A free boundary problem with multiple boundaries for a general class of anisotropic equations

Applied Mathematics and Computation 362 (2019) 124551

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A free boundary problem with multiple boundaries for a general class of anisotropic equationsR L. Barbu a, C. Enache b,∗ a

Ovidius University, Department of Mathematics, 124 Mamaia Blvd, Constant¸ a 900527, Romania American University of Sharjah, Department of Mathematics and Statistics, University City Road, P.O. Box, Sharjah 26666, United Arab Emirates

b

a r t i c l e

i n f o

a b s t r a c t

Keywords: Anisotropic equations Maximum principles Overdetermined problems Wulff shapes

In this paper we are going to investigate a free boundary problem for a class of anisotropic equations on a multiply-connected domain  ⊂ RN , N ≥ 2. Our aim is to show that if the problem admits a solution in a suitable weak sense, then the underlying domain  is a Wulff shaped ring. This result represents the anisotropic extension of a result obtained by Payne and Philippin (1991) [13]. For the proof, we make use of a maximum principle for an appropriate P-function, a Rellich type identity, an anisotropic form of Minkowski inequality for convex sets and some geometric arguments involving the anisotropic mean curvatures of the free boundaries. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Let F : RN → [0, ∞ ), N ≥ 2, be a positive homogeneous function of degree 1, with the following properties:





3,α F ∈ Cloc RN {0} , with

α ∈ (0, 1 ),  F (ξ ) > 0 for any ξ ∈ R {0} . 

N

(1.1)

Obviously, since F is homogeneous and defined at the origin, we have F (0 ) = 0. Furthermore, we also assume that F satisfies the following convexity condition:

WF (1 ) := {x ∈ RN ; F (x ) < 1} is strictly convex. 3 ,α Next, let G ∈ Cloc ( 0, ∞ )



(1.2)

C 1 [0, ∞ ) be a function such that

G ( 0 ) = G ( 0 ) = 0,

G(t ) > 0, G (t ) > 0, G (t ) > 0 for any t ∈ (0, ∞ ). As in Cozzi et al. [8], we assume that either conditions (I) or (II) is verified, where: (I) There exists p > 1, k ∈ [0, 1 ), γ > 0,  > 0 such that, for any ξ ∈ RN \ {0}, ζ ∈ RN , we have R ∗

This paper is dedicated to Prof. Gérard A. Philippin, on the occasion of his 75th birthday. Corresponding author. E-mail addresses: [email protected] (L. Barbu), [email protected] (C. Enache).

https://doi.org/10.1016/j.amc.2019.06.065 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

(1.3)

2

L. Barbu and C. Enache / Applied Mathematics and Computation 362 (2019) 124551

[Hess(G ◦ F )(ξ )]i j ζi ζ j ≥ γ (k + |ξ | ) p−2 |ζ | , 2

N    [Hess(G ◦ F )(ξ )]i j  ≤ (k + |ξ | ) p−2 .

(1.4)

i, j=1

3 ,γ

(II) The composition G ◦ F is of class Cloc (RN ) and for any R > 0 there exists a positive constant ϱ > 0 such that, for any ξ , ζ ∈ RN , with |ξ | ≤ R, we have

[Hess(G ◦ F )(ξ )]i j ζi ζ j ≥  |ζ | . 2

(1.5)

Now, let us introduce the following quasilinear anisotropic operator:

Qu :=

N  i=1

 ∂   G (F (∇ u ) )Fξi (∇ u ) . ∂ xi

(1.6)

p We immediately notice that Q coincides with the anisotropic p-Laplace operator in the particular case G(t ) = t /p, while

when G(t ) = 1 + t 2 − 1, Q represents the so-called anisotropic mean curvature operator. Let k , k = 1, . . . , m, be m strictly convex regions in RN strictly contained in a bounded domain 0 ⊂ RN . In this paper we are concerned with the following free boundary problem:

⎧ m ⎨Qu = −2 in  := 0 \ ∪ k , k=1 ⎩u = 0, F (∇ u ) = a0 = const. > 0 on ∂ 0 , u = ck , F (∇ u ) = ak = const. < a0 on ∂ k , k = 1, . . . , m,

(1.7)

where the boundaries ∂ k , k = 0, . . . , m, are assumed to be of class C2,α . We note that in problem (1.7) the values of the positive constants ck are not given, but they are determined from the further conditions:

∂ k

G (F (∇ u ) )F (ν )dσ = 2|k |, k = 1, . . . , m,

(1.8)

where |k | is the N-volume of k and ν is the inward unit normal to ∂ k . We also note that the assumptions on F and G considered in this paper guarantee that u ∈ C 1,α () ∩ C 3,α ( \ C ), where C := {x ∈ ; ∇ u(x ) = 0} (see Cozzi et al. [8], Propositions 3.1 and 3.2). Therefore, the partial derivatives of u(x), up to third order, are all well defined on  \ C. The main result of this paper states the following: Theorem 1.1. If problem (1.7) and (1.8) has a weak solution u ∈ W 1,p () ∩ C 1 () satisfying

∇ u = 0 in ,

(1.9)

then 0 contains a single hole 1 and, up to a translation, 0 and 1 are concentric Wulff shapes whose radii are given by R0 = N2 G (a0 ), R1 = N2 G (a1 ). Moreover, the solution u(x) is given explicitly by the following formula:

u (x ) =

R0 F ◦ (x )

G

 2s  N

ds

for any

x ∈ ,

where G is the inverse of the map t → G (t) on (0, ∞) and c1 =

(1.10)  R0 R1

G ( 2Ns )ds.

The outline of this paper is as follows. In Section 2 we briefly give the definition of the Wulff shape of F and mention some properties of F and the anisotropic mean curvature (or F-mean curvature) of the level surfaces of u(x). More precisely, we will first define the anisotropic mean curvature of a level surface of u(x) and give a representation of the anisotropic operator Q in terms of the anisotropic mean curvature of the underlying level surface of u. Furthermore, an anisotropic version of Minkowski inequality and a Newton type inequality for symmetric matrices will be also recalled. The proof of the main result is given in Section 3. The main ingredients of the proof are as follows: a maximum principle for an appropriate functional combination of u(x) and ∇ u(x), i.e. a P–function in the sense of Payne (see the book of Sperb [14]), a Rellich type identity and some properties of the given P-function and geometric arguments involving the anisotropic curvature of the free boundary. We note that this idea of proof, based on maximum principles for P-functions, was developed first by Weinberger in [15] and used recently to investigate other anisotropic problems by several authors (see Wang and Xia [16], Lv et al. [12] or Barbu and Enache [1–3]). For an account on other similar results for anisotropic free boundary problems, using different method of proof, we refer the reader to Cianchi and Salani [7], the survey paper of Farina and Valdinoci [9], respectively the recent articles of Bianchini and Ciraolo [5] and Bianchini et al. [6]. Finally, for convenience, notice that throughout this paper the comma is used to indicate differentiation and the summation from 1 to N is understood on repeated indices. Moreover, we adopt the following notations:

L. Barbu and C. Enache / Applied Mathematics and Computation 362 (2019) 124551

F = F ( ∇ u ),

Fi = Fξi ,

G = G (F (∇ u )),

3

G = G (F (∇ u )),

∂2 (G ◦ F )(∇ u )(x ) = G Fi j + G Fi Fj , ∂ ξi ∂ ξ j ∂3 ai jk (∇ u )(x ) : = xsx (G ◦ F )(∇ u )(x ). ∂ ξi ∂ ξ j ∂ ξk ai j (∇ u )(x ) : =

(1.11)

where i, j, k ∈ {1, . . . , N }. 2. Preliminaries In this section we are going to recall some properties which play an important role in the proof of our result. First, let F◦ be the dual norm of F, that is

F ◦ (x ) = sup

x , ξ 

ξ =0 F (ξ )

for any x ∈ RN ,

(2.1)

also called the polar of F. We denote by WF (1) and WF ◦ (1 ) the unitary balls with respect to the F, respectively F◦ , that is

WF ◦ (1 ) := {x ∈ RN ; F ◦ (x ) < 1},

WF (1 ) := {x ∈ RN ; F (x ) < 1}.

(2.2)

In general, for r > 0, we say that WF ◦ (r ) := {x ∈ < r} is the Wulff shape (or equilibrium crystal shape) of F, of radius rand center 0. A set D ⊂ RN is a Wulff shape of F if there exist r > 0 and x0 ∈ RN such that D = {x ∈ RN ; F ◦ (x − x0 ) < r}. Further details about Wulff shapes may be found in Ferone and Kawohl [11] and Belloni et al. [4] The following two lemmas come with some properties satisfied by F and its polar F◦ , useful in our later computations: RN ; F ◦ ( x )

Lemma 2.1. (see Cianchi and Salani [7], Lemma 3.1): If F ∈ C 1 (RN \ {0} ) and WF (1) is strictly convex, then

F (∇ F ◦ (x )) = 1 for any x = 0, F ◦ (∇ F (ξ )) = 1, for any

ξ = 0.

Lemma 2.2. (see Farina and Valdinoci [10], Appendix, Lemma 3) If F ∈ 1, then we have

Fi (ξ )ξi = F (ξ ),

Fi j (ξ )ξi = 0,

C3



RN

(2.3)

 \ {0} is a positive homogenous function of degree

Fi jk (ξ )ξi = −Fjk (ξ ),

(2.4)

for any ξ ∈ RN \ {0}, i, j, k ∈ {1, ..., N }. Next, following Wang and Xia [17], we naturally define the anisotropic mean curvature of ∂ , denote by KF (∂ ), as follows:

KF (∂ ) := −div



 ∇ξ F ◦ ν = −Fi j νij ,

(2.5)

(ν 1 , . . . , ν N )

where ν = is the inward unit normal to . The next lemma gives a geometric representation of the anisotropic operator Q, along a level surface of u(x), in terms of the anisotropic mean curvature of the underlying level surface. More precisely, we have the following identity: Lemma 2.3. (see Wang and Xia [17], Lemma 2.4): If u(x) is a C2 function with a regular level surface St := {x ∈ ; u(x ) = t } and KF (St ) is the anisotropic mean curvature of the level surface St , then

Qu(x ) = G (F (∇ u ))Fi Fj ui j − G (F (∇ u ))KF (St ) on St .

(2.6)

The following result represents an anisotropic version of the Minkowski inequality for convex sets: Lemma 2.4. (see Bianchini et al. [6], Proposition 2.9): If F is a norm of RN of class C 2 (RN \ {0} ) and D ⊂ RN is a bounded strictly convex domain, with boundary of class C2,α , then

∂D

KF (∂ D )F ◦ ν dσ ≤

N−1 PF (D )2 , N

(2.7)

where PF (D) is the anisotropic perimeter of D, that is PF (D) := ∫∂ D F◦ν dσ . Finally, the following version of Newton’s inequality for symmetric matrices will also play a role in the proof of the main result: Lemma 2.5. (see Cianchi and Salani [7], Lemma 3.2) Let B and C be symmetric matrices in RN×N and B be positive semidefinite. Set W = BC. Then

  1 2 [T r (W )] ≤ T r W 2 . N

(2.8)

Moreover, if Tr(W) = 0 and equality holds in (2.8), then

W =

T r (W ) IN . N

(2.9)

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L. Barbu and C. Enache / Applied Mathematics and Computation 362 (2019) 124551

3. The proof of Theorem 1.1 Let us first consider the following P-function:

P (u; x ) := G (F (∇ u(x ) ) )F − G(F (∇ u(x ) ) ) +

2 u ( x ), N

(3.1)

where u(x) is a solution to Eq. (1.7)1 . We then have the following maximum principle: Lemma 3.1. The auxiliary function P(u; x) defined in (3.1) takes its maximum value on ∂ , unles P ≡ const. in . Proof of Theorem 1.1. The main idea of the proof is the construction of an elliptic second order differential inequality for the auxiliary function P(u; x). The conclusion of the theorem will then follow immediately, as a direct consequence of Hopf’s first maximum principles (see Sperb [14]). First of all we remind that u(x) is in C 1,α () ∩ C 3,α (), so the partial derivatives of u(x), up to third order, are well defined on . We then compute successively in  the followings:

Pi = G F Fk uki +

2 ui , N

(3.2)

Pi j = G(3) F Fl Fk uki ul j + G Fl Fk ul j uki + G F Fkl ul j uki + G F Fk uki j +

2 ui j . N

(3.3)

Next, making use of notation (1.11), we can rewrite (1.7)1 as follows:





ai j ui j = G Fi j + G Fi Fj ui j = −2.

(3.4)

From Lemma 2.2 we have

Fi ui = F ,

Fi j u j = 0,

Fi jk ui = −Fjk ,

(3.5)

for i, j, k ∈ {1, . . . , N}. Therefore, from (3.3)–(3.5), we get

ai j Pi j = G(3) G F Fl Fk Fi j uki ul j + G(3) G F Fl Fk Fi Fj uki ul j + G G Fi j Fl Fk ul j uki + (G )2 Fk Fl Fi Fj ul j uki + G G F Fi j Flk ul j uki + (G )2 F Fi Fj Flk ul j uik + G F Fk ai j uki j −

4 . N

(3.6)

On the other hand, from (3.2) one may easily derive the following identities:

Fk uki =

2 ui 2 ui Pi − =− + ..., G F N G F N G F

(3.7)

2 Fi ui 2 1 Pi Fi − =− + ..., G F N G F N G

(3.8)

Fi Fk uki =

where here and in the remaining part of this paper dots stand for terms containing Pi . Moreover, making use of (3.8) in (3.4), we obtain

G Fi j ui j = −2 − G Fi Fj ui j = −2 +

2 + .... N

(3.9)

Differentiating (3.4) with respect to xk , we also have

0 = 2G Fil Fj ulk ui j + G Fl Fi j ulk ui j + G Fi jl ulk ui j + G(3) Fi Fl Fj ulk ui j + ai j ui jk .

(3.10)

Inserting now aij uijk from (3.10) into (3.6) we obtain

ai j Pi j = G(3) G F Fl Fk Fi j uki ul j + G(3) G F Fl Fk Fi Fj uki ul j + G G Fi j Fl Fk ul j uki + (G )2 Fk Fl Fi Fj ul j uki + G G F Fi j Flk ul j uki + (G )2 F Fi Fj Flk ul j uik − (G )2 F Fk Fl Fi j ulk ui j − G G F Fk Fi jl ulk ui j − G(3) G F Fk Fl Fi Fj ulk ui j − 2(G )2 F Fk Fj Fil ulk ui j −

4 . N

(3.11)

On the other hand, from (3.5)3 and (3.9) we have

G Fi jl ui j ul = −G Fi j ui j = 2 −

2 + .... N

Making use of (3.7)–(3.9) and (3.12) in (3.11), after some simplifications we get

(3.12)

L. Barbu and C. Enache / Applied Mathematics and Computation 362 (2019) 124551



5



2 4   2 −2 + G + ... + G G F Fi j Fkl uki ul j N NG  −2u      2 1  −2 2 l − G G F + ... F u − G F + ... −2 + + ... i j i jl NG F G N ( p − 1 )G N 2  2 G F  2  4 2 4 = G G F Fi j Fkl uki ul j + 2 − − −2 + −2 N N N N G N N   2 G F 2 = G G F Fi j Fkl uki ul j + − 2 + .... N G N

ai j Pi j = −

(3.13)

Next, making use of (3.4), (3.7) and (3.8), we evaluate separately the term (G )2 Fij Fkl ulj uki , as follows:

  2 G



Fi j Fkl ul j uki = ai j − G Fi Fj





akl − G Fk Fl ul j uki

 2

= ai j akl ul j uki + G

Fi Fj Fk Fl ul j uki − 2ai j G Fk Fl ul j uki



2 ui 4 − 2G − + ... N G F N2 4 = ai j akl ul j uki − 2 + .... N = ai j akl ul j uki +

 2 u j −

N G F

+ ...





G Fi j + G Fi Fj + ... (3.14)

Inserting now (3.14) into (3.13), we obtain

ai j Pi j + ... =





4 G F ai j akl ul j uki − . G N

(3.15)





Next, in order to evaluate the term aij akl ulj uki in (3.15), we denote by W = wi j the matrix whose coefficients are given by wi j := aik uk j , i, j ∈ {1, . . . , N}. Then, we have

T r (W ) = ai j ui j = Qu = −2

in .

T r (W 2 ) = ai j akl ul j uki

(3.16)

Making use of Lemma 2.5 for B = (ai j ), C = (ui j ) and W = BC, we obtain the following estimate:

ai j akl ul j uki =

N 

1 4 2 [T r (W )] = , N N

w2ik = T r (W 2 ) ≥

i,k=1

Using now (3.17) in (3.15) we obtain

ai j Pi j + · · · =



(3.17)



4 G F ai j akl ul j uki − ≥ 0. G N

(3.18)

Since ∇ u = 0 in , the operator aij Pij is strongly elliptic in  and the result follows from Hopf’s first maximum principles (see, for instance, Sperb [14], Theorem 2.4). The proof of Lemma 3.1 is thus achieved.  In what follows we establish a Rellich type identity for P. More precisely, we have Lemma 3.2. The auxiliary function P verifies the following identity:



P (u; x ) dx = P (∂ 0 )|0 | −

m 

P (∂ k )|k |.

(3.19)

k=1

Proof of Lemma 3.2. By divergence theorem we derive

∂

G(F (∇ u ) )x, ν dσ =

Now, we compute



div(G(F (∇ u ) )x )dx =



div(G(F (∇ u ) )x )dx.

(3.20)

[NG(F (∇ u ) ) + xi (G(F (∇ u ) ))i ]dx

    =N G(F (∇ u ) )dx + xi G (F (∇ u ) )Fk (∇ u )ui − ui G (F (∇ u ) )Fk dx k k  

 =N G(F (∇ u ) )dx + x, ∇ uG (F (∇ u ))Fk (∇ u )νk dσ  ∂

  − G (F (∇ u ))Fk (∇ u )uk dx − x, ∇ udiv G (F (∇ u ) )Fξ (∇ u ) dx 





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L. Barbu and C. Enache / Applied Mathematics and Computation 362 (2019) 124551

=

 NG(F (∇ u ) ) − G (F (∇ u ) ) dx + x, νG (F (∇ u ))F (∇ u )dσ  ∂

+ 2 x, ∇ udx.





On the other hand, we have



x, ∇ u dx = −N



u dx +

∂

ux, νdσ .

(3.21)

(3.22)

Combining (3.20)–(3.22) we obtain



 NG(F (∇ u ) ) − G (F (∇ u ) ) dx − 2N udx + 2 ux, νdσ   ∂

  = G(F (∇ u ) ) − G (F (∇ u ) ) x, νdσ ,



(3.23)

∂

or





−NG(F (∇ u ) ) + G (F (∇ u ) ) + 2Nu dx + 

= P (u; x )x, νdσ .

2 N

−2

 ∂

ux, νdσ (3.24)

∂

Multiplying (1.7)1 by u and integrating the result over  we get

−





div G (F (∇ u ) )Fξ (∇ u ) udx 

=− G (F (∇ u ) )Fi (∇ u )ui dx + G (F (∇ u ) )Fi (∇ u )νi udσ  ∂

=− G (F (∇ u ) )F (∇ u )dx + G (F (∇ u ) )Fi (∇ u )νi udσ .

2u dx =



(3.25)

∂



From (3.24) and (3.25) we obtain that

2  P (u; x )dx + (1 − N ) uG (F (∇ u ) )F (ν )dσ + −2 ux, νdσ N  ∂ ∂

= P (u; x )x, νdσ .

N

(3.26)

∂

On the other hand, by the divergence theorem we have the equalities

x, νdσ = N|0 |,

− x, νdσ = N|k | for any k = 1, ..., m,

∂ 0

(3.27)

∂ k

which combined with (1.7)2,3 , (1.8) and (3.26) lead to the conclusion of Lemma 3.2.



The following lemma establishes a comparison between the values of P(u; x) on the different boundary components

∂ k , k = 0, . . . , m.

Lemma 3.3. The auxiliary function P satisfies the inequality

P (∂ 0 ) ≤ max P (∂ k ).

(3.28)

1≤k≤m

Proof of Lemma 3.3. Let us suppose, for contradiction, that

P (∂ 0 ) > max P (∂ k ).

(3.29)

1≤k≤m

This inequality and Lemma 3.1 imply that

P (u; x ) ≤ P (∂ 0 ) for any x ∈ .

(3.30)

Integrating (3.30) over  and making use of Lemma 3.2 we obtain that

P (∂ 0 )|0 | −

m  k=1

P (∂ k )|k | =





P (u; x )dx ≤ P (∂ 0 )

|0 | −

m  k=1

 |k | ,

(3.31)

L. Barbu and C. Enache / Applied Mathematics and Computation 362 (2019) 124551

7

which implies m 

P ( ∂ 0 )

|k | ≤

k=1

m 

P (∂ k )|k |.

(3.32)

k=1

Therefore,

max P (∂ k )

1≤k≤m

m 

|k | ≥

k=1

m 

P (∂ k )|k | ≥ P (∂ 0 )

k=1

m 

|k |.

(3.33)

k=1

Inequality (3.33) contradicts the assumption (3.29), so the proof of Lemma 3.3 is thus achieved.



Proof of Theorem 1.1. According to Lemmas 3.1 and 3.2, the auxiliary function P(u; x) is either identically constant on , or it takes its maximum value on some interior boundary component, say ∂ 1 , where Hopf’s second maximum principle implies that Pν > 0 (here, ν is the exterior unit normal to ∂ , while Pν is the normal derivative of P). Let us first consider the case when P is not identical constant, so that we have Pν > 0 on ∂ 1 . Since ν F := Fξ ◦ν , and

νF , ν = F ◦ ν > 0 on ∂ , ν F must point outward. From the Dirichlet boundary conditions u = a1 on ∂ 1 , ν = we have νF = Fξ (∇ u ) on ∂ 1 . Therefore, PνF := ∇ P, Fξ  > 0 or, equivalently, Pi Fi = G F Fi Fk uik +

2 F > 0 on N

∂ 1 .

∇u |∇ u| , so that

(3.34)

On the other hand, according to Lemma 2.3 (see (2.6)), we have

Qu(x ) = G (F (∇ u ))Fi Fj ui j − G (F (∇ u ))KF (∂ 1 ) = −2 on Using (3.35) in (3.34), we obtain



G (F (∇ u ))KF (∂ 1 ) > 2 −

∂ 1 .

(3.35)



2 F, N

(3.36)

which leads to

G (F (∇ u ))KF (∂ 1 )F ◦ ν > 2

N−1 F ◦ ν on N

Integrating (3.37) over ∂ 1 we obtain

G ( a1 )

∂ 1

KF (∂ 1 )F ◦ ν dσ > 2

∂ 1 .

N−1 F ◦ ν dσ . N ∂ 1

(3.37)

(3.38)

On the other hand, from (1.8) we have the equality

2|1 | = so G (a1 ) =

∂ 1

∂ 1

2|1 | . PF (1 )

G (F (∇ u ) )F (ν )dσ = G (a1 )PF (1 ),

(3.39)

Therefore (3.38) may be rewritten as

KF (∂ 1 )F ◦ ν dσ >

N−1 PF (1 )2 . N

(3.40)

Inequality (3.40) contradicts (2.7). From this analysis we conclude that P(u; x) is identically constant in . This implies that we must have equality in (3.18), that is

ai j akl ul j uki =

4 . N

(3.41)

Equality (3.41) implies equality in (3.17), so W = − N2 IN , which means that

wi j = aik uk j = −

2 δi j in , N

(3.42)

or, equivalently,





G Fi j + G Fi Fj uk j = −

  2 2 δi j ⇔ G (F (∇ u ) )Fi (∇ u ) j = − δi j in , N N

(3.43)

for any i, j ∈ {1, . . . , N}. An integration of (3.43) with respect to xj leads to

G (F (∇ u ) )∇ F (∇ u ) = −

2 x for any x ∈ , N

(3.44)

for a suitable choice of the origin. Therefore, making use of (2.3) we obtain that

G (F (∇ u(x ))) =

2 0 F (x ) for any x ∈ . N

(3.45)

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L. Barbu and C. Enache / Applied Mathematics and Computation 362 (2019) 124551

Now, the strict monotonicity of the map t → G (t) on (0, ∞) (see (1.3)) can be used to write F(∇ u(x)) explicitly as a function of F◦ (x) on . Since the auxiliary function P(u; x) is constant in , we conclude, from (3.1) that u(x) depends only on F ◦ (x ). Therefore, the level sets {u = const } are the boundaries of concentric Wulff shapes (up to a translation), implying that there is no more than one single interior boundary component ∂ 1 , and that 0 is an Wulff shape concentric to 1 . Next, we are going to prove that u(x), the solution to problem (1.7) and (1.8), is given explicitly by formula (1.10). Let R0 and R1 , 0 < R1 < R0 the radii of the Wulff shapes 0 and 1 , respectively. As a consequence of the results obtained before, there exists at most one solution u(x) of our problem of the form u(x ) = v(r ), where r = F ◦ (x ), decreasing as function of r. We remark that the corresponding Euler equation of our problem is





r N−1 G −v (r )



= 2r N−1 on [R1 , R0 ].

On the other hand, we can rewrite (1.8) for k = 1 as

2|1 | =



G (F (∇ u ) )F (ν ) dσ = G −v (R1 )

∂ 1

(3.46)



1 ∂ 1

| ∇ F ◦ (x ) |

dσ .

(3.47)

Using the co-area formula we obtain that



2|1 | = G −v (R1 ) so





 ∂  N |1 |    1dx = G −v (R1 ) , ∂ R 1 1 R1

2R1 . N

G −v (R1 ) =

(3.48)

(3.49)

Now, integrating (3.46) we get





G −v (r ) = 2r /N + k/r N−1 on [R1 , R0 ].

(3.50)

Moreover, taking into account (3.49) we get k = 0, so

−v (r ) = G

 2r  N

on [R1 , R0 ],

(3.51)

where G is the inverse of the map t → G (t) on (0, ∞). Integrating now (3.51) from F◦ (x) to R0 , we get

u (x ) =

R0 F ◦ (x )

G

 2s  N

ds for any x ∈ .

(3.52)

Evaluating (3.52) for F ◦ (x ) = R1 and making use of the boundary condition u = c1 on ∂ 1 , we get

c1 =

R0

R1

G

 2s  N

ds.

(3.53)

In addition, from (3.52) we also get



∇ u(x ) = −G

2F ◦ ( x ) N

 ∇ F ◦ (x ) for any x ∈ .

(3.54)

Therefore, making use of (2.2) we conclude that



F (∇ u (x ) ) = G

2F ◦ ( x ) N



for any x ∈ .

(3.55)

Using (3.55) and the boundary conditions (1.7), for F ◦ (x ) = R0 and F ◦ (x ) = R1 , we get

N  G ( a0 ) = R0 , 2

N  G ( a1 ) = R1 . 2

This achieves the proof of Theorem 1.1.

(3.56) 

Acknowledgements The second author was supported by a Seed Grant from the American University of Sharjah. References [1] L. Barbu, C. Enache, Maximum principles, Liouville-type theorems and symmetry results for a general class of quasilinear anisotropic equations, Adv. Nonl. Anal. 5 (4) (2016) 395–405. [2] L. Barbu, C. Enache, A Liouville type theorem for a class of anisotropic equations, An. S¸ tiint¸ . Univ. “Ovidius” Constanta Ser. Mat. 24 (3) (2016) 47–59. [3] L. Barbu, C. Enache, On a free boundary problem for a class of anisotropic equations, Math. Methods Appl. Sci. 40 (6) (2017) 2005–2012. [4] M. Belloni, V. Ferone, B. Kawohl, Isoperimetric inequalities, wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angev. Math. Phys. 54 (2003) 419–431.

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