Also we extend the existence result to the multi-layer case

Electronic Journal of Differential Equations, Vol. 2006(2006), No. 128, pp. 1–23.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE OF SOLUTIONS FOR THE ONE-PHASE AND THE MULTI-LAYER FREE-BOUNDARY PROBLEMS WITH THE

P-LAPLACIAN OPERATOR

IDRISSA LY, DIARAF SECK

Abstract. By considering the p-laplacian operator, we show the existence of a solution to the exterior (resp interior) free boundary problem with non constant Bernoulli free boundary condition. In the second part of this article, we study the existence of solutions to the two-layer shape optimization problem. From a monotonicity result, we show the existence of classical solutions to the two- layer Bernoulli free-boundary problem with nonlinear joining conditions. Also we extend the existence result to the multi-layer case.

1. Introduction

In part I, we study the exterior and interior free-boundary problem with non- constant Bernoulli boundary condition. GivenK a C²-regular bounded domain in R^N and a positive continuous functiong, such thatg(x)> α >0 for all x∈R, we find for the exterior problem a domain Ω and a functionu_Ωsuch that

−∆puΩ= 0 in Ω\K, 1< p <∞ uΩ= 0 on∂Ω

uΩ= 1 on∂K

−∂uΩ

∂νe

=g(x) on∂Ω

(1.1)

Here ∆p denotes the p-Laplace operator, i.e. ∆pu:= div(k∇uk^p−2∇u) andνe is the normal exterior unit of Ω. And for the interior problem, we look for a domain Ω and a functionuΩ such that

−∆_pu_Ω= 0 inK\Ω,¯ 1< p <∞ uΩ= 1 on∂Ω

u_Ω= 0 on∂K

∂uΩ

∂νi

=g(x) on∂Ω

(1.2)

2000Mathematics Subject Classification. 35R35.

Key words and phrases. Bernoulli free boundary problem; starshaped domain;

shape optimization; shape derivative; monotonicity.

c

2006 Texas State University - San Marcos.

Submitted March 6, 2006. Published October 11, 2006.

1

whereνi is the normal interior unit of Ω. The problem arises when a fluid flows in porous medium around an obstacle. In certain industrial problems such as shape optimization, galvanization, we seek to find level lines of the potential function with prescribed pressure.

Inspired by the pioneering work of Beurling, where the notion of sub and su- persolutions in geometrical case is used, Henrot and Shahgohlian [16] studied this problem . They proved that when K ⊂ R^N is a bounded and convex domain a generalization of [14, 15] to the case of the non-constant Bernoulli boundary con- dition.

By combining a variational approach and a sequential method, we establish an existence result by generalizing the problems studied in [25, 26] to the case of the non-constant Bernoulli boundary condition.

The structure of Part I is as follows: In the first part, we present the main result which generalizes results in [25, 26], to the case of non-constant Bernoulli boundary condition. In the second section, we give auxiliary results. The third part deals with the study of the shape optimization problem and the existence of Lagrange multiplierλΩfor the exterior (respectively interior) case. First, we study the existence result for the shape optimization problem for the exterior case: Find

min{J1(w), w∈ O¹},

whereO¹={w⊃K:wis an open set satisfying the-cone property,R

w g^p

c^p(x)dx= V0}, whereV0 is a given positive value. The functionalJ1 is

J1(w) := 1 p

Z

w\K

k∇uwk^pdx, whereuwis a solution to the Dirichlet problem

−∆puw= 0 in w\K, 1< p <∞ u_w= 0 on ∂w

uw= 1 on ∂K.

(1.3)

Second, we study the existence result for the shape optimization problem. For the interior case: Find

min{J2(w), w∈ O²},

whereO²={w⊂K:wis an open set satisfying the-cone property,R

w g^p

c^p(x)dx= W0}, where W0is a given positive value. The functional J2is

J2(w) := 1 p

Z

K\w¯

k∇uwk^pdx, whereuwis a solution to the Dirichlet problem

−∆puw= 0 inK\w,¯ 1< p <∞ u_w= 1 on ∂w

uw= 0 on ∂K.

(1.4)

Next, we obtain an optimality condition for the exterior (respectively interior) case

−∂u

∂νe

= ( p

1−pλ_Ω)^1/p (respectively ∂u

∂νi

= ( p

p−1λ_Ω)^1/p) on∂Ω

Then we conclude this section with a monotonicity result for the exterior (respec- tively interior) case. The last part is the proof of the main result for the exterior (respectively interior) case.

In Part II, we study the multi-layer case. LetD₀^∗andD₁^∗beC²-regular, compact sets inR^N and star shaped with respect to the origin such thatD^∗₁strictly contains D₀^∗. We look for (D, v, u) whereD isC²-regular domain such thatD₀^∗⊂D⊂D¯ ⊂ D₁^∗ andvand uare solutions of the problems

−∆pv= 0 in D₁^∗\D v= 1 on∂D v= 0 on∂D^∗₁

−∆pu= 0 inD\D^∗₀ u= 0 on∂D u= 1 on∂D₀^∗

(1.5)

respectively, and satisfy the nonlinear joining condition

k∇vk^p− k∇uk^p=λ on∂D, λ∈R. (1.6) This joining condition is justified by the method which we are using.

The described above problem appears in several physical situations and can be appropriately interpreted in many industrial applications. In the case p= 2, we refer the reader to [1, 2] and the references therein.

Inspired by the pioneering work of Beurling, where the notion of sub and su- per solutions in the geometrical case is used, Acker et al [7] studied this problem with a more general non linear joining junction. They assumed that D₀^∗ and D^∗₁ are bounded convex domains and D₁^∗ contains D^∗₀ strictly. By considering the p- laplacian operator, we show the existence of a solution to the exterior (resp interior) free boundary problem with non constant Bernoulli free boundary condition. In the second part of this article, we study the existence of solutions to the two-layer shape optimization problem. From a monotonicity result, we show the existence of classical solutions to the two-layer Bernoulli free-boundary problem with nonlinear joining conditions. Also we extend the existence result to the multi-layer case..

They proved there exists a convex C¹ domain D, D₀^∗ ⊂⊂ D ⊂⊂ D^∗₁ which is a classical solution of the two-layer free-boundary problem (1.5)-(1.6).

Using convex domains, Acker [1, 2] proved the existence for multi-layer free boundary problems by using the operator method in the case wherep= 2. Laurence and Stredulinsky [18, 19], use convex domains, when proving an existence result in the casep= 2, N = 2.

Now, by combining a variational approach and a sequential method, we establish an existence result for non necessarily convex domains.

The structure of the Part II is as follows. In the first part, we present the main result. By considering the auxiliary results in the Part I, we study in the second part the shape optimization problem and the existence of Lagrange multiplier function λ. The existence result for the shape optimization problem consists of finding a domainD such that

J(D) = min{J(w), w∈ O},

whereO={w⊂R^N, D₀^∗⊂⊂D⊂⊂D^∗₁, wverifying the-cone property , vol(w) = m0}, wherevol denotes the volume,m0is a fixed value inR^∗+. The functionalJ is defined onO by

J(D) = 1 p

Z

D^∗₁\D

k∇vk^p+1 p

Z

D\D^∗₀

k∇uk^p, 1< p <∞,

wherev anduare solutions of

−∆pv= 0 in D₁^∗\D v= 1 on∂D v= 0 on∂D^∗₁

−∆pu= 0 inD\D^∗₀ u= 0 on∂D u= 1 on∂D₀^∗

(1.7)

respectively. Next, we obtain the joining condition as an optimality condition:

k∇vk^p− k∇uk^p= p

p−1λ_D on∂D, λ_D∈R . (1.8) Then we conclude this section with a monotonicity result. The third section is devoted to the proof of the main result. And the last part is devoted to the extension to the multi-layer case.

2. main results of Part I

For the exterior case, let K be a C²-regular, star-shaped with respect to the origin and bounded domain.

Theorem 2.1. IfΩsolution of the shape optimization problemmin{J1(w), w∈ O¹} isC²-regular domain, then the free boundary problem (1.1) admits a classical unique solution Ω.

For the interior case, letKbe aC²-regular, star-shaped with respect to the origin and bounded domain. Let

α(R_K, p, N) :=











e/R_K ifp=N

|^p−N_p−1|

(_N^p−1₋₁)^N−1N−p −(_N^p−1₋₁)^N−pp−1

1

R_K ifp6=N.

where RK = sup{R >0 : B(o, R)⊂K}, Here cK is the minimal value for which the interior Bernoulli problem (1.2) admits a solution.

Theorem 2.2. If the solutionΩof the shape optimization problemmin{J(w), w∈ O²} isC²-regular, then for all constant c >0 satisfying c≥α(RK, p, N),Ωis the classical solution of the free-boundary problem (1.2). Moreover The constant cK

satisfies0< cK ≤α(RK, p, N).

To prove these theorems we need the following results.

3. Auxiliary results

For the rest of this article, we consider a fixed, closed domainD which contains all the open subsets used.

Letζ be an unitary vector ofR^N, be a real number strictly positive andy be in R^N. We call a cone with vertex y, of directionζ and angle to the vertex and height, the set defined by

C(y, ζ, , ) ={x∈R^N :|x−y| ≤and|(x−y)ζ| ≥ |x−y|cos}.

Let Ω be an open set ofR^N, Ω is said to have the-cone property if for allx∈∂Ω then there exists a directionζ and a strictly positive real numbersuch that

C(y, ζ, , )⊂Ω, for ally∈B(x, )∩Ω.¯

LetK1 andK2 be two compact subsets ofD. Let d(x, K1) = inf

y∈K2

d(x, y), d(x, K2) = inf

y∈K1

d(x, y).

Note that

ρ(K₁, K₂) = sup

x∈K₂

d(x, K₁), ρ(K₂, K₁) = sup

x∈K₁

d(x, K₂).

Let

dH(K1, K2) = max[ρ(K1, K2), ρ(K2, K1)], which is called the Hausdorff distance ofK1 andK2.

Let (Ωn) be a sequence of open subsets ofDand Ω be an open subset ofD. We say that the sequence (Ωn) converges on Ω in the Hausdorff sense and we denote by Ωn

→H Ω if lim_n→+∞dH( ¯D\Ωn,D\Ω) = 0.¯

Let (Ωn) be a sequence of open sets of R^N and Ω be an open set ofR^N. We say that the sequence (Ωn) converges on Ω in the sense of L^p, 1 ≤p <∞ ifχΩ_n

converges onχΩin L^p_loc(R^N), χΩ being the characteristic functions of Ω.

Let (Ωn) be a sequence of open subsets ofDand Ω be an open subset ofD. We say that the sequence (Ωn) converges on Ω in the compact sense if:

(1) Every compact Gsubset of Ω, is included in Ωn fornlarge enough, (2) every compactQsubset of ¯Ω^c, is included in ¯Ω^c_n fornlarge enough.

Lemma 3.1. Let Ω1 andΩ2 be two different domains star-shaped with respect to the origin and bounded. IfΩ¯1⊂Ω¯2then there exists0< t0<1such thatt0Ω2⊂Ω1

andt0∂Ω2∩∂Ω16=∅.

The proof of the above lemma can be found in [22].

Lemma 3.2. Let (f_n)_n∈N be a sequence of functions of L^p(Ω), 1 ≤ p < ∞ and f ∈L^p(Ω). We suppose fn converges onf a.e. and lim_n→∞kfnkp =kfkp. Then we have lim_n→∞kfn−fkp= 0.

For the proof of the above lemma see for example [17].

Lemma 3.3 (Brezis-Lieb). Let (f_n)_n∈N be a bounded sequence inL^p(Ω),1≤p <

∞. We suppose that f_n converges onf a.e., then f ∈L^p(Ω) and kfkp= lim

n→∞(kfn−fkp+kfnkp).

For the proof of the above lemma, see for example [17].

Lemma 3.4. Let (Ωn)_n∈N be a sequence of open sets in R^N having the -cone property, with Ω¯n ⊂F ⊂D,F a compact set and D a ball, then, there exists an open set Ω, included in F, which satisfies the ₂-cone property and a subsequence (Ωn_k)_k∈Nsuch that

χΩ_nk L¹

→χΩ, Ωn_k

→H Ω

∂Ω_nk→^H ∂Ω, Ω¯_nk→^H Ω.¯

The above lemma is a well known result in functional analysis related to shape optimization; its proof can be found for example in [26].

4. Shape optimization result and monotonicity result

For the exterior case, we have the following result, whose proof can be found in [25].

Proposition 4.1. The problem: FindΩ∈ O¹ such that J1(Ω) = min{J1(w), w∈ O¹} admits a solution.

For the interior case, we have the following result, whose proof can be found in [26].

Proposition 4.2. The problem: FindΩ∈ O² such that J2(Ω) = min{J2(w), w∈ O²} admits a solution.

Remark 4.3. Let us define another class of domains:

O0={w⊃K:wis an open set ofR^N.C^k-regular domain, Z

w

g^p

c^p(x)dx=V₀} wherek≥3. It is possible to use the oriented distance, the results in [9, theorems 5.3 5.5,5.6], [10] and the Ascoli theorem to prove the existence of a domain at least of classC^k−1which is minimum for the shape optimization problem.

For the rest of this article, we assume that Ω isC²-regular in order to use the shape derivatives. The next theorems give a necessary condition optimality condi- tion. We follow the approach of Sokolowski-Zolesio [29] to define the shape deriva- tives (see also [28]).

For the exterior case, we have the following result.

Proposition 4.4. If Ωis the solution of the shape optimization problem min{J1(w) :w∈ O¹},

then there exists a Lagrange multiplierλΩ<0 such that−_∂ν^∂u

e = (_1−p^p λΩ)^1pg_c(x)on

∂Ω.

Proof. LetJ₁ be a functional defined onO¹ by J1(w) := 1

p Z

w\K

k∇uwk^pdx, whereu_wis a solution to the Dirichlet problem

−∆puw= 0 in Ω\K, 1< p <∞ u_w= 0 on ∂w

uw= 1 on ∂K.

(4.1)

We use classical Hadamard’s formula to compute the Eulerian derivative of the functionalJ1 at the point Ω in the directionV. A standard computation, see [22], shows

dJ1(Ω;V) = Z

∂Ω

k∇uk^p−2∂u

∂νe

u⁰ds+1 p Z

∂Ω

k∇uk^pV(0).νeds whereu⁰=−_∂ν^∂u

eV(0).νeon∂Ω. This implies dJ(Ω;V) =1−p

p Z

∂Ω

k∇uk^pV(0).νeds.

Let us takeJ(Ω) =R

Ω g^p

c^p(x)dx. Then dJ(Ω;V) =

Z

Ω

div(g^p

c^p(x)V(0))dx= Z

∂Ω

g^p

c^p(x)V(0).νeds.

Ω is optimal then there exists a Lagrange multiplierλ_Ω∈Rsuch thatdJ₁(Ω, V) = λ_ΩdJ(Ω;V). We obtain

Z

∂Ω

(1−p

p k∇uk^p−λΩ

g^p

c^p(x)V(0).ν)ds= 0 for all V.

Then

k∇uk^p= p 1−pλ_Ωg^p

c^p(x) on∂Ω, k∇uk= ( p

1−pλΩ)^1pg

c(x) on∂Ω Since Ω isC²-regular andu= 0 on∂Ω, we get

−∂u

∂νe

= ( p

1−pλΩ)^1pg

c(x) on∂Ω.

For the interior case, we have the following result.

Proposition 4.5. If Ωis the solution of the shape optimization problem min{J₂(w) :w∈ O²},

then there exists a Lagrange multiplier λΩ >0 such that _∂ν^∂u

i = (_p−1^p λΩ)^1pg_c(x) on

∂Ω.

For the proof of the above proposition, we use the same technics as in proposition 4.4. To conclude this section, we state a monotonicity result. For the exterior case, we have the following result, whose proof can be found in [25].

Proposition 4.6. Suppose thatK is star-shaped with respect to the origin. LetΩ₁ andΩ₂be two different solutions to the shape optimization problemmin{J₁(w), w∈ O¹}, star-shaped with respect to the origin such that Ω¯₁⊂Ω¯₂. The mapping which associates to everyΩthe corresponding Lagrange multiplierλ_Ωis strictly increasing i.eλΩ₂> λΩ₁.

For the interior case, we have the following result, whose proof is found in [26].

Proposition 4.7. Suppose thatK is star-shaped with respect to the origin. LetΩ₁ andΩ₂be two different solutions to the shape optimization problemmin{J2(w), w∈ O²}, star-shaped with respect to the origin such that Ω₁⊂Ω₂ and ∂Ω₁∩∂Ω₂6=∅.

The mapping which associates to everyΩthe corresponding Lagrange multiplierλ_Ω is decreasing i.e λ_Ω1 ≥λ_Ω2.

5. Proof of the main results of Part I

We use the preceding properties to prove the main result. Exterior case:

Proof of the Theorem 2.1. We choose a ball B(O, R) centered at the origin and radiusR and a ball B(O, r) such thatB(O, r)⊂K⊂B(O, R). First, we have to look for a solutionu0to the problem

−∆pu= 0 in B_R\Br

u= 0 on∂BR

u= 1 on∂B_r.

(5.1)

The solutionu0 is explicitly determined by

u₀(x) =











lnkxk −lnR

lnr−lnR ifp=N kxk^p−Np−1 −R^p−Np−1

r^p−Np−1 −R^p−Np−1

ifp6=N,

(5.2)

and

k∇u0(x)k=











1

kxk²(lnR−lnr) ifp=N

|^p−N_p−1|kxk^{−N−p+2p−1}

|r^p−Np−1 −R^p−Np−1|

ifp6=N.

In particulark∇u₀k< c on∂B_RforR big enough.

Now consider the problem

−∆_pu= 0 in B_R\K u= 0 on∂BR

u= 1 on∂K.

(5.3)

This problem admits a solution denoted byu_R. This solution is obtained by mini- mizing the functionalJ₁ defined on the Sobolev space

V⁰ ={v∈W₀^1,p(BR), v= 1on∂K}

andJ1(v) =¹_pR

B_R\Kk∇vk^pdx.

Consider the problem

−∆pv= 0 inBR\K v= 0 on∂B_R v=u0 on∂K.

(5.4) It is easy to see that v = ur is a solution to problem (5.4). By the comparison principle [30], we obtain 0 ≤ u0 ≤1 and 0 ≤ uR ≤ 1. On ∂(BR\K), we obtain uR≥u0 and then,uR≥u0in BR\K. Finally, we havek∇uRk ≥ k∇u0kon∂BR. Case p=N. IfR1< R0, we getk∇u0k_|∂BR

0 ≤ k∇u0k_|∂BR

1 then the mapping for allRassociatesk∇u0k_|∂BR is decreasing.

Initially, we choose a radiusR0big enough and we compute k∇u0k_|∂BR

0 and if k∇u0k_|∂Br

0 −c

> δ, whereδ > 0 is a fixed and sufficiently small number. We continue the process by varying R in the increasing sense, we will achieve a step denotedN such that

k∇u0k_|∂BRN −c < δ.

ConsiderON the class of admissible domains defined as follows O_N =

w∈ O:w⊂B_RN, Z

w

g^p c^p =V₀

,

whereV0denotes a fixed positive constant. We look for Ω∈ ON andλΩa real such that

−∆pu= 0 in Ω\K u= 0 on∂Ω u= 1 on∂K

−∂u

∂ν =cΩ on∂Ω

(5.5)

where cΩ = (_p−1^−pλΩ)^p1g_c(x). Applying proposition (4.1), the shape optimization problem min{J1(w), w∈ ON}admits a solution and by proposition 4.4, Ω satisfies the overdetermined boundary condition−^∂u_∂ν =cΩ.

We have Ω∈ ON, then Ω⊂BR_N, according to the lemma 3.1 there existst0<1 such that t0BR_N ⊂Ω, and t0∂BR_N ∩∂Ω6=∅. Let us takex0∈t0∂BR_N ∩∂Ω and setut₀(x) =uR_N(_t^x

0),_t^x

0 ∈BR_N\K. ut₀ satisfies

−∆_pu_t0 = 0 in t₀(B_RN\K) ut₀ = 0 ont0∂BR_N

ut₀ = 1 ont0∂K.

(5.6)

On the other hand, we havet₀B_RN ⊂Ω, let us takew₃=u_|t0B

RN, thenw₃satisfies

−∆pw3= 0 int0BRN\K w3=u_|t0∂BRN ont0∂BR_N

w3= 1 on∂K.

(5.7)

Let us consider the problem

−∆pz= 0 int0BR_N\K z= 0 on∂t0∂BR_N

z=ut₀|∂K on∂K.

(5.8)

It is easy to see that z = u_t0 is a solution to the problem (5.8). And we get 0≤u_t0 ≤1 and 0≤u≤1. On∂(t₀B_RN\K), we haveu_t0 ≤u, by the comparison principle [30], we obtainu_t0 ≤u in(t₀B_RN\K). We have

t→0lim

ut₀(x0−νet)−ut₀(x0)

t ≤lim

t→0

u(x0−νet)−u(x0)

t ,

which is equivalent to

−∂ut₀

∂ν_e(x₀)≤ −∂u

∂ν_e(x₀). This implies

k∇uR_N(x0)k ≤ − ∂u

∂ν_e(x₀) Let us consider Ω = Ω0as the first iteration and

O¹_N =

w∈ O:w⊂Ω0⊂BR_N, Z

w

g^p

c^p =V1 , (V1< V0) whereV₁ denotes a fixed positive constant.

We iterate by looking for Ω1∈ O¹_N andλΩ₁ such that such that

−∆pu1= 0 in Ω1\K u₁= 0 on∂Ω₁

u1= 1 on∂K

−∂u

∂ν =c_Ω1 on∂Ω₁

(5.9)

where cΩ₁ = (_p−1^−pλΩ₁)^1pg_c(x). Applying proposition (4.1), the shape optimization problem min{J2(w), w∈ O_N¹}admits a solution and by proposition 4.4, Ω1satisfies the overdetermined boundary condition −^∂u_∂ν¹ = cΩ₁. We have Ω ∈ O¹_N, then Ω1⊂BR_N, according the lemma 3.1 there existst1<1 such thatt1BR_N ⊂Ω, then t1∂BRN ∩∂Ω16=∅.

Let us take x₁ ∈ t₁∂B_RN ∩∂Ω₁ and set u_t1(x) = u_RN(_t^x

1),_t^x

1 ∈ B_RN\K. u_t1 satisfies

−∆put₁ = 0 in t1(BR_N\K) u_t1 = 0 ont₁∂B_RN

ut₁ = 1 ont1∂K.

(5.10) On the other hand, we havet1BR_N ⊂Ω1, let us takew4=u_|t1BRN, thenw4satisfies

−∆pw₄= 0 int₁B_RN\K w4=u1_|t1∂BR

N

ont1∂BR_N

w4= 1 on∂K.

(5.11) Let us consider the problem

−∆pz= 0 int1BR_N\K z= 0 ont1∂BR_N

z=ut₁|∂K on∂K.

(5.12)

It is easy to see that z=ut₁ is a solution to (5.12). And we get 0≤ut₁ ≤1 and 0≤u1≤1. On∂(t1BR_N\K), we haveut₁ ≤u1, by the comparison principle [30], we obtainut₁≤u1 in(t1BR_N\K). We have

t→0lim

u_t1(x₁−ν_et)−u_t1(x₁)

t ≤lim

t→0

u₁(x₁−ν_et)−u₁(x₁)

t ,

which is equivalent to

−∂ut₁

∂ν_e(x₁)≤ −∂u1

∂ν_e(x₁).

This implies

k∇uR_N(x1)k ≤ −∂u1

∂νe

(x1).

We can continue the process until a step denoted byk such that

−∂u_k

∂νe

(x_k) =c_Ωk

For all s ∈ ∂BR_N, we get k∇u0(s)k ≤ k∇uR_N(s)k then there exists s0 ∈∂BR_N, such thatk∇u0(s0)k> cΩ_k.

The sequence (cΩj)_(0≤j≤k) is strictly decreasing and positive, then (_p−1^−pλΩj)^1p converges on c. Then there exists Ω solution to problem (1.1), the sequence

(Ωj)_(0≤j≤k) gives a good approximation to Ω. The uniqueness of the solution Ω is given by the monotonicity result.

Case p 6= N. If R1 < R0, we get k∇u0k_|∂BR

1 ≥ k∇u0k_|∂BR

0 then the mapping for all R associates k∇u₀k_|∂BR is decreasing. Initially, we choose a radiusR₀ big enough and we computek∇u0k_|∂BR

0 and if

k∇u0k_|∂BR

0 −c

> δ, δ >0 fixed and sufficiently small number. We continue the process by varyingR in the increasing sense, we will achieve a step denotedN such that

k∇u0k_|∂BRN −c

< δ. Here the

reasoning is identical to the casep=N.

Interior case.

Proof of the Theorem 2.2. Let RK = sup{R >0 :B(o, R)⊂K}. Letr >0 such thatB(o, r)⊂B(o, RK). First, we have to look for a solutionu0 to the problem

−∆pu= 0 in BR_K\Br

u= 0 on∂B_RK u= 1 on∂Br.

(5.13)

The solutionu₀ is explicitly determined by

u0(x) =















lnkxk −lnRK

lnr−lnR_K ifp=N

−kxk^p−Np−1 +R

p−N p−1

K

R

p−N p−1

K −r^p−Np−1

ifp6=N,

(5.14)

and

k∇u0(x)k=











1

r(lnRK−lnr) ifp=N

|^p−N_p−1|kxk^−N+1p−1

|r^p−Np−1 −R^p−Np−1|

ifp6=N.

In particular k∇u0k > c on ∂Br for r small enough. Now let us consider the problem

−∆pu= 0 inK\Br

u= 1 on∂B_r u= 0 on∂K.

(5.15) Then problem (5.15) admits a solution denoted byur. This solution is obtained by minimizing the functionalJ defined on the Sobolev space

V⁰ ={v∈W^1,p(K\Br), v= 1on∂Br andv= 0 on∂K}

andJ(v) = ¹_pR

K\Brk∇vk^pdx. Consider the problem

−∆_pv= 0 inB_RK\B_r v= 1 on∂Br

v=u_r on∂B_RK.

(5.16)

It is easy to see thatv =ur is a solution to (5.16). By the comparison principle [30], we obtain 0≤u₀ ≤1 and 0≤u_r ≤1. On∂(B_RK\Br), we obtain u_r ≥u₀ and then,u_r≥u₀ inB_RK\Br. Finally, we havek∇urk ≤ k∇u0kon∂B_r.

Case p=N.

k∇u0k_|∂Br= 1

r(lnRK−lnr) =h(r), ∀r∈]0, RK[.

It is easy to see thath(r) is a strictly decreasing function on ]0,^R_e^K[ and a strictly increasing function on ]^R_e^K, RK[. Then for all r ∈]0, RK[, k∇u0k_|∂Br ≥h(^R_e^K) =

e RK.

(1) For g(x) = e/RK, let δ > 0 be a fixed and sufficiently small number. To initialize we choose r₀ ∈]0,^R_e^K[∪]^R_e^K, R_K[ such that

k∇u0k_|∂Br

0 −c

> δ. To fix ideas let us considerr0∈]0,^R_e^K[. The process will be identical ifr0∈]^R_e^K, RK[.

By varyingrin the increasing sense, we will achieve a step denotednsuch that rn∈]0,RK

e [and

k∇u0k_|∂Brn−c < δ.

ConsiderOn the class of admissible domains defined as follows O_n=

w∈ O, B_rn⊂w, ∂B_rn∩∂w6=∅, and Z

w

g^p

c^p =V₀ ,

whereV₀denotes a fixed positive constant. We look for Ω∈ O_n andλ_Ωsuch that

−∆pu= 0 in K\Ω¯ u= 1 on∂Ω u= 0 on∂K

∂u

∂ν =cΩ on∂Ω

(5.17)

wherecΩ= (_p−1^p λΩ)^1pg_c(x). Applying the proposition (4.2), the shape optimization problem min{J2(w), w∈ On}admits a solution and by proposition 4.5, Ω satisfies the overdetermined boundary condition ^∂u_∂ν = cΩ. Then problem (5.5) admits a solution.

Since Ω∈ On, we haveBr_n⊂Ω,∂Br_n∩∂Ω6=∅ andur_n satisfies

−∆pur_n= 0 in K\Br_n

u_rn= 1 on∂B_rn ur_n= 0 on∂K.

(5.18)

Let us consider the problem

−∆pz= 0 inK\Ω¯ z=ur_n on∂Ω

z= 0 on∂K.

(5.19)

It is easy to see that z=ur_n is a solution to (5.19), and we get 0≤ur_n ≤1 and 0≤u≤1. On∂(K\Ω), we have¯ ur_n≤u. Since∂Ω∩∂Br_n6=∅, letx0∈∂Ω∩∂Br_n, we have

t→0lim

ur_n(x0−νt)−ur_n(x0)

t ≤lim

t→0

u(x0−νt)−u(x0)

t ,

This is equivalent to

∂ur_n

∂ν (x₀)≥ ∂u

∂ν(x₀) =c_Ω.

Let Ω = Ω0 as the first iteration. We iterate by looking for Ω1∈ O¹_n such that

−∆pu1= 0 in K\Ω¯1

u₁= 1 on∂Ω₁ u1= 0 on∂K

∂u₁

∂ν =c_Ω1 on∂Ω₁.

(5.20)

wherec_Ω1 = (_p−1^p λ_Ω1)^1pg_c(x), and O¹_n=

w∈ O: Ω₀⊂w, ∂w∩∂B_rn 6=∅ Z

w

g^p

c^p =V₁ ,

where V₁ is a strictly positive constant and V₀ < V₁. By the same reasoning as above, we conclude that

∂ur_n

∂ν (x1)≥∂u1

∂ν (x1) =cΩ₁

where x1 ∈ ∂Ω1∩∂Br_n. We can continue the process until a step denoted by k which we will determine and we have

∂u_rn

∂ν (xk)≥ ∂u_k

∂ν (xk) =cΩ_k and xk∈∂Ωk∩∂Br_n.

Finally, we have constructed an increasing sequence of domain solutions: Ω₀ ⊂ Ω₁⊂Ω₂· · · ⊂Ω_k. By the monotonicity result, we havec_Ω0≥c_Ω1 ≥c_Ω2· · · ≥c_Ωk. Sincek∇urnk ≤ k∇u0kon∂B_rn,kis chosen as follows: At each points₀∈∂B_rn, we have

cΩ_k≤ ∂u0

∂ν (s0)≤cΩk−1. Then we obtain the inequality

cΩ_k− e

R_K ≤∂u0

∂ν (s0)− e

R_K ≤cΩ_k−1− e

R_K. (5.21)

The sequence (cΩ_j)_(0≤j≤k) is decreasing and strictly positive, then it converges on l. Passing to the limit in (5.21), we obtain thatl=_R^e

K and there exists Ω solution to problem (1.2). The sequence (Ω_j)_(0≤j≤k)gives a good approximation to Ω. The uniqueness of the solution Ω is given by the monotonicity result.

(2) Forg(x)>_R^e

K andr∈]0,^R_e^K[∪]^R_e^K, R_K[. We have the same reasoning and we show that the problem (1.2) admits a solution.

Case p6=N. Here the reasoning is identical to the case p=N. We note that k∇u0k_|∂Brn =

p−N

p−1

1 1−(_R^r

K)^N−pp−1 1 r =h(r)

and h is strictly increasing on ](_N^p−1₋₁)^N−pp−1R_K, R_K[ and a strictly decreasing on ]0,(_N−1^p−1)^N−pp−1RK[. For all

g(x)≥ |p−N

p−1| 1

|(_N−1^p−1)^N−1N−p−(_N−1^p−1)^N−pp−1| 1 RK

=h((p−1

N−1)^N−pp−1RK), problem (1.1) admits a solution.

It is easy to have, 0< cK ≤α(RK, p, N). IfKis a ball of radius R, an explicit computation gives cK = α(R, p, N) and for all 0< c < cK problem (1.2) has no

solution.

6. Main result of Part II

LetD₀^∗ andD^∗₁ be C²-regular, compact sets inR^N and starshaped with respect to the origin such thatD^∗₁ strictly containsD^∗₀. We want to find (v, u) solutions of

−∆pv= 0 in D₁^∗\D v= 1 on∂D v= 0 on∂D^∗₁

−∆pu= 0 inD\D^∗₀ u= 0 on∂D u= 1 on∂D₀^∗

(6.1)

respectively, and satisfy the non linear joining condition

k∇vk^p− k∇uk^p=λ on∂D, (6.2) whereλis a given real∈R.

Theorem 6.1. Let D₀^∗ andD^∗₁ beC²-regular, compact sets inR^N and starshaped with respect to the origin such that D₁^∗ strictly contains D^∗₀. One supposes in add that there is R0= sup{R >0 :B(O, R)∈D₁^∗}andD₀^∗∈B(O, R0) IfD,C²-regular domain solution to the shape optimization problem min{J(w), w ∈ O} such that D₀^∗ ⊂⊂ D ⊂⊂ D^∗₁, then D is a solution of the two-layer free boundary problem (6.1)-(6.2).

To prove the main result of the Part II, we need to establish some results such as shape optimization and monotonicity results.

7. Shape optimization and monotonicity result

Theorem 7.1. The problem: FindD∈ O such that J(D) = min{J(w), w∈ O} admits a solution

Proof. LetEbe a functional defined on W^1,p(D₁^∗)×W^1,p(D^∗₁) by E(˜v,u) =˜ 1

p Z

D^∗₁

k∇˜vk^p+1 p

Z

D^∗₁

k∇˜uk^p, 1< p <∞,

where ˜vis the extension ofv inD0 and ˜uis the extension by 0 inD^∗₁\Dofu. And v anduare solutions of

−∆pv= 0 in D₁^∗\D v= 1 on∂D v= 0 on∂D^∗₁

−∆pu= 0 inD\D^∗₀ u= 0 on∂D u= 1 on∂D₀^∗

(7.1)

Let J(D) :=E(˜v,u). It is easy to see that˜ J(D)≥0, this implies inf{J(w), w ∈ O}>−∞. Let α= inf{J(w), w∈ O}. Then there exists a minimizing sequence (Dn)_(n∈N)⊂ O such that J(Dn) converges toα. Since the sequence is bounded, there exists a compact setF such thatD₀^∗⊂⊂D¯n ⊂F ⊂⊂D^∗₁. By the lemma 3.4, there exists a subsequence (Dn_k)_(nk∈N) and D verifying the -cone property such that

χD_nk L¹

→χD andDn_k

→H D.

It is easy to see the sequence (v_n, u_n) is bounded inW^1,p(D₁^∗) see [25, 27]. Since W^1,p(D^∗₁) is a reflexive space, there exists a subsequence (v_nk, u_nk) and (v^∗, u^∗) such

that vn_k converges weakly on v^∗ in W^1,p(D₁^∗) and un_k converges weakly on u^∗ in W^1,p(D^∗₁). The norm is lower semi continuous for the weak topology inW^1,p(D^∗₁), then we have

1 p

Z

D^∗₁\D

k∇v^∗k^p+1 p

Z

D\D^∗₀

k∇u^∗k^p

≥lim inf(1 p

Z

D^∗₁\D_nk

k∇vnkk^p+1 p

Z

D_nk\D^∗₀

k∇unkk^p).

From the above we getJ(D)≥α, thenJ(D) = min{J(w), w∈ O}.

Remark 7.2. On the one hand, see [25] [26], it is easy to verify thatv=v^∗, u=u^∗ andv^∗, u^∗ satisfy

−∆pv^∗= 0 inD⁰(D₁^∗\D) v^∗= 1 on∂D v^∗= 0 on∂D^∗₁

−∆pu^∗= 0 inD⁰(D\D^∗₀) u^∗= 0 on∂D u^∗= 1 on∂D₀^∗

respectively. On the other hand, we have regularity for v, u as solutions to (7.1);

see [11, 21, 31].

Remark 7.3. The remark 4.1 can be stated for the multilayer case. The theorem 4.3 and the lemma 4.4 proved in [26] are valid too for the multilayer case.

For the rest of this article, we assume thatD isC²-regular domain in order to use the shape derivatives. We follow the approach of Sokolowski-Zolesio to define the shape derivatives [29] (see also [28]).

Theorem 7.4. IfDis a solution to the shape optimization problemmin{J(w), w∈ O}, then there exists a Lagrange multiplier functionλD∈Rsuch that

k∇vk^p− k∇uk^p= p

p−1λD on ∂D. (7.2)

Proof of the theorem 7.4.

J(D) = 1 p

Z

D^∗₁\D

k∇vk^p+1 p

Z

D\D^∗₀

k∇uk^p, 1< p <∞, wherev anduare solutions of

−∆pv= 0 in D₁^∗\D v= 1 on∂D v= 0 on∂D^∗₁

−∆pu= 0 inD\D^∗₀ u= 0 on∂D u= 1 on∂D₀^∗

(7.3)

A standard computation , see [22], shows the Euleurian derivative of the functional J at the pointD in the directionV isdJ(D, V) =A+B, where

A= Z

D^∗₁\D

k∇vk^p−2∇v⁰∇vdx+1 p

Z

D^∗₁\D

div(k∇vk^p)V(0))dx B =

Z

D\D^∗₀

k∇uk^p−2∇u⁰∇udx+1 p

Z

D\D₀^∗

div(k∇uk^p)V(0))dx

By the Green formula, we have A=−

Z

D^∗₁\D

div(k∇vk^p−2∇v)v⁰dx+1 p

Z

∂(D^∗₁\D)

k∇vk^p−2 ∂v

∂ν₁v⁰ds +1

p Z

∂(D^∗₁\D)

k∇vk^pV(0).ν1ds.

InD^∗₁\D, we have div(k∇vk^p−2∇v) = 0, then A= 1

p Z

∂(D₁^∗\D)

k∇vk^p−2∂v

∂ν₁v⁰ds+1 p Z

∂(D^∗₁\D)

k∇vk^pV(0).ν1ds.

By the same reasoning, we obtain B= 1

p Z

∂(D\D^∗₀)

k∇uk^p−2∂u

∂ν2

u⁰ds+1 p

Z

∂(D\D₀^∗)

k∇uk^pV(0).ν2ds.

Let us takeν1=−ν2 whereν2 is the exterior normal unit toD. By the computa- tions, see [22], we obtain

u⁰=−∂u

∂ν2

V(0).ν2 on∂D, v⁰ =−∂v

∂ν1

V(0).ν1 on∂D.

This implies

A=− Z

∂D

k∇vk^pV(0).ν1ds+1 p

Z

∂D

k∇vk^pV(0)ν1ds, B=−

Z

∂D

k∇uk^pV(0).ν₂ds+1 p

Z

∂D

k∇uk^pV(0).ν₂dx . Then we have

dJ(D, V) = 1−p p

Z

∂D

(−k∇vk^p+k∇uk^p)V(0).ν2ds.

Let us takeJ2(D) =R

Ddx=V0, then dJ2(D, V) =

Z

D

div(V(0))dx= Z

∂D

V(0).ν2ds.

There exists a Lagrange multiplier λ_D ∈ R such that dJ(D, V) =λ_DdJ₂(D, V).

We obtain Z

∂D

[1−p

p (−k∇vk^p+k∇uk^p)−λ_D)]V(0).ν₂ds= 0 for all V,

thenk∇vk^p− k∇uk^p= _p−1^p λ_D on∂D.

Remark 7.5. The consequence (D, v, u) in theorems (7.1) and (7.4) satisfies

−∆pv= 0 in D₁^∗\D v= 1 on∂D v= 0 on∂D^∗₁

−∆pu= 0 inD\D^∗₀ u= 0 on∂D u= 1 on∂D₀^∗ and satisfy the nonlinear joining condition

k∇vk^p− k∇uk^p= p

p−1λ_D on∂D.

To conclude this section, we state a monotonicity result, in the following sense.

Theorem 7.6. Let D₀^∗ andD^∗₁ beC²-regular, compact sets inR^N and starshaped with respect to the origin such thatD^∗₁ strictly containsD₀^∗. LetD1 andD2be two different solutions to the shape optimization problemmin{J(w), w∈ O}starshaped with respect to the origin such thatD₁⊂D₂ and∂D₁∩∂D₂6= ∅thenλ_D1 ≥λ_D2. Proof. For anyi∈ {1,2}, ifDi is the solution to the shape optimization problem, we have (vi, ui) satisfy the problem

−∆pvi= 0 inD₁^∗\Di

vi= 1 on∂Di

vi= 0 on∂D₁^∗

−∆pui= 0 inDi\D₀^∗ ui= 0 on∂Di

ui= 1 on∂D₀^∗ and the nonlinear joining condition

k∇vik^p− k∇uik^p= p

p−1λ_Di on∂D_i, λ_Di∈R.

Consider the problem

−∆pv3= 0 inD^∗₁\D2

v₃=v₁ on∂D₂ v₃= 0 on∂D^∗₁

(7.4) It is easy to see that v3 = v1 is a solution to (7.4). We get 0 ≤ v2 ≤ 1 and 0≤v1≤1. On∂(D^∗₁\D2), we havev2≥v1. By the comparison principle [30], we obtainv2≥v1in D₁^∗\D2.

Letx0∈∂D1∩∂D2and ν be the exterior unit normal inx0, then we get v₂(x₀+νh)−v₂(x₀)

h ≥ v₁(x₀+νh)−v₁(x₀)

h ,

By passing to the limit,

h→0lim

v2(x0+νh)−v2(x0)

h ≥ lim

h→0

v1(x0+νh)−v1(x0)

h ,

which implies

∂v2

∂ν (x0)≥ ∂v1

∂ν (x0).

It suffices to remark that ^∂v_∂νⁱ(x0)<0 (i= 1,2) to conclude that

k∇v1(x0)k^p≥ k∇v2(x0)k^p. (7.5) Consider the problem

−∆pu3= 0 inD1\D^∗₀ u₃=u₂ on∂D₁

u₃= 1 on∂D₀^∗

(7.6) It is easy to see that u3 = u2 is a solution to (7.6 ). We get 0 ≤ u1 ≤ 1 and 0≤u2≤1. On∂(D1\D₀^∗), we haveu2≥u1. By the comparison principle [30], we obtainu2≥u1 inD1\D₀^∗.

Letx0∈∂D1∩∂D2, then

u₂(x₀−νh)−u₂(x₀)

h ≥ u₁(x₀−νh)−u₁(x₀)

h .

By passing to the limit,

h→0lim

u2(x0−νh)−u2(x0)

h ≥ lim

h→0

u1(x0−νh)−u1(x0) h