A characterization result for the existence of a two-phase material minimizing the first eigenvalue

Available online at www.sciencedirect.com ScienceDirect Ann. I. H. Poincaré – AN 34 (2017) 1215–1226 www.elsevier.com/locate/anihpc A characterization result for the existence of a two-phase material minimizing the first eigenvalue Juan Casado-Díaz Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain Received 31 October 2015; received in revised form 9 July 2016; accepted 19 September 2016 Available online 13 October 2016 Abstract Given two isotropic homogeneous materials represented by two constants 0 <α<βin a smooth bounded open set  ⊂RN, and a positive number κ<||, we consider here the problem consisting in finding a mixture of these materials αχω+β(1 −χω), ω⊂RNmeasurable, with |ω| ≤κ, such that the first eigenvalue of the operator u ∈H1 0() →−divαχω+β(1 −χω)∇u reaches the minimum value. In a recent paper, [6], we have proved that this problem has not solution in general. On the other hand, it was proved in [1] that it has solution if is a ball. Here, we show the following reciprocate result: If  ⊂RNis smooth, simply connected and has connected boundary, then the problem has a solution if and only if is a ball. ©2016 MSC: 49J20 Keywords: Two-phase material; Control in the coefficients; Eigenvalue; Non-existence 1. Introduction We consider a bounded open set  ⊂RN, N≥2, and two constants 0 <α<β, representing two homogeneous isotropic materials (thermic, electric, elastic,...). A classical problem in optimal design consists in mixing these materials in order to minimize a certain functional. Such as it is proved in [17] and [18], this type of problems has not solution in general and then it is usual to deal with relaxed formulations which can be obtained by using the homogenization theory (see e.g. [2,7,19,22,23]). Between the most studied problems of this type (see e.g. [2,6,14,15,19]) we emphasize the following one ⎧ ⎪ ⎨ ⎪ ⎩ min   (αχω+βχ\ω)|∇u|2dx −div(αχω+βχ\ω)∇u=fin , u =0on∂, |ω|<κ, (1.1) E-mail address: [email protected]. http://dx.doi.org/10.1016/j.anihpc.2016.09.006 0294-1449/©2016 L'Association Publications de l'Institut Henri Poincaré. Published by Elsevier B.V. All rights reserved. L'Association Publications de l'Institut Henri Poincaré. Published by Elsevier B.V. All rights reserved. 1216 J. Casado-Díaz / Ann. I. H. Poincaré – AN 34 (2017) 1215–1226 with f∈H−1() and κ∈(0, ||)(for κ≥||, the solution is the trivial one ω=). A special attention has been paid for f=1 and N=2, where it represents the optimal distribution of two materials in the cross section a beam in order to minimize the torsion. In this case, it has been proved in [19] that if is simply connected and smooth and there exists a smooth solution ω, then is a ball. This result has been improved in [6] by showing that the result holds true without any smoothness assumptions on ω(the case N>2is also considered). The proof is based on certain regularity results for the solution of the relaxed formulation of (1.1) also obtained in [6]. A related problem consisting in replacing the minimum in (1.1) by a maximum has been considered in [5]. It has also been observed in [6] that problem (1.1) is strongly related to another classical optimization design problem for a two-phase material. It consists in finding a measurable set ω⊂with |ω| ≤κ(0 <κ<||as above) such that the first eigenvalue of the operator u∈H1 0() → −div (αχω+βχ\ω)∇u∈H−1() (1.2) becomes minimal. Namely, it is proved that the relaxed formulation of this problem is equivalent to solve the relaxed formulation of (1.1) for every f∈L2() with fL2() =1 and then to minimize in f. Thus, the regularity results proved in [6] for (1.1) also hold for the minimization of the eigenvalue. As an application, it has been shown that the problem has not solution if is a rectangle or an ellipsis. On the other hand, it was proved in [1] that the eigenvalue problem has a solution in the particular case where is a ball and then the optimal set ωhas a radial structure. Some discussions about the exact structure of ωwhen is a ball can be found in [8,9,16] and [20]. The purpose of the present paper is to show that, similarly to the result stated above for problem (1.1) with f=1, if is a smooth simply connected open set with connected boundary such that the minimization of the first eigenvalue of the operator (1.2) has a solution, then is a ball. As for problem (1.1), the proof uses the results obtained in [6] but the reasoning is more involved. For problem (1.1) with f=1 one has that the optimal solutions (ω, u) are such that there exist an analytic function wand a positive number μ, satisfying (αχω+βχ\ω)∇u=∇w, {|∇w|>μ}⊂ω⊂{|∇w|≥μ}.(1.3) Moreover, the Laplacian of |∇w|2in is nonnegative. For the eigenvalue problem, statement (1.3) still holds true but now wis non-analytic and the Laplacian of |∇w|2can change its sign in . Thus, many of the ideas used in [6] (and [19]) cannot be used here. 2. The characterization result For a smooth bounded open set  ⊂RN, N≥2, and three constants 0 <α<β, 0 <κ<||, we consider the problem consisting in finding a measurable subset ωof with |ω| ≤κ, such that the first eigenvalue of the problem −div (αχω+βχ\ω)∇u=λu in  u=0on∂ (2.1) reaches its minimum value. This can also been formulated as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ min  αχω+βχ\ω|∇u|2dx u∈H1 0(),   |u|2dx =1 ω⊂measurable, |ω|≤κ. (2.2) We remark that if we do not assume the volume restriction |ω| ≤κ, then the solution is the trivial one given by ω=. However in the applications, the material αcan be more expensive than βand thus, we can only dispose of a certain quantity κof material α. The question then is how to distribute it in an optimal way. As an application of (2.2) we can consider the following problem for the heat equation ⎧ ⎪ ⎨ ⎪ ⎩ ∂tu−div (αχω+βχ\ω)∇u=0in×(0,∞) u=0on∂×(0,∞) u|t=0=u0, J. Casado-Díaz / Ann. I. H. Poincaré – AN 34 (2017) 1215–1226 1217 where u0is a given function in L2(). Such as it is hoped from the physical point of view, it is well known that u(t, x) tends to zero when ttends to infinity, i.e. for tlarge, the temperature in becomes equal to the temperature outside . Namely, the following estimate holds max u(·,t)L2() ≤e−λt u0L2(),∀t≥0, with λthe first eigenvalue of problem (2.1). This estimate is optimal in the sense that we can choose u0such that it is an equality. This means that if we want to choose the non-homogeneous material in in such way that the temperature goes to zero as slowly as possible, we must take λsmall. Thus, for heat conduction, problem (2.2) is equivalent to get the most insulated material in by using a quantity of material αsmaller or equal to κ. Such as it has been proved in [6], problem (2.2) has not solution if is a rectangle or an ellipsis. On the other hand, in the particular case where is a ball, it has been proved in [1] that (2.2) has a solution and it is radial. Our purpose in the present paper is to give the following reciprocate result. Theorem 2.1. Assume  ⊂RNof class C1,1, simply connected, with connected boundary. If problem (2.2) has an optimal solution (ω, u) then is a ball and χω, uare radial functions. The proof of Theorem 2.1 will be given in Subsection 2.2, for this purpose we will need some previous results which expose in subsection 2.1 2.1. Preliminary results In this subsection, we recall some previous lemmas which we need to prove Theorem 2.1. We start by recalling some results which have been proved in [6], where it is considered the following relaxed formulation for problem (2.2) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ min   |∇u|2 1+cθ dx u∈H1 0(),   |u|2dx =1 θ∈L∞(;[0,1]),   θdx ≤κ, (2.3) with c=(β −α)/α. It consists in replacing the materials of the form αχω+β(1 −χω)by the harmonic mean value of αand βwith proportions θand 1 −θ. These new materials are constructed via homogenization (see e.g. [2,19,23]) by using a lamination of αand βwith proportions θand (1 −θ) in the direction of ∇u. Following Theorem 4.2 in [6], we have the following regularity result for 2.2. Lemma 2.2. Assume that  ⊂RNis C1,1and consider a solution (θ, u) of problem (2.3), then we have •The function ubelongs to W1,∞(). •The function σ:= ∇u 1+cθ (2.4) is in H1()Nand denoting by νthe outward unitary normal to on ∂, we have σ=(σ ·ν)ν on ∂. (2.5) •The function θsatisfies ∂iθσj−∂jθσi∈L2(), 1≤i, j ≤N. (2.6) Moreover, if θonly take the values 0 and 1, then 1218 J. Casado-Díaz / Ann. I. H. Poincaré – AN 34 (2017) 1215–1226 ∂iθσj−∂jθσi=0,1≤i, j ≤N(2.7) and curl(σ) =0in . (2.8) Taking into account Theorem 2.2 in [6], we also have the following optimality condition for the solutions of (2.3). Lemma 2.3. Assume that (θ, u) is a solution of problem (2.3) and define σby (2.5), then,   θdx=κ(2.9) and there exists a constant μ >0such that θ(x)=1if |σ(x)|>μ 0if |σ(x)|<μ. (2.10) Remark 2.4. In the previous Lemma we have asserted that μis strictly positive which is a sufficient condition in Theorem 2.2 in [6] in order to have (2.9). Indeed, if μ =0, then by (2.10) and the restriction of the integral of θin (2.3) we get ||>κ≥  θdx≥|{x∈:|σ(x)|>0}|, and therefore σvanishes on a set of positive measure but then, taking into account that σis in H1()Nby (2.2) we get that λu =−div σvanishes on a set of positive measure, which is in contradiction with the maximum principle. In the present paper, we are interested in the existence of a solution for problem (1.1), which is equivalent to the existence of a solution (θ, u) for problem (2.3), such that θ=χωwith ωa measurable subset of . In this case, Lemmas 2.2 and 2.3 imply Corollary 2.5. Assume  ∈C1,1, simply connected, with connected boundary. We suppose there exist u ∈H1 0() and ω⊂measurable such that (χω, u) is a solution for problem (2.3), then, ubelongs to W1,∞() and defining was the solution of −w =λu in  w=0on ∂, (2.11) with λ= αχω+β(1−χω)|∇u|2dx, (2.12) we have αχω+β(1−χω)∇u=∇w. (2.13) Moreover, |ω|=κ, (2.14) and there exists μ >0such that {x∈:|∇w(x)|>μ}⊂ω⊂{x∈:|∇w(x)|≥μ}(2.15) J. Casado-Díaz / Ann. I. H. Poincaré – AN 34 (2017) 1215–1226 1219 Proof. Lemma 2.3, provides u ∈W1,∞(), and since θ=χω, it also provides σ:= ∇u 1+cθ =α βχω+χ\ω∇u in H1()Nwith vanishing curl. Since is simply connected, this implies the existence of w∈H2() such that βσ =∇w. Now, using (2.5) we get that wis constant on ∂, but wis defined up to a constant. Therefore, we can take wvanishing on ∂. Using then that ugives the minimum in min v∈H1 0()   |∇v|2 1+cθ dx =min v∈H1 0()  α βχω+χ\ω|∇v|2dx and the definition of σ, we get −w =−βdivσ=λu, with λdefined by (2.12). This proves that wis the solution of (2.11). To finish the proof we use (2.9) and (2.10), which imply that (2.14) and (2.15) hold, with the constant μwhich appears in Corollary 2.5 replaced by βμ.2 In order to prove Theorem 2.1, we will also need the following result. Lemma 2.6. Assume  ⊂RNopen, K⊂compact and connected and c∈R. We consider two functions u ∈ W1,p(), 1 ≤p≤∞and w∈C1() such that w=cin K, ∇wdoes not vanish in Kand ∇uis proportional to ∇wa.e. in . Then, there exist a neighborhood Uof Kcontained in , τ>0and h ∈W1,p(c −τ, c+τ) such that w(U) =(c −τ,c +τ), u(x)=h(w(x)), ∀x∈U, ∇w= 0in U. Proof. We start assuming that Kreduces to a point x0∈. In this case, the result is well known, at least if uis also in C1(). If u ∈W1,p(), 1 ≤p≤∞the reasoning is similar to the classical one but for more clarity, let us detail it. Consider N−1 vectors ξ2, ..., ξN∈RNsuch that {∇w(x0), ξ2, ··· , ξN}is a basis in RNand define F= (F1, ··· , FN) : →RNby F1(x) =w(x), Fi(x) =ξi·x, 2≤i≤N, ∀x∈. By the inverse function theorem, there exists δ>0 with B(x0, δ) , and V⊂RNopen, bounded, such that F: B(x0, δ) →Vis invertible and F−1belongs to C1(V )N∩W1,∞(V )N. In particular, since DF(x) is invertible for x∈B(x0, δ), we have that its first row, ∇wdoes not vanish in B(x0, δ). Defining h =u ◦F−1∈W1,p(V ), we have that u(x) =h(F (x)) for every x∈B(x0, δ) and then ∇u(x) =∂y1h(F (x))∇w(x) + N  i=2 ∂yih(F (x))ξi, but by assumption, ∇u(x) is proportional to ∇w(x) and therefore, recalling that ∇w(x), ξ2, ··· , ξNare linearly independent, the previous inequality shows ∂yih(F (x)) =0,2≤i≤N. (2.16) Using that Fis a isomorphism from B(x0, δ) into V, we then get ∂yih=0inV, 2≤i≤N. Taking τsmall enough to have the cube Q:= (w(x0)−τ,w(x0)+τ)×(ξ2·x0−τ,ξ2·x0+τ)×...×(ξN·x0−τ,ξN·x0+τ), contained in V, we deduce from (2.16) that honly depends on the first variable in Qand therefore, using the definition of F, we have 1220 J. Casado-Díaz / Ann. I. H. Poincaré – AN 34 (2017) 1215–1226 u(x) =h(F (x)) =h(w(x)), ∀x∈F−1(Q). Defining U=F−1(Q) which is an open set contained in B(x0, δ) and observing that by definition of Fand Q, wis onto from Qinto (w(x0) −τ, w(x0) +τ), we get the result for K={x0}. Let us now consider the general case: By the above proved, for every z∈K, there exists a neighborhood Uzof z, τz>0 and hz∈W1,p(c −τz, c+τz)such that w(Uz)=(c −τz,c+τz), u(x) =h(w(x)), ∀x∈Uz,∇w= 0inUz. Since Kis compact, we can take the open sets Uzsuch that there exists ε>0 satisfying that B(z, 2ε) ⊂Uz, for every z∈K. Thus B(z1,ε)∪B(z2,ε)⊂Uz1∩Uz2,∀z1,z 2∈K, |z1−z2|<ε, and then hz1(w(x)) =u(x) =hz2(w(x)), ∀z1,z 2∈K, |z1−z2|<ε, ∀x∈B(z1,ε)∪B(z2,ε). (2.17) Now, we use that Kcompact, w=cin K, w∈C1() and ∇w= 0in K, imply the existence of τ>0 such that (c −τ, c+τ) ⊂w(B(z, ε)), for every z∈K, and then (2.17) shows hz1=hz2in (c −τ,c +τ), ∀z1,z 2∈K, |z1−z2|<ε. This allows us to show that for ˆz∈Kfixed, the set of z∈Ksuch that hz=hˆzin (c −τ,c +τ) is open and closed in the topology relative to K, and thus, since Kis connected hz=hˆzin (c −τ,c +τ), ∀z∈K. Defining then h=hˆz,U=w−1(c −τ,c +τ) z∈K B(z,ε), we get the result. 2 To finish this subsection we recall the coarea formula (see e.g. Theorem 2 section 3.4.3 in [11]) and as Corollary the version we use in the proof of Theorem 2.1. The definition of Jacobian of a derivable function f:RN→RM, J(f)is given in [11], section 3.2. Theorem 2.7. Let f∈W1,∞(RN)Mbe, N≥M. Then, for every measurable function g∈L1(RN), we have that g|f−1(y) is integrable with respect to the Hausdorf (N −M)-dimensional measure, HN−M, for a.e. y∈RM, the function y∈RM→  f−1(y) gdH N−M is integrable in RMand  RN gJ(f)dx = RM f−1(y) gdH N−Mdy. (2.18) Corollary 2.8. Assume U⊂RNopen and w∈W1,∞(U) such that ∇w(x) = 0, then, for any g∈L1(U), we have that (g/|∇w|)|w−1(s) is integrable with respect to HN−1, for a.e. s∈R, the function J. Casado-Díaz / Ann. I. H. Poincaré – AN 34 (2017) 1215–1226 1221 s∈R→  f−1(s) gdH N−1 is integrable in w(U) and  U gdx= w(U)  w−1(s) g |∇w|dHN−1ds. (2.19) Proof. For ε>0, we define Uε=x∈U:|∇w(x)|≥ε. Thanks to Kirszbraun’s Theorem ([11], chapter 3.1.1, [12], chapter 2.10.43), we know there exists wε∈W1,∞(RN), such that wε=win Uε. Applying (2.18) to gε=g |∇w|χUε,f=wε, and taking into account that J(w ε) =|∇wε|, we deduce that (gε)|w−1(s) is integrable with respect to the HN−1, for a.e. s∈R, the function s→  w−1(s) gεdHN−1 is integrable in Rand  Uε gdx= RN g |∇w|χUε|∇wε|dx = R w−1 ε(s) g |∇w|χUεdHN−1ds = w(Uε) w−1(s) g |∇w|dHN−1ds. Thanks to the Lebesgue dominated convergence theorem, we can now pass to the limit in this equality when εtends to zero to deduce the result. 2 2.2. Proof of the main result In the present subsection, let us use the results of the previous one in order to prove Theorem 2.1, which is the aim of the present paper. Proof of Theorem 2.1.By Corollary 2.5, we know that if there exists a solution (ω, u) of (2.2) and λis the corresponding eigenvalue, then ubelongs to W1,∞() and (2.13) holds, with w, the solution of (2.11) and λgiven by (2.12). Moreover, (2.14) and (2.15) are satisfied, with μstrictly positive Using the classical smoothness results for elliptic equations and  ∈C1,1, we also have win W2,p() ∩W3,p loc (), for every p≥1. On the other hand, it is well known that uand then wcan be chosen strictly positive in . The following three steps are devoted to show that these properties of ω, uand wimply that is a ball and that χω, uand ware radial functions. Step 1. Assume x0∈such that |B(x0,r)∩ω|>0,|B(x0,r)∩( \ω)|>0,∀r>0.(2.20) Let us prove that there exists a connected open set Oof class W3,p for every p∈[1, ∞), with connected boundary such that x0∈∂O, |∇w|=μon ∂O, w =w(x0)on ∂O. (2.21) 1222 J. Casado-Díaz / Ann. I. H. Poincaré – AN 34 (2017) 1215–1226 Moreover, all the points in ∂O satisfy condition (2.20) and there exists a connected neighborhood Uof ∂O, τ>0 and h ∈W1,∞(w(x0) −τ, w(x0) +τ) such that w(U) =(w(x0)−τ,w(x0)+τ), u(x)=h(w(x)), ∀x∈U. (2.22) For this purpose, we first observe that (2.15), (2.20) and ∇wcontinuous imply |∇w(x0)| =μ = 0 and thus, the implicit function theorem, w∈W3,p loc () for every p∈[1, ∞), and w(x0) = 0 prove that the set ˆ ϒ=x∈:w(x) =w(x0), |∇w(x)|>μ 2(2.23) is a (N −1)-dimensional submanifold of RNstrictly contained in of class W3,p for every p∈[1, ∞), containing x0. We define ϒas the connected component of ˆ ϒcontaining x0. By (2.13), we can apply Lemma 2.6 to deduce that for every compact set K⊂ϒwhich contains x0, there exists an open neighborhood U⊂of K, τ>0, and a function h ∈W1,∞(w(x0) −τ, w(x0) +τ) such that w(U) =(w(x0)−τ,w(x0)+τ), u(x)=h(w(x)), ∀x∈U, ∇w= 0inU. (2.24) In particular, we have ∇u(x) =h(w(x))∇w(x), a.e. x∈U. (2.25) Taking into account (2.13) and (2.25), we can apply (2.19) to deduce 0= Uαχω+βχ\ω∇u−∇wdx = w(x0)+τ  w(x0)−τ U∩w−1(s) |αχω+βχ\ω∇u−∇w |∇w|dHN−1ds, 0= U∇u−h(w)∇wdx = w(x0)+τ  w(x0)−τ U∩w−1(s) |∇u−h(s)∇w |∇w|dHN−1ds, and therefore w(x0)+τ  w(x0)−τ U∩w−1(s)  h(s) −1 αχω+1 βχ\ω dHN−1ds =0. Thus, there exists a null measure set N⊂(w(x0) −τ, w(x0) +τ) such that s∈(w(x0)−τ,w(x0)+τ)\N=⇒ ∃ h(s) ∈{α−1,β−1} and h(s) =α−1=⇒ HN−1((w−1(s) ∩U)\ω) =0 h(s) =β−1=⇒ HN−1(w−1(s) ∩U∩ω) =0.(2.26) On the other hand, we observe that for every r>0 with B(x0, r) ⊂Uformula (2.19) gives |B(x0,r)∩ω|= w(x0)+τ  w(x0)−τ B(x0,r)∩ω∩w−1(s) 1 |∇w|dHN−1ds, (2.27) |B(x0,r)∩( \ω)|= w(x0)+τ  w(x0)−τ B(x0,r)∩(\ω)∩w−1(s) 1 |∇w|dHN−1ds, (2.28) which thanks to (2.20) shows the existence of s1 n, s2 n/∈N, converging to w(x0)when ntends to infinity such that HN−1B(x0,1/n) ∩ω∩w−1(s1 n)>0,H N−1B(x0,1/n) ∩( \ω) ∩w−1(s2 n)>0. J. Casado-Díaz / Ann. I. H. Poincaré – AN 34 (2017) 1215–1226 1223 Therefore, by (2.26), we get HN−1(w−1(s1 n)∩U)\ω=0,H N−1w−1(s2 n)∩U∩ω=0 and then, by (2.15) |∇w|≥μin w−1(s1 n)∩U, |∇w|≤μin w−1(s2 n)∩U. Then, passing to the limit in n, we deduce |∇w|=μin w−1(w(x0)) ∩U⊃K. (2.29) Now, we observe that (2.27), (2.28), (2.26) and (2.20), imply that for every δ>0 there exist two disjoint measurable sets S1 δ, S2 δ⊂(w(x0) −δ, w(x0) +δ) such that 0<|S1 δ|,0<|S2 δ|,|S1 δ|+|S2 δ|=2δ, HN−1((w−1(s) ∩U)\ω) =0,∀s∈S1 δ,H N−1(w−1(s) ∩U∩ω) =0,∀s∈S2 δ, while w=w(x0)in Kimplies that for every r>0, there exists δ>0 such that HN−1(B(x, r) ∩w−1(s)) > 0,a.e. s∈(w(x0)−δ,w(x0)+δ). Then, using again (2.19), we have for every r>0, with B(x, r) ⊂U |B(x,r) ∩ω|= w(x0)+τ  w(x0)−τ B(x,r)∩ω∩w−1(s) 1 |∇w|dHN−1ds ≥ S1 δ B(x,r)∩w−1(s) 1 |∇w|dHN−1ds > 0, |B(x,r) \ω|= w(x0)+τ  w(x0)−τ B(x,r)∩(\ω)∩w−1(s) 1 |∇w|dHN−1ds ≥ S2 δ B(x,r)∩w−1(s) 1 |∇w|dHN−1ds > 0. This proves that every point in Ksatisfies (2.20). We have thus proved that every compact and connected set K⊂ϒwhich contains x0is such that |∇w| =μin Kand that every point in Ksatisfies (2.20). Since ϒis connected and locally connected by paths, because it is a manifold, it is connected by paths (see e.g. [10], chapter 5.5). Therefore, for every point xin ϒthere exists a compact set K⊂ϒcontaining xand x0. This proves |∇w|=μin ϒ, (2.30) and that every point in ϒsatisfies condition (2.20). Let us use (2.30) to show that ϒis closed in RN. In particular this will imply that we can take K=ϒabove and then that there exists a neighborhood Uof ϒ, τ>0 and h ∈W1,∞(w(x0) −τ, w(x0) +τ) satisfying (2.24). To prove ϒclosed, we take xn∈ϒconverging to a certain point x∈¯ . Then, since w∈C1(¯ ), we have w(x) = w(x0), |∇w(x)| =μ. This shows in particular that xbelongs to ˆ ϒbut since ϒis a connected component of ˆ ϒ, it is closed (and open) for the topology of ϒ. Therefore, xn∈ϒconverging to x∈ˆ ϒimplies that xbelongs to ϒ, and so ϒis closed in RN. The above proved shows that ϒis a connected compact submanifold of class W3,p, for every p∈[1, ∞)and dimension N−1. By the Jordan–Brouwer theorem (see e.g. [13] Chapter 5.2) and ϒstrictly contained in (because w(x) =w(x0) >0in ϒ), we have that ϒis the boundary of a bounded connected open set Oof class W3,p, for every p∈[1, ∞). Step 2. Let us prove that uand ware radial in a ball contained in . First we observe that a point x0satisfying (2.20) always exists because in another case, the sets {x∈:∃r>0 with |B(x0,r)∩ω|=0},{x∈:∃r>0 with |B(x0,r)∩( \ω)|=0}, are two disjoint nonempty open sets with positive measure (because 0 <|ω| <||) whose union agrees with , which is a contradiction with connected. Therefore, we can consider a bounded open set Oin the conditions of Step 1.