot | The measurement and modelling of the mechanical properties of human skin in vivo -II. The model[END_REF][START_REF] Clark | Mechanical properties of normal skin and hypertrophic scars[END_REF][START_REF] Vescovo | Contribution à l'étude des propriétés mécaniques de la peau humaine[END_REF][START_REF] Gahagnon | Skin anisotropy in vivo and initial natural stress effect: A quantitative study using high-frequency static elastography[END_REF][START_REF] Flynn | Measurement of the force-displacement response of in vivo human skin under a rich set of deformations[END_REF]. Both Evans and Boyer combined mechanical and optical measurements in their tensile devices [START_REF] Evans | On the implementation of a wrinkling, hyperelastic membrane model for skin and other materials[END_REF][START_REF] Boyer | Assessment of the in-plane biomechanical properties of human skin using a finite element model updating approach combined with an optical fullfield measurement on a new tensile device[END_REF]. A three-pad extensometer was also designed with the aim to protect the measuring zone from peripheral forces [START_REF] Lim | New extensometer to measure in vivo uniaxial mechanical properties of human skin[END_REF].

In all those instruments hold by a stand, the parasitic forces due to interactions between the tested skin and the experimental device have to be kept sufficiently small to be neglected or measured to be taken into account during measurement analysis. This point is critical since movements due to either breathing or blood flow cannot be canceled.

To solve this problem, various hand-portable devices operating directly on a free skin were developed to be insensitive to small body movement disturbances [START_REF] Alexander | Accounting for natural tension in the mechanical testing of human skin[END_REF][START_REF] Jacquet | A new experimental method for measuring skin's natural tension[END_REF][START_REF] Prevorovsky | Ultrasonic wave propagation and mechanical properties of human skin stretched in vivo in 2[END_REF]. Those extensometers are suited for application on almost any body zone compatible with the device size.

The rigid attachment of the extensometer pads to the skin is required by all the devices mentioned above. No fully satisfactory solution does exist and liquid (cyanocrylate) or double-sided adhesive tapes are the most commonly used means.

The skin response to any mechanical loading depends on diverse environmental and instrumental parameters that have to be monitored. Even in best measurement conditions, huge differences are observed in the human-skin mechanicalcharacteristics as reported in literature. The human skin is indeed not made of a standardized material with constant specifications. It corresponds rather to a generic material of which mechanical properties vary between individuals as well as between body zones [START_REF] Flynn | Measurement of the force-displacement response of in vivo human skin under a rich set of deformations[END_REF]. This variability raises a methodological challenge since averaged values cannot be used to predict the skin response to a specific solicitation.

With the aim to shift toward a personalized medicine, the actual properties of the skin to be treated have to be measured and suitable instruments are required. This paper presents an ultra-light standalone device developed for that purpose and reports on its ability to characterize the mechanical response of the human skin in vivo to diverse mechanical tests. The extensometer design and control are described and its functioning is validated from reference measures.

Its ability to assess the mechanical properties of the human skin in vivo are illustrated by typical results. Finally the device performances are analyzed and discussed.

Extensometer development, control and validation

Aim and design

The extensometer was designed to fulfill the following specifications :

• usable in most of the human sites in vivo,

• light enough to work as a standalone instrument,

• easy to use and consistent with standards and biomedical requirements,

• performing extension tests with either force or displacement control (loading and unloading), • permitting large quasi-static displacements as well as low-frequency dynamic solicitations on the human skin in vivo,

• keeping the tested skin area visible to permit simultaneous image acquisition.

The whole experimental setup is shown in figure 1 whereas the details of the extensometer are shown in figure 2. The specifications of the device built are summarized in table 1. A speckle pattern is applied on the skin for the sake of accurate local-motion detection by means of digital image processing (cf. 2.3).

Mechanical part

The device is made of a rectangular frame (100 × 40 × 6 mm 3 ) at which are fixed an electric motor, two guiding pillars and two sets of complementary pads: one static, one mobile (cf. Fig. 2). The mobile set of pads is held by a traverse beam translated by the motor by means of a screw-nut system. Each set of pads is made of a measuring pad ( by a "U-shaped " pad (24 × 17 mm 2 called guarding pad ). The guarding pads protect the measuring zone from lateral and peripheral forces by shifting the border between the solicited and unsolicited zones towards the outside [START_REF] Lim | New extensometer to measure in vivo uniaxial mechanical properties of human skin[END_REF]. The central and guarding pads are moved together but the force is measured from the central pad only; i.e. one third of the solicited width. The ratio between length and width of the force-measurement region is larger than 3. The force sensor is made of a specific bronze-berrylium sensor equipped with strain gauges stuck on the sensor where the strain due to flexion is maximal. The device is supplemented by a LVDT-type displacement sensor providing the mobile pad position.

Imaging part

A vision system is associated to the extensometer to allow the measurement of the field of displacements on the solicited area during the mechanical test by means of digital image correlation (DIC). The camera is hold by a six degrees of freedom stand independent of the extensometer and allowing to focus on the tested skin area. Images are recorded at a 30 fps rate and post-processed using Matlab.

Since the extensometer is fixed on the skin through its four pads of which two are moving, some slight displacements of the device on the skin occur during mechanical tests in addition to slight body movements. To avoid biases due to these disturbances, the field of observation of the vision system is chosen large enough to make the whole device visible. All images are registered within a single (X, Y ) coordinate reference from the positions of the pads that provide also a size reference to convert pixels into actual sizes without further calibration.

A random pattern is applied onto the skin using a water-based paint with a brush as required by the DIC method used [START_REF] Eberl | Digital Image Correlation and tracking with MATLAB[END_REF]. The digital method consists in comparing the current image to the initial one to retrieve the averaged displacement values of sub-images of 32 × 32 pixels. The software looks for the displacement value that minimizes the correlation function built from the products of the functions describing both the reference and current images.

The method is applied to the whole sequence of images to provide the fields of displacements for the entire duration of the mechanical test.

Electronic control

An electronic controller based on a microcontroller (Arduino Mega 2560) was developed to drive the extensometer motor and to acquire the data from the

User assistance interface

Two complementary software applications were developed under Labview.

One is dedicated to the easy generation of motor driving command files. The latter are built only once and available to proceed to identical tests on different subjects or different skin areas. Any kind of test is allowed (quasi-static, harmonic, ...) as long as within the device specifications (speed < 1 mm/s, displacement < 15 mm and force < 5 N) without other limitations.

The second software application is dedicated to the overall control of the device and experiments (Fig. 3). It is made of three independent and complemen- Finally the trial documentation block allows the storage of the data relevant to the current experiment. Predefined fields are proposed for generic data (age, sex, weight, ...) whereas additional fields can be created and filled according to the user needs. A trial report is automatically generated at the end of every trial thus making the all data available for further consultation or analysis. The low device mass in regard to its surface of contact (pads plus frame) results in applying a very weak pressure onto the tested skin. The induced disturbances are therefore considered as negligible, thus allowing the use of the device without a stand provided that patient movements remain sufficiently small and slow.

Experiments on the human skin in vivo

3.1. Experimental protocol

Ethical aspects

Tests on living tissues are submitted to permission by the ethical board of the hospital as well as to the subject's consent once the person was fully informed regarding the test objectives and experimental protocol. To avoid any risk to the patient's safety, a risk table was established. The major element able to produce damages was identified as an uncontrolled extension applied to the skin. This possibility was discarded by means of an emergency stop button that triggers the instantaneous release of the loading (or unloading) forces applied to the skin. Furthermore the interface screen plots in real-time the measured values of both the force and the displacement thus allowing an additional monitoring of the ongoing test by the operator.

Mode of operation

Some test preparation is necessary on both patient and operator sides and the full sequence of test operation takes place as indicated below:

• 20 mn acclimatisation period to ensure constant environmental conditions.

• Speckle pattern realization.

• Test documentation on the software interface (subject identifier, age, sex, body zone, test orientation, etc.).

• Loading of the test command file, either already existing or just created.

• Sticking of the extensometer on the patient's skin (hardening in 20-30 s)

and camera focusing adjustment.

• Launching of simultaneous test and image recording. Data are stored in a specific directory and summarized in an automatic test report.

• Extensometer removal and cleaning.

• Data available for DIC computations and result analysis.

Typical results

A huge diversity of tests with either force or displacement control are allowed by the device. Figure 4 presents three typical results obtained with displacement control. The 1 st row presents the displacements applied to the skin as measured The recorded data are available for further analysis and material behavior modeling using either hyperelastic or viscoelastic models.

Performance analysis

Ability to report on the skin behavior

The simultaneous measurement of forces and displacements can be analyzed as stress-strain curves since the test can be considered as a uniaxial traction test (cf. 3.3.2 and 3.3.3). The proposed device allows the demonstration of the non-linear elastic behavior of the tested skin and the parameters of an isotropic hyperelastic model can be identified from such uniaxial extension tests.

The analysis of the unloading and holding phases requires the introduction of viscosity terms in the behavior models.

The use of the complementary imaging unit provides the field of displacements over the entire field of view as represented in Fig. 5. We observe that the deformed zone extends beyond the area between the measuring pads and even beyond the guarding pads. For a given X-position, the X-displacement parallel to the tensile axis is uniform over almost the whole transverse direction. The deformation field xx is almost constant over the all area thus confirming that the X-displacement is linearly distributed along the tensile axis. The field of transverse Y-displacements shows a shrinking of the zone between the pads as expected. The field of deformations yy is also uniform and negative and the shear deformation field xy is close to zero over the whole area. These results fit the behavior of a homogeneous material under uniaxial traction test.

Beyond these results, the analysis of digital imaging data reports on a possible heterogeneity in the skin behavior. Tests with holding phases can be compared with cutometry tests commonly used by dermatologists. The continuous and simultaneous measurement of both force and displacement allows to gather data necessary to address the skin viscosity. Finally harmonic loadings provide a means for identifying the intrinsic parameters of the material independently of its environment thanks to the decoupling of elasticity and viscosity parameters.

As reported in literature, mechanical measurements on the human skin present a high degree of variability. Even on the same zone of the same subject, results may differ as a function of time or conditions. The skin response to the same loading is not the same at the first, second or third cycle. Beyond the third cycle, we experimentally observed that a preconditioning is obtained and that the material response becomes reproducible.

Efficiency of the guarding tabs

The measurement zone is located between the measurement pads and, thanks to the guarding pads, the surrounding area is solicited in the same way. Disturbances due to distant unsolicited skin affect thus mostly the zone located between the guarding pads and are thus minimized on the central, actual measurement zone. The protective effect of the guarding pads can be assessed by comparing Fig. 6 obtained without sticking the guarding pads with Fig. 5. This comparison shows that the transverse uniformity is lost or significantly reduced for four represented variables amongst five. For Y-displacement, the transversal decrease of the displacement is smoother with the guarding tabs stuck. It should be noticed that the contour map of X-displacement is perpendicular to the tensile direction when the guarding pads are stuck (Fig. 5) whereas they are curved without the guarding pads (Fig. 6). This transverse uniformity is highlighted by Fig. 7 presenting the X-displacement over the all area between the pads whereas the zone between the measuring pads is marked by dotted lines. These figures confirm the efficiency of the guarding pads in minimizing the influence of the surrounding skin. Thanks to this protective effect, the extension test performed by the device can be considered as equivalent to a pure traction test with an excellent approximation.

At the level of the skin surface, the test produces an uniform longitudinal deformation on the skin located between the measuring pads. It is assumed that the traction test applies homogeneously over the whole skin thickness (≈ 1 -2 mm). This hypothesis seems reasonable in regard to the distance between the pads and the deformation rates applied.

Useful complementary measurements

The skin thickness is measured after the extension test by ultrasound (Dermacup ATYS Medical, 25MHz probe). This allows the conversion of force data into stress data as applied to a tensile test-piece of width equal to the measuring pads one and with a thickness equal to the measured one.

The mechanical skin response to an extension test is also known to be sensitive to the degree of stratum corneum humidity. This parameter is kept relatively constant by respecting the biometrology tests requirements, especially the resting time in a room with controlled temperature and hygrometry.

First in vivo results

The device was applied to diverse series of tests in controlled conditions on the human skin in vivo, in particular on a cohort of 20 healthy people in our laboratory. Different anatomic zones were explored following different angular directions. The results obtained provide a wide source of data to analyze the human skin mechanical behavior. We notably document non-linear elasticity, anisotropy, preconditioning and initial tension of the human skin as well as its intra-and inter-individual variability. Fig. 8 illustrates the huge variability obtained on the force -displacement curves. In depth analysis and discussion of the results obtained on that cohort are presented in a dedicated paper [START_REF] Jacquet | Intra-and inter-individual variability in mechanical properties of human skin: Results from in vivo measurements with a portative extensometer[END_REF] including the conversion of raw data into non-linear parameters characteristic of the human skin behavior.

Discussion and conclusion

A device performing uniaxial tensile tests with effort or displacement control has been presented in this paper. A displacement speed up to 1 mm/s and a deformation up to 50% are permitted by the hardware. The pads are stuck on the in vivo skin and the device lightness avoids the need for a stand. The advantage is that the patient can be either sitting, standing or laying and that his small movements do not disturb measurements without any control of the tangential efforts of the device onto the skin. This is especially important in regard to slight displacements induced by either breathing or blood flow. These specifications make thus the device operational on most body zones.

The guarding tabs minimize the effects of surrounding skin and the performed extension test can be considered as a traction test with a satisfactory approximation. Complementary thickness measurements allow the determination of stress-strain curves. In parallel, an independent imaging unit enables the image recording of the solicited skin area for the digital retrieval of the strain fields. The latter report on the heterogeneity of the solicited area. They also allow the identification of the intrinsic parameters of a material behavior model from the 2D strain distribution measured in response to an input control command in either moving-pad displacement or applied effort.

The software interface for generating input control command files offers a wide variety of solicitation types; either quasi-static or harmonic, with preload or holding steps. The material viscosity can thus be taken into account by identifying the parameters of a viscoelastic model. Successive tests can be oriented differently with respect to the Langer's lines to address the local skin anisotropy. As a prospect, it is intended to develop a similar biaxial device to address the coupling between the axes of anisotropy of the material.

Figure 1 :

 1 Figure 1: Left: Electronic and mechanical parts of the device. Right: Setup in measuring situation on a forearm.

Figure 2 :

 2 Figure 2: Top (a) and bottom (b) views of the extensometer.

  embedded sensors.Two Arduino shields based on H-bridge circuits command the motors by means of a Pulse-Width-Modulate (PWM) signal. A Linear Variable Displacement Transducer (LVDT) provides a tension proportional to the position of the pad that is detected thanks to an adaptation card (EL-35C-Analog by Singer Instrument). A supplementary electronic card (based on INA2126) shifts the LVDT signal in always-positive tensions as required by the microcontroller. The signal provided by the strain gauge is amplified by a conditioning card (based on LM358). The user accesses the microcontroller parameters by means of a software interface developed under Labview.

Figure 3 :

 3 Figure 3: Extensometer control.

  tary blocks: extensometer control, (optional) camera control, documentation of the ongoing trial. The extensometer can be controlled either automatically by means of a driving command file or manually by means of virtual potentiometers. The displacements of the mobile pad are servo-controlled either in displacement or in force. The camera control block aims to the driving of an optional camera. It allows the recording of a set of images of the tested skin area synchronously with the motion applied to the pad. The images obtained are stored in a folder attached to the ongoing trial for further digital image processing.

2. 6 .

 6 Validation of the instrument Specific experiments were carried out to validate the correct functioning of the extensometer by comparing the force and displacement values provided by the strain gauges and the LVDT sensor to reference ones. The correct functioning of the instrument was also checked by comparing the results provided by our device on a thin sample of elastomer to those obtained from a traction machine (BOSE Electroforce 3200) on the same material. Despite the large area of the tested sample (analog to actual experiments on the skin), the agreement observed validated the correct functioning of our extensometer. A standalone device is easy to use and does not limit the patient's movements. Force and displacement sensors are embedded and follow naturally the global motion. At contrary the camera is fixed and detect rigid-body patient's displacements. The correct registration of the sequences of images was validated by the successful comparison of the data provided by the LVDT sensor during a loading-unloading test with those obtained by DIC.

Figure 4 :

 4 Figure 4: Typical results: Column a): three successive loadings-unloadings, Column b): harmonic loading after a pre-load; Column c): loading-holding-unloading. 1 st row: Displacement returned by the LVDT. 2 nd row: measured force; 3 rd row: force-displacement curves.

Figure 5 :

 5 Figure 5: From top to bottom: Displacement fields following the tensile axis X and the transverse axis Y. Strain fields xx, yy and xy obtained from a test with displacement control. (Scales in pixels with 5.2 mm per 100 pixels).

Figure 6 :

 6 Figure 6: From top to bottom: Displacement fields following the tensile axis X and the transverse axis Y. Strain fields xx, yy and xy obtained from a test with displacement control and without guarding tabs. (Scales in pixels with 6.0 mm per 100 pixels).

Figure 7 :

 7 Figure 7: X-Displacement field following the tensile axis X obtained. (Displacement control with guarding pads. The zone between the dotted lines corresponds to the measuring pads.).

Figure 8 :

 8 Figure 8: Illustration of the variability observed from a series of in vivo skin extension measurements. Tests on different people, different anatomic areas and different directions of solicitation. (Initial pad distance: 30 mm).

Table 1 :

 1 8 × 8 mm 2 called central pad ) surrounded Specifications of the device.

		Central Pads and guarding pads
	General specifications	Force or displacement control
		Bonding pads
		Dimensions with motor	100 × 40 × 10 mm 3
		weight	36 g
		Observable area	< 45 × 24 mm 2
		Effort sensor	5 N
	Technical specifications
		Effort accuracy	5 mN
		Maximal displacement	15 mm
		Maximal strain	0, 5
		Maximal displacement speed	1 mm/s
		Sampling rate	100 samples/s

Acknowledgements

We are thankful to Eric Joseph for technical support and to the European Union for a FEDER grant from Région de Franche-Comté, France (n • 36381).


Jan Jaap

R Van Assen

Pascal Barla

Roland W Fleming

Shape features underlying the perception of liquids

Visually identifying and estimating properties of materials is critical for many tasks, from reaching for a steaming kettle to walking on a wet marble staircase. From a computational point of view, liquids represent a particularly interesting class of materials because of their highly mutable shapes and physical complexity. The ways liquids move and change shape is affected both by external forces (e.g. object interactions, gravity) and intrinsic properties (e.g. viscosity, density). Estimating viscosity therefore requires disentangling the relative contributions of external and internal factors. Previous studies have shown that despite the wide spectrum of possible liquid shapes, observers are surprisingly good at estimating viscosity across variations in the external conditions. In this study we test the extent to which perceived viscosity can be predicted using mid-level shape and motion features (e.g. 'spread', 'piling up', 'angularity').

We conducted experiments using simulated liquids. Observers either rated viscosity or twenty shape features (e.g., volume, folding, spiraling), separately in three different sets: (1) A liquid pouring on a plane simulated with 32 steps of viscosity. (2) Eight random variations of this scene with seven different viscosities, created by perturbing the flow of the liquids with different force fields, leading to substantial shape differences over time. (3) Eight different scenes of various kinds of interaction (e.g., stirring, rain, waterfall) at seven different viscosities.

Results indicate that viscosity ratings are well predicted through a linear combination of small numbers of mid-level features, as rated by other participants. In the first stimulus set, all twenty predictors yields R 2 = 0.99 and with the six main predictors R 2 = 0.98. Since the liquids are computer simulated we know the true 3D shape for each frame. From these we derived shape measurements that correlate with each of the six main predictors, resulting in predictions of perceived viscosity with R 2 = 0.92. Applying the same model with the same feature weights to the ratings on the second stimulus set (variations of pouring liquids) yields R 2 = 0.80, but the model loses much of its predictive power for the diverse scene variations of the third stimulus set (R 2 = 0.15). Looking at the results of shape feature ratings for the eight different scenes it becomes clear that rather than relying on a fixed set of features, the visual system flexibly identifies and re-weights mid-level cues depending on the context to achieve high levels of constancy. Combining data across all experiments we have isolated four shape features (Angular, Spread, Piling up, Complexity) that generalize quite well across contexts (mean R 2 = 0.97) but this requires a model that can re-weight the features according to constraints derived from each particular context. How our visual system estimates these constraints on a scene-by-scene basis remains the important open question.

Figure 1 :

 1 Figure 1: The graphs show the proportional reduction in error, or contribution, for each shape feature for two scenes (Smear, Push). The liquid is simulated with a viscosity of 0.6 Pa•s. Research supported by EU FP7 Marie Curie Initial Training Networks (ITN) project PRISM, Perceptual Representation of Illumination, Shape and Material (PITN-GA-2012-316746); DFG: SFB-TRR-135 "Cardinal mechanisms of perception"; ERC Consolidator Award (ERC-2015-CoG-682859: "SHAPE"); Google Faculty Research Award.


Dr Emmanuelle Jacquet
email: emmanuelle.jacquet@univ-fcomte.fr

Keywords: Mechanical skin behavior, Extension tests, In vivo measurements, Variability, Hyperelastic behavior, Characteristic parameters

Background/Purpose: The mechanical properties and behavior of the human skin in vivo are of medical importance, particularly to surgeons who have to consider the skin extension capabilities in the preparation of surgical acts. Variable data can be found in literature that result from diverse kinds of tests (in vivo, ex vivo, post-mortem) performed with different instruments.

Methods: This paper presents the results of in vivo measurements performed on a cohort of 20 healthy volunteers with an ultralight homemade uniaxial extensometer. Different anatomical zones were explored under different directions of solicitation in order to document inter-and intra-individual variability as well as skin anisotropy.

Results: The experimental data obtained are fitted with a phenomenological exponential model allowing the identification of three parameters characteristic of the tested skin behavior. These parameters can be related to the concept of skin extensibility used by surgeons.

Conclusion:

The inter-and intra-variability observed on that cohort confirms the need for a patient-specific approach based on the in vivo measurement of the mechanical behavior of the human skin of interest. Even the direction of higher skin stiffness is found to be individualdependent. The capability of the extensometer used in this study to fulfill such measurement needs is also demonstrated.

Introduction

The human skin constitutes an interface between the human body and the surrounding environment. It stands for a chemical and mechanical protection against external adversities. When the skin is damaged either accidentally or surgically, the skin continuity has to be reconstructed. This is especially problematic when a significant area of skin has been lost, in case of skin tumors for instance, since the surrounding skin has to be extended to cover the wound [START_REF] Pauchot | Bilateral latissimus dorsi V-Y musculocutaneous rotation flap for closure of a large dorsal radionecrosis. Geometric analysis and interest. About one case[END_REF]. In such circumstances, the mechanical properties of the skin correspond to a key-characteristic determining up to which rate the skin can be stretched. The closing of the wound will indeed be much easier with a soft and elastic skin than with a rigid and inextensible skin [START_REF] Remache | Numerical analysis of the V-Y shaped advancement flap[END_REF]. Surgeons are used to appraise empirically the skin elasticity and the subcutaneous fat layer thickness by applying manually a pinch test [START_REF] Borges | Relaxed skin tension lines, Z-plasties on scars, and fusiform excision of lesions[END_REF][START_REF] Waldorf | Planning incisions[END_REF]. Such a qualitative evaluation is subjective and depends on surgeon's know-how. From a clinical point of view, the pinch test is used to determine the best suited surgical act, notably to spread the closing efforts on the right area of surrounding skin to optimize wound healing. In case of excessive skin extension, internal fluid flow can be disturbed up to induce complications such as necrosis. In this context, the development of objective tools able to specify the mechanical properties of the skin would make this decision process more rational, safer, and thus beneficial to the patient.

Numerical simulations of the wound closing efforts have been proposed but they are highly dependent on the mechanical skin behavior [START_REF] Jr | A finite element model of skin deformation III. The finite element model[END_REF][START_REF] Gambarotta | In vivo experimental testing and model identification of human scalp skin[END_REF][START_REF] Flynn | Finite element models of wound closure[END_REF][START_REF] Capek | The analysis of forces needed for the suturing of elliptical skin wounds[END_REF]. The latter is known to depend on both individuals and body zones and no standardized data can be used satisfactorily. It is therefore necessary to characterize the mechanical skin behavior in an individualized and localized way to determine the actual parameters to take into account [START_REF] Annaidh | Characterization of the anisotropic mechanical properties of excised human skin[END_REF][START_REF] Groves | An anisotropic, hyperelastic model for skin: Experimental measurements, finite element modelling and identification of parameters for human and murine skin[END_REF]. Once the actual mechanical parameters are determined, numerical simulations of the efforts and of the stress fields induced by diverse surgical options can help surgeons in deciding the best surgical strategy for each act and patient. This paper presents the results of such mechanical tests performed on a cohort of healthy volunteers with a light and portable instrument described elsewhere [START_REF] Jacquet | Ultra-light extensometer for the assessment of the mechanical properties of the human skin in-vivo[END_REF]. We highlight the intra-individual and inter-individual variability also observed in suction tests [START_REF] Ryu | Influence of age and regional differences on skin elasticity as measured by the Cutometer[END_REF], in out-ofplane deformation tests [START_REF] Flynn | Measurement of the force-displacement response of in vivo human skin under a rich set of deformations[END_REF], and in dynamic indentation [START_REF] Boyer | Dynamic indentation on human skin in vivo: ageing effects[END_REF]. We also interpret those results by means of three parameters characterizing the tested skin mechanical behavior, thus providing objective parameters to help surgeons in their decision processes.

Material and Methods

Portative extensometer for in vivo measurement

The mechanical tests were carried out using the tensile testing device shown in Fig. 1 and developed specifically for measurements on the human skin in vivo. It is a portable lightweight extensometer usable without stand on most anatomical zones. It consists of a rectangular frame to which are attached an electric motor, two guide columns and two crossheads; one fixed, the other mobile and each comprising two tabs. Measurements are performed on the central tab whereas the secondary surrounding tab serves for guarding by protecting the measuring zone from lateral and peripheral forces. The central and guarding tabs are moved together and the extension is applied to both the measuring zone and to the surrounding skin. The movable crosshead is driven by an electric motor whose rotary movement is transformed into translation by a screw-nut system. The movable crosshead is equipped with a specific bronze-berrylium beam which serves as force sensor on which strain gauges are stuck at the location where the bending deformation is maximal. The device is supplemented by a LVDT-type displacement sensor that provides the moveable tab position. The device, servo-controlled in either force or displacement, allows the performance of large extensions tests on the human skin in vivo. The gap between the fixed and mobile tabs ranges from 30 mm to 45 mm, allowing a maximum extension of 50 %. The maximal force that can be applied is limited to 5 N. Diverse kinds of controlled tests are permitted, quasistatic, harmonic, impulse or relaxation tests, on most of the human sites in vivo with a maximum displacement speed of 1 mm/s. The device is light enough to work as a standalone instrument and is consistent with standards and biomedical requirements. The secondary, guarding tabs minimize the parasitic effects of the area surrounding measurement zone, thus making the stress field homogeneous as in a classic tensile test. Full description of the device used can be found in [START_REF] Jacquet | Ultra-light extensometer for the assessment of the mechanical properties of the human skin in-vivo[END_REF].

Volunteer cohort and complementary biological parameter assessment

The mechanical tests were carried out in the Department of Applied Mechanics of FEMTO-ST Institute at several anatomical sites (forearm, abdomen, thigh and calf) of 20 recruited volunteers. The subjects are 12 men, 8 women distributed between 18 and 57 years. Their body mass index (BMI) is between 18 and 29. The tested skin thickness was measured at the Center for Studies and Research on the Integument of the regional University Hospital Center of Besançon with Figure 1: Extensometer device stuck on forearm a high-resolution ultrasound skin Dermcup Atys ® . For each measurement, three ultrasound images were taken and for each image three thickness measurements were carried out. For each site, the average value of the nine measurements is taken. The average thickness is 1.96 ± 0.26 mm on the abdomen, 1.6 ± 0.36 mm on the thigh, 1.28 ± 0.14 mm on the calf and 1.39 ± 0.30 mm on the forearm. These thickness values are similar to the ones in [START_REF] Reihsner | Two-dimensional elastic properties of human skin in terms of an incremental model at the in vivo configuration[END_REF]. Additional measurements of stratum corneum hydration rate were taken using a Corneometer ® CM 825 (Courage and Khazaka electronic GmbH) evaluating the dielectric capacitance of the skin surface layers (epidermis + stratum corneum). After a rest of 15 minutes in a room at constant temperature and humidity and before the tensile tests, three successive measurements of the hydration index of the tested zones were performed.

Protocol and experiments

The tests followed the rules imposed by the Hygiene and Safety Service of the University.

Prior to measurements, the test protocol is submitted in full details to volunteers who give their written approval. It involves the following steps :

• A preparation of the tested area. The area should not contain any hair and not being hydrated and treated with a cream since the day before the test.

• A rest period in an elongated position with straight legs, arms alongside the body and acclimatization of the skin to the air at constant temperature and humidity of 15 minutes during which the subject answers the questionnaire (age, BMI and so on).

• The measuring of the hydration index of the area to be tested.

• The mechanical tests according to the procedure detailed below. The volunteer is lying down in order to present the area of interest steady and horizontal upwards.

The zone is defined based on the cranio-caudal and medio-lateral lines and precise anatomical landmarks : At the front face of the arm, three zones are defined on the forearm to 1/4 (distal forearm), 1/2 (middle forearm) and 3/4 (proximal forearm) of the wrist -elbow segment (Fig. 2).

At the abdomen, the center of the tested area is offset from the cranio-caudal line a third from the navel-iliac crest segment (Fig. 2). At the leg, a front thigh area is about 10 cm above the knee, a calf area at the widest area of the calf and a back thigh area to the back of leg area about 10 cm above the knee bend (cf Fig. 2). The solicited axes are also identified from the cranio-caudal line : At the arm, the 0 • -axis is along the arm. At the abdomen, the 90 • -axis is parallel to the cranio-caudal line. At the leg, the 0 • -axis is along the leg. The tabs of the extensometer are attached to the skin surface thanks to a surgical glue. Upload consists of three cycles followed by a holding time to take into account the preconditionning of the skin [START_REF] Ryu | Influence of age and regional differences on skin elasticity as measured by the Cutometer[END_REF]. This phenomena has been widely discussed by several authors [START_REF] Fung | Biomechanics: mechanical properties of living tissues[END_REF][START_REF] Boyer | Dynamic indentation on human skin in vivo: ageing effects[END_REF]. After three repeating cycles a steady state tends to be reached and the material behavior is more reproducible. In the following results, only the third loading sequence has been analyzed. The loading rate remains constant, equal to 1 mm/s. The test is then repeated until a maximum of 42% of extension when the elasticity of the skin allows. The take-off of the device is immediate and painless. The skin is then cleaned with alcohol and the glue residues on the tabs of the device are removed with a solvent and then the tabs are rinsed with alcohol. This process is repeated for each loading direction.

Data analysis

The raw data are obtained by measuring the force recorded on the measuring tab when it is moved during the test. Incremental time, command, displacement and force data are registered during the whole test. The raw data are processed using a Butterworth filter operating a moving average on the force data obtained for an imposed displacement speed. An order two lowpass digital Butterworth filter with normalized cutoff frequency 0.1 Hz is successively used in the forward and backward directions [START_REF] Gustafsson | Determining the initial states in forward-backward filtering[END_REF]. Filtering the raw data is necessary particularly for low levels of force.

We neglect the existing shear stress due to the fact that the skin is loaded on a surface and not directly on the section of the considered area. The stress field in the measuring area is then defined by the unique component T of the first Piola-Kirchhoff stress tensor in the traction direction,

T = F A 0 = F d • e 0 , ( 1
)
where F is the measured force, A 0 is the initial cross-sectional area of the specimen defined by the product of the measuring tab width d and the skin thickness e 0 at the beginning of the test.

The uniaxial engineering strain ε in the loading direction is obtained from the imposed distance between the two main tabs, i.e the displacement of the tab whose movement is controlled,

ε = ∆L L 0 = L -L 0 L 0 , ( 2
)
where ∆L is the displacement of the measuring tab; L and L 0 are, respectively, the actual and initial length between the two tabs.

For the characterization of the mechanical behavior of soft tissues, the phenomenological approach consists in fitting mathematical expressions to experimental stress-strain curves. In the literature, the nonlinear behavior of skin tissue is mostly characterized by an exponential form of stress-strain relationship when only the elastin fibers are being stretched for small deformations, and by a power equation when the collagen fibers are being mainly loaded for large deformations [START_REF] Wijn | The alinear viscoelastic properties of human skin in vivo for small deformations[END_REF]. For various elastin-dominant soft tissues, Fung [START_REF] Fung | Stress-strain-history relations of soft tissues in simple elongation[END_REF] has shown a linear relationship between the stress T and the tangent modulus dT /dε, which leads to an exponential form of the stressstrain curve. Fung [START_REF] Fung | Stress-strain-history relations of soft tissues in simple elongation[END_REF] has also proposed a piecewise linear function of dT /dε versus T curve in order to distinguish the initial phase at elastin-dominant from the final one at collagen-dominant.

For the sake of simplicity, we propose the use of an exponential equation for the whole stressstrain curve :

T = A [exp(B ε) -1] , (3)
where A and B are fitting parameters.

By following Raghavan et al. [START_REF] Raghavan | Ex vivo biomechanical behavior of abdominal aortic aneurysm: Assessment using a new mathematical model[END_REF]'s analysis on aorta, we propose to identify three physical parameters useful for the surgeon (see Fig. 3) :

• the initial tangent modulus E 1 (initial slope of the stress-strain curve) which corresponds to the contribution of the elastin fibers alone ;

• the final tangent modulus E 2 (final slope of the stress-strain curve) which stands for the combined contribution of elastin and collagen fibers ;

• the intercept ε T of the strain-axis with the final slope of the stress-strain curve.

The three physical parameters are determined by applying the following consecutive steps (see Fig. 3). First, the values of parameters A and B in Eq. ( 3) are fitted to experimental stressstrain data by using an inverse method based on Levenberg [START_REF] Levenberg | A method for the solution of certain non-linear problems in least squares Quart[END_REF] -Marquardt [START_REF] Marquardt | An algorithm for least-squares estimation of nonlinear parameters[END_REF] algorithm. The coefficient of determination R 2 is computed to quantify the goodness-of-fit. Second, the value of E 1 -parameter is determined by fitting the first values of the stress-strain experimental curve with a linear regression technique, and computing the resultant slope. Third, the value of E 2parameter is obtained from the derivative of Eq. ( 3) taken at the strain ultimate value. Fourth, the value of ε T -parameter is determined by intercepting the strain-axis with the final tangent of the stress-strain curve.

According to [START_REF] Daly | The biomechanical characteristics of human skin[END_REF] and [START_REF] Brown | A scanning electron microscope study of the effects of uniaxial tension on human skin[END_REF], a typical stress-strain curve for in vivo skin in uniaxial tension, as shown in Fig. 3, can be separated into three consecutive parts : the initial phase (the elastin fibers are stretched and the collagen fibers orientate into the direction of extension), the transitional [START_REF] Gibson | Directional variation in extensibility of human skin in vivo[END_REF] have interpreted this parameter as a measure of extensibility of the skin in that direction. As in [START_REF] Millington | Structural and mechanical aspects of connective tissue[END_REF], this state defines a physiological load limit that enables to maintain the integrity of the skin. The ε T -parameter can be interpreted as the strain threshold that should not be exceeded during a surgery to prevent medical complications. Considering the anisotropy of the skin behavior, Gibson et al. [START_REF] Gibson | Directional variation in extensibility of human skin in vivo[END_REF] have shown the close correlation between the direction of minimum ε T -parameter value and Langer's line [START_REF] Langer | On the anatomy and physiology of the skin: I. The cleavability of the cutis[END_REF]. That confirms the interest of ε T -value for the surgeon.

Results and discussion

Most of the results are presented by stress-strain curves as previously explained with physical parameters presented in the corresponding tables.

Model relevance for measurement on human skin

In order to analyse the experimental results, the constitutive parameters of the mathematical model (see Eq. ( 3)) have been identified on several stress-strain curves and collected in tab. 1.

(a) Reproducibility -Parameters values corresponding to Fig. 4 As shown in Fig. 4

Intra-individual variability

Stress-strain curves in Fig. 5 have been obtained on different anatomical areas for the same volunteer. These areas are the three parts of the forearm (distal, middle and proximal), three parts of the leg (posterior thigh, anterior thigh and calf) and the abdomen (Fig. 2). The results highlight the large variability associated with the different anatomical zones and directions of extension. For three neighboring areas on forearm, the E 2 -variability (see Table 1b) is large. In 1a for the corresponding values of E 1 , E 2 , ε T and R 2 .

addition, for the abdomen, the ε T -values are much higher than the values on the forearm that are below 15 %. This is observed despite on the abdomen, the loading has been interrupted before reaching the final phase of stress-strain curve, leading therefore to underestimated values of E 2 and ε T for this zone. It should be noticed that the front thigh E 2 -value (22%) is much larger than the back thigh one (14%). Note that these tests were performed with a straight leg (Fig. 2).

Anisotropy consideration

Table 1c-d summarizes the mean values and the corresponding standard deviation of the physical parameters E 1 , E 2 and ε T for three loading directions on abdomen and proximal forearm, respectively. Note that the mean value and the standard deviation of E 1 (respectively, E 2 ) are of the same order, which demonstrates the high variability and the need for a patient-specific approach to predict the mechanical behavior of the skin in vivo.

In regard to the tests on abdomen Table 1c, we note that the maximum stiffness is obtained the most frequently for 45 • -direction. However, the maximum stiffness is found for 90 • -direction in one test among fourteen, and 0 • -direction in three tests among fourteen. Moreover, the mean value of physical parameters (Table 1c) is highly dependent on the direction of extension. Some authors explain that the direction of maximum stiffness depends on the direction of collagen fibers. These works are attributed to the pretension analysis by punching [START_REF] Langer | On the anatomy and physiology of the skin: I. The cleavability of the cutis[END_REF][START_REF] Cox | The cleavage lines of the skin[END_REF] or by observing 1b for the corresponding values of E 1 , E 2 , ε T and R 2 .

the wrinkles [START_REF] Borges | Relaxed skin tension lines, Z-plasties on scars, and fusiform excision of lesions[END_REF]. Our work agrees with [START_REF] Groves | Quantifying the mechanical properties of skin in vivo and ex vivo to optimise microneedle device design[END_REF], and differs from [START_REF] Cox | The cleavage lines of the skin[END_REF] about the assumption that the maximum stiffness is systematically reached for 0 • -direction (medio-lateral).

A similar analysis on the leg and on the forearm shows that the direction of maximum stiffness is, approximately, evenly distributed among the three directions (0 • , 45 • and 90 • ). For the proximal forearm (Table 1d), the E 2 -parameter is about three times greater for 0 • -direction than for 90 • -direction. Note that the test at 45 • -direction with the smallest value of ε T belongs to the results with the greater values of E 2 . For the front thigh area, the E 1 -parameter is nearly three times larger for 45 • -direction than for 0 • -direction. The differences in the ε T -value reach a factor of two when compared with these two directions of extension. Thus, these results confirm that the human skin behavior is anisotropic and emphasize an inter-individual variability as discussed in the next section.

No histological analysis has been carried out on the mechanically-tested skin tissues in the present study, the link between the anisotropy of the cutaneous tissue and its microstructure of fibers networks is not denied : Indeed, the preferential fibers direction in the skin results from not only functional constraints in the concerned zone but also the particular loading history of the material and its environment.

The results in Fig. 6a reveals a large difference between the mechanical behavior of front thigh versus both back thigh and calf concerning the strain threshold ε T in particular and the transitional phase between low and high stresses. The same is true for E 1 -value which is much smaller at 0 • -direction on front thigh area than at other directions. The stress levels measured on front thigh at 0 • do not exceed 55 kPa, even though they reach 100 kPa on calf and back thigh. These observations are not true at 45 • -direction (Fig. 6b) or at 90 • -direction (Fig. 6c). The 0 • -direction is also the one for which the leg may be stretched out or tucked under, extending the skin area just above the knee as a reserve supply of skin, wrinkled/unwrinkled when straight/bent. This functionality is not necessary for 45 • -and 90 • -directions, and the mechanical behavior is quite similar for close areas.

Inter-individual variability

As shown by [START_REF] Boyer | Dynamic indentation on human skin in vivo: ageing effects[END_REF] and [START_REF] Flynn | Measurement of the force-displacement response of in vivo human skin under a rich set of deformations[END_REF], a large variability of skin mechanical properties has been observed for the same anatomical site of several volunteers regardless of the gender, the age and the body mass index (BMI) [START_REF] Henry | Striae distensae of pregnancy. An in vivo biomechanical evaluation[END_REF]. Similarly, we have noticed a large inter-individual variability on abdomen, on forearm and on leg (Fig. 6). The resulting values of the physical parameters displayed in Fig. 7 exhibit this variability for men and women between 18 and 52 years old. The BMI-values are ranging between 19 and 26.7. No correlation between BMI-value and physical parameters has been noticed. As illustrated in Fig. 7c, the ε T -value tends to increase with the age especially for the men. In addition we observe a downward trend of E 1 -value with age whatever the gender as also reported by [START_REF] Wijn | The alinear viscoelastic properties of human skin in vivo for small deformations[END_REF]. The E 1 -values range between 5 and 35 kPa, the E 2 -values between 500 and 2000 kPa and the ε T -values between 17 and 27%. Fig. 7b do not show any relationship with sex and age. This inter-individual variability also occurrs on the other sites (forearm and leg), whatever the direction.

Conclusion

This article combines the experimental results of uniaxial extension tests performed on different anatomical sites for numerous volunteers and several directions. These tests have been carried out with a servo-controlled homemade extensometer. This ultra-light and portable experimental extensometer is an efficient device to lessen the influence of the surrounding skin on the measurement and to approximate a classical uniaxial tensile test. This enables to obtain stress-strain curves without assuming an a priori behavior law. The phenomenological exponential model is used to determine the mechanical parameters of the intrinsic behavior. A good fit between the exponential equation and experimental measurements is obtained provided that the loading reaches the final phase of skin behavior. By using inverse method, three physical parameters useful for the surgeon can be identified:

• The initial tangent modulus E 1 corresponding to the slope of the stress-strain curve at low strain levels in the initial phase. This modulus is interpreted by various authors as elastin modulus, whereas the collagen fibers are not highly mechanically-stressed by extension and therefore does not yet oppose the deformation.

• The final tangent modulus E 2 obtained for high levels of strain in the final phase of the stress-strain curve when collagen fibers are associated with elastin fibers to resist deformation. This modulus results from the combination of collagen and elastin fibers.

• The strain threshold ε T , as obtained by the intersection of the tangent to the stress-strain curve at the final phase with the abscissa of strain. This threshold value in the transitional phase can be interpreted as a level of strain achieved when the major part of the collagen fibers are aligned in the direction of extension and begin to resist deformation. This strain threshold, which refers to the concept of skin extensibility, is a useful indicator for planning surgery to avoid increasing tensions in the skin tissue and prevent medical complications.

Note that some tests have been analyzed even if the final phase of stress-strain curve has not been reached. In such cases, the corresponding final tangent modulus has been underestimated and consequently the strain threshold has been slightly underestimated too. From the medical point of view, this weak discrepancy in the ε T -value does not present any risk to the patient. If a personalized numerical simulation of surgical operations is aimed, the experimental characterization of the tissue must cover a large range of strain in order to obtain reliable parameters. Moreover, a multiaxial characterization should be undertaken to improve such an approach.

From a mechanical point of view, the tests undertaken within this paper demonstrate the benefit of the direct analysis of measurements and the adaptation of loading to reach the final phase of the mechanical behavior in order to obtain the physical parameters representative of the skin behavior. This allows to combine direct measurements of the mechanical properties soft tissues with patient-specific computational methods developed in computed-assisted surgery.

Figure 2 :

 2 Figure 2: Anatomical areas that were studied. Distal, middle and proximal forearm, abdomen and anterior or posterior thigh and calf.

Figure 3 :

 3 Figure 3: Experimental data (cross symbols) obtained on a right forearm sollicited along 0 • -axis; corresponding exponential model fit (solid line) from Eq. (3) and geometrical description of physical parameters. Obtained values : A = 1.3 kPa, B = 27.4, E 1 = 62.7 kPa, E 2 = 4250 kPa, ε T = 13.9 %. and R 2 = 99.95 %

  , three successive full tests have been performed under the same conditions, on the same volunteer, on the same anatomical area (proximal forearm), with three directions (0 • , 45 • and 90 • ) and ten minutes between each test. The comparison of obtained results has permitted to evaluate the reproducibility of the test with regard to the variability of the skin tissue. The experimental device has been unglued only when the direction of extension has been changed. Except for the 90 • -tests where the final phase has not been reached, E 2 -values steadily decrease with the test number.

Figure 4 :

 4 Figure 4: Reproducibility phenomenon for the same volunteer, on the same forearm zone, with ten minutes between each test and three directions (0 • , 45 • and 90 • ). See Table1afor the corresponding values of E 1 , E 2 , ε T and R 2 .

Figure 5 :

 5 Figure 5: Intra-individual variability observed on the same volunteer for different anatomical zones and extension directions (0 • , 45 • and 90 • ). See Table1bfor the corresponding values of E 1 , E 2 , ε T and R 2 .

  (a) Angle of 0 • from x-axis (5 volunteers). (b) Angle of 45 • from x-axis (4 volunteers). (c) Angle of 90 • from x-axis (4 volunteers).

Figure 6 :

 6 Figure 6: Extension tests on the leg.

Figure 7 :

 7 Figure 7: Physical parameters values for abdomen sites of men (M) and women (W). Influence of age on E 1 (a) ; E 2 (b); ε T (c).

Table 1 :

 1 Parameters values. The symbol * is used for underestimated values. N = number of tests.

	(one volunteer).

Acknowledgements

We thank all the volunteers who participated in the tests and the Centre for Studies and Research on the Integument of the Regional University Hospital Center of Besançon for allowing us to perform the dermis thickness measurements. We are thankful to Eric Joseph and Djamel Remache. This work was supported by European Union FEDER grant from Région de Franche-Comté, France (N • 36381). The authors would like to warmly thank Marine Lévêque for the graphical drawing corresponding to Fig. 2.

Conflict of interest statement

No conflict of interest.


Marilys Pradel
email: marilys.pradel@irstea.fr

LCA APPLIED TO RESIDUAL ORGANIC FERTILIZING MATERIALS -AN OVERVIEW OF SUBSTITUTION METHODOLOGICAL OPTIONS AND QUANTITATIVE SUBSTITUTION RATES

Keywords: state of the art, LCA, substitution, residual organic fertilizing materials, mineral fertilizers

LCA applied to residual organic fertilizing materials. An overview of substitution methodological options and quantitative substitution rates

Marilys Pradel, C. Déchaux

LCA case studies applied to waste management options of organic products face several LCA methodological choices and among them, the multifunctionality question. Indeed, studies dealing with waste management aim to assess the environmental impacts of different organic waste treatments that enhance the organic waste fertilizing and amending properties. This is expressed in LCA studies by a primary function (to treat the waste) and secondary functions (to fertilize or provide an organic amendment to agricultural soils).The induced multifunctionality is often treated by system expansion by subtracting the secondary functions, leading to potentially avoided impacts. When dealing with residual organic fertilizing materials, scientific papers are used to substitute the function "fertilizing with organic wastes" by an alternative function "fertilizing with mineral fertilizers". Occasionally, the function "amending with organic wastes" can be substituted by an alternative function "amending with peat". This paper aims to present the results of a literature review dealing with methodological choices and quantitative rates to substitute residual organic fertilizing materials (digestates, composts issued from various substrate origins: sludge, crop residues, manure…) with mineral fertilizers.

Based on a corpus of 132 peer-reviewed papers dealing with LCA case studies of residual organic fertilizing materials including land spreading, a set of 41 papers detailing system expansion by subtraction (substitution) was selected among this corpus. Those selected papers were used to state how the authors solve the abovementioned multifunctionality problem and to highlight the quantitative rates to substitute residual organic fertilizing materials with mineral fertilizers.

The literature review highlights nine different forms to consider substitution. In most of these papers, substitution is based on the fertilizing potential1 of the residual organic fertilizing material and/or on its utilization rate based on the nutrient availability2 (referred later on as substitution coefficient). Regarding P and K, the substitution coefficients are often close and most of the papers consider a 100% substitution coefficient whatever the organic waste. The substitution coefficient for N ranges 20 to 30% for compost and 40 to 90% for digestate. These range differences depend on the substrate origin, the type of treatment and its performances. When considering the alternative function (i.e. to fertilize with mineral fertilizers), most of reviewed papers only take into account the mineral fertilizer production but do not consider their transport, spreading operation and spreading emissions. The main mineral fertilizers substituted are ammonium nitrate for N, superphosphate or triple superphosphate for P and potassium chloride for K.

Whereas the 41 case studies of the review have different system boundaries, functional units and hypothesis, the impact on the results of the substituted mineral fertilizers can be barely compared between the studies. The impact of substitution is not significant in most of the case studies; however some authors highlight a significant contribution of the substituted mineral fertilizers. This is pointed out regarding mainly climate change, acidification and eutrophication, and generally these impacts are respectively caused by the N 2 O, NH 3 , nitrate and phosphate emissions of the potentially avoided spread mineral fertilizers. Therefore, the way system expansion is led has a decisive influence on results. Impacts on smog, particulate matter formation, toxicity, ecotoxicity, resources, ozone layer depletion and primary energy barely emerged.

The review led to an overview of the LCA habits when considering substitution of residual organic fertilizing materials used as agricultural fertilizers. It pointed out the need to superimpose the boundaries of the substituted activities. The way authors deal with the N, P and K content of digestates has been examined, leading to an overview of substitution practices. It has also been demonstrated that the most often substituted mineral fertilizers reflect the most often sold mineral fertilizers. Thus, conclusions of the literature review led to recommendations for practitioners using LCA in the organic waste management field.

Defined as the amount of N, P and K (g/kg) contained in the residual organic fertilizing material

Defined as the availability of nutrients to be taken up and used by living biological organisms (agricultural crops)Life Cycle Assessment and Other Assessment Tools for WasteManagement and Resource Optimization, June 5-10, 2016, Cetraro, Italy