Density estimates for vector minimizers and applications

We extend the Caffarelli-Cordoba estimates to the vector case in two ways, one of which has no scalar counterpart, and we give a few applications for minimal solutions.

The basic estimate for such solutions is The key hypothesis in our theorems is This assumption excludes examples (b) and (c) above. It is well known that the phase transition model is linked to minimal surfaces (m = 1) and Plateau Complexes (m ≥ 2). In particular in the vector case entire solutions to (1.1) are linked to singular minimal cones which unlike planes have additional hierarchical structure ( Alikakos [3] ).
The main purpose of this paper is the various extensions of the Caffarelli-Cordoba density estimates [14] to the vector case. In the scalar case, among other things, these estimates refine the linking of the phase transition model to minimal surfaces and have played a major role in the resolution of De Giorgi conjecture in higher dimensions ( Savin [23] ). Other extensions to the density estimates in different contexts have been provided by Farina and Valdinoci [18], Savin and Valdinoci [24], [25], and Sire and Valdinoci [26].
where L n stands for the n-dimensional Lebesgue measure. Note that A R satisfies A R ≤ CR n−1 by (1.4). In the context of diffuse interfaces A R measures interface area while V R enclosed volume ( [14]).
The new points in the proof of Theorem A are the polar form u(x) = a + q u (x)ν u (x), the choice of the test functions which are limited to perturbations of the modulus q u and keep ν u fixed, (1.8) σ = a + q σ ν u , q σ = min{q h , q u }, and the resulting identity where minimality on balls was used in the last inequality. The proof of Theorem A otherwise follows closely the argument in Caffarelli-Cordoba [14]. We give a number of applications of Theorem A. We mention here a few and refer the reader to the main body of the paper for the precise statements.

(i) Lower Bound
For the phase transition model (a) above, under the hypotheses of Theorem A, and provided u is not a constant, the lower bound holds (1.10) We recall that for all nonconstant solutions to (1.1) and any W ≥ 0 which allows u ∈ W 1,2 loc ∩ L ∞ the estimate holds, and that (1.11) can not in general be improved (Alikakos [2]). In light of (1.4) estimate (1.10) is optimal.
(ii) Liouville-Rigidity Theorem This was proved in Fusco [20] with a different, though related method.
Entire equivariant (minimal) solutions to (1.1) correspond to minimal cones and possess a hierarchical structure at least for a class of symmetries. They were established by Bronsard, Gui and Schatzman [13] for triple junctions, n = m = 2, and by Gui and Schatzman [22] for quadruple junctions (n = m = 3) and for general n, m in a series of papers [5] [4] [19]. In the papers [13] [22] the hierarchical structure is built in, while in [5] [4] [19] can be deduced a posteriori (see [8]).
Our next theorem concerns an aspect that has no scalar counterpart. We look at the simplest possible set up for this kind of result. Consider (1.1) in the class of symmetric solutions where for z ∈ R d we denote byẑ the reflection of z in the plane {z 1 = 0}, and we take W a C 3 potential, symmetric W (u) = W (û), u ∈ R m , and with exactly two minima W (a − ) = W (a + ) = 0, W > 0 on R m \ {a + , a − }. Under hypotheses of nondegeneracy for a + , a − there is such a symmetric solution, minimal in the symmetric class, and satisfying the estimate Consider the Action The key hypotheses in our theorems is that A has a hyperbolic global minimum e in the symmetric class. Following [8] we define the Effective-Potential and thus we have that The basic estimate in the present context is and by analogy to (1.6) (1.15) Note that A R ≤ CR n−2 by (1.14).
(i) Assume that the Action A has exactly two global minima e − , e + , W(e − ) = W(e + ) = 0, W > 0 otherwise, where e − , e + satisfy the hypotheses of e above. Assume for u the hypotheses of Theorem B. Then for 0 < θ < e − − e + the following is true: , with a similar statement for e + .
(ii) Assume the hypothesis of Theorem B and suppose that either This was proved in [8] under the hypothesis {W = 0} = {e} with a different though related method.
We recall that Alama, Bronsard and Gui in [1] have established, under the hypothesis of (i) above, the existence of a solution u : R 2 → R 2 converging to a ± as s → ±∞, and converging to e ± as y → ±∞. Thus there are solutions genuinely higher dimensional connecting e + and e − . The paper is structured as follows. In Part I Theorem A is stated and proved and its applications are presented in individual sections. Similarly in Part II Theorem B is stated and proved, followed by its applications.

PART I
2 Theorem A

Hypotheses and Statement
(HA) The potential W : R m → R is nonnegative and W (a) = 0 for some a ∈ R m .
If 0 < α < 2 we assume where · denotes the Euclidean inner product in R m , C * a positive constant.
If α = 2 we assume, for some constant C 0 > 0, The figure below shows the behavior of W for different values of α.
Note: In the proof of Theorem A we utilize minimality only on balls. For each z ∈ R k , k ≥ 1 and r > 0 we let B r (z) ⊂ R k be the open ball of center z and radius r and B r the ball centered at the origin. We denote by L k (E) the k-dimensional Lebesgue measure of a measurable set E ⊂ R k .
As in [14] Theorem A has the following important consequence Theorem 2.1. Assume there are a 1 = a 2 ∈ R m such that W (a 1 ) = W (a 2 ) = 0, W (u) > 0, for u ∈ {a 1 , a 2 } and assume that (HA) holds at a = a j , j = 1, 2. Let u : R n → R m is a minimizer in the sense of (HB). Then, given 0 < θ < |a 1 − a 2 | the condition where C > 0 depends only on µ 0 , θ and M . An analogous statement applies to a 2 .
Therefore Theorem A yields To conclude the proof we observe that and therefore W (u) > 0, for u ∈ {a 1 , a 2 } and Lemma 2.
Note: We note that the argument above when applied to potentials W that vanish at more than two points: W (a 1 ) = · · · = W (a N ) = 0, N ≥ 3 , provides estimates (2.9) only for two of the minima, even if (2.8) holds for all N of them. The selection of the particular two minima depends in general on R.

1.The Polar Form
We will utilize the polar form of a vector map u ∈ W 1,2 (A; R m ) ∩ L ∞ (A; R m ), A ⊂ R n open and bounded, (2.11) We have [9] q u ∈ W 1,2 (A) ∩ L ∞ (A) and ∇ν u is measurable and such that q u |∇ν u | ∈ L 2 (A) and (2.12) Moreover for q h ∈ W 1,2 (A) ∩ L ∞ (A), q u ≥ 0 the vector function σ defined via and satisfies the corresponding (2.12).
By the polar form (2.12) of the energy and the minimality of u assumed in (HB) it follows that where we have also used the definition (2.13) of σ which implies q σ ≤ q u .

2.The Isoperimetric Inequality for Minimizers
We will assume that q h ≥ q u on ∂B R and therefore by (2.13) that q σ = q u on ∂B R , q h to be further specified later. Define We also define the cut-off function which is related via the map a + βν u to the variation σ in (2.13). The modification in the definition of A with the integration over the sub-level set together with the definition of the function β in the context of the Caffarelli-Cordoba [14] set-up was introduced in Valdinoci [28]. By applying the inequality in [16] pag.141 to β 2 we obtain where C > 0 is a constant independent of R and we have used β = 0 on ∂B R and the fact that ∇β = 0 a.e. on q u − q σ > λ. By Young's inequality, for A > 0 we have From (2.18) and (2.14) it follows

The Upper Bound
The objective is to estimate the right and side of (2.19) by the first term involving the potential. Naturally the third term can be handled more easily for α < 2. For handling the second term one needs a very particular choice of q h . The splitting of the integrations over B R−T and the rest aims at deriving a difference inequality involving the quantities in (2.15), as in (2.33). A major difference between α < 2 and α = 2 is in the choice of q h , that can vanish on B R−T for α < 2, while can only be exponentially small (in T ) for α = 2. We begin with B R−T . Since q σ = 0 on B R−T the right hand side I of (2.19) on B R−T reduces to and therefore for A > αλ 2−α /C * we obtain −CA This and (2.22) conclude the proof of the claim.
Next we consider the right hand side of (2.19) on B R \ B R−T . Set and therefore from (2.5) it follows where W M = max |u−a|≤M W (u). As in the proof of Claim 1, for q σ ≤ q u ≤ λ ≤ min{ρ 0 , 1}, we get This and (2.25) establish Claim We now complete the definition (2.20) of q h by setting as in [14] (2.26) where c H is a constant that depends on H. Since τ < 2 implies τ − 1 < τ 2 , (2.28) yields

Claim 3
There existsĈ > 0 independent of R such that Proof. From (2.27) and q u = q σ on ∂B R and integration by parts it follows where we have observed that q u = q σ on the set {q h ≥ q u } and that q σ = q h on the set {q h < q u }. From (2.29) and q h ≤ q u it follows (2.32) As before we split the integration over {q u ≤ λ} and {q u > λ}. To conclude the proof we observe that λ ≤ min{ρ 0 , 1} and (HA) imply We are now in the position of completing the proof of Theorem A for the case 0 < α < 2.
Therefore as in [14], using also the assumption (2.6), we deduce that there are C(λ, µ 0 ) > 0 and k 0 ≥ 1 such that To complete the argument we recall the basic estimate (2.35) below (c.f. Lemma 1 in [14] for the scalar case. The proof is similar for the vector case) Lemma 2.2. Assume that W satisfies (HA) and assume that u is minimal as defined in (HB). Then there is a constant C > 0, depending on M , independent of ξ and such that This concludes the proof of Theorem A in the case 0 < α < 2 for λ > 0 small. The restriction on the smallness of λ is easily removed via (2.35).

4.The case α = 2.
We let ϕ : B R → R the solution of the problem where c 1 < c 0 will be chosen later and c 0 is the constant in (HA). It is well known that ϕ satisfies the exponential estimate for some c 2 > 0. Define and as before From (2.19), q σ = q u on ∂B R , and an integration by parts we get where we have used that q u > q σ implies q σ = q h , h = a + q h ν u . By (HA) there is λ * > λ sufficiently small (and fixed from now on) so that the maps s → W (a + sν) and  Set R = (k + 1)T where T > 0 is a large number to be chosen later. Set where c 2 is the constant in (2.37) and C 0 > 0 is a constant, C 0 = C 0 (A, λ, M ).
Proof. On B kT we have q h ≤ M e −c 2 T and therefore we can choose T > 0 so large that We begin by estimating part of the right hand side of (2.41) over B R \ B R−T by utilizing (2.47) and (2.32) where we have set W = max |u−a|≤M W (u) and C * = 2CA(W + c 1 M 2 ) + C A λ 2 . Next we estimate the remaining part of (2.41) over B R−T . The smoothness of W implies that there are C 0 > 0 andq > 0 such that We can assume T > 0 so large that M e −c 2 T ≤q. Then we have where we have set C • = 2CAM 2 ( 1 2 C 0 + c 1 ) and ǫ = e −c 2 T . From (2.50) we obtain Combining ( To estimate the left hand side of (2.52) from below we observe that (2.47) implies (2.53) Combining this with (2.52) we obtain (2.46). The proof of Claim 5 is complete Proof. We proceed by induction. For k = 1 (2.54) holds by (2.6) for any 0 < c * ≤ µ 0 , T ≥ 1. Thus we assume that (2.54) holds true for j ≤ k and show that it is true for k + 1. From the inductive assumption we have Therefore for the left hand side of (2.46) we have the lower bound Observe now that we have the obvious bound where η is the measure of the unit sphere in R n . Therefore we can derive for the right hand side of (2.46) the upper bound From this and (2.56) we get Since ǫ = e −c 2 T we can choose T > 0 so large that Then from (2.59) we obtain Therefore to complete the induction it suffices to observe that we can choose c * so small that

Let [R/T ] the integer part of R/T and observe that
[ From (2.54) and (2.55) we have Claim 6 concludes the case α = 2 and completes the proof of Theorem A for small λ > 0. As in the case α < 2 the restriction on the smallness of λ is removed via (2.35).
The proof is complete.
The following exponential estimate ( see [20] Theorem 1.3) can be considered a consequence of the density estimate in Theorem A. Proof. First we note that it is sufficient to establish that, given a small number λ > 0, there is d λ > 0 such that d(x, ∂D) ≥ d λ ⇒ |u(x) − a| ≤ λ since then linear theory renders the result. From Theorem 3.1 it follows that we can take d λ = R(λ). The proof is complete.

On the Linking with the Minimal Surface Problem
We will consider partitions with Dirichlet conditions for simplicity. The volume constraint case is more involved but similar. Assume that W is as in Theorem 2.1 and that therefore 0 = W (a 1 ) = W (a 2 ) < W (u), u ∈ {a 1 , a 2 } for a 1 = a 2 ∈ R m . Let {u ǫ k } be a sequence of global minimizers of J ǫ D (u) = D ( ǫ 2 2 |∇u| 2 + W (u))dy subject to the Dirichlet condition u ǫ k = g on ∂D, g : ∂D → {a 1 , a 2 }.
We assume D ⊂ R n open bounded with C 1 boundary and consider a partition of the boundary B j = g −1 ({a j }), j = 1, 2 with H n−1 (∂D \ (B 1 ∪ B 2 )) = 0. We also assume that u ǫ k L ∞ (D;R m ) < M uniformly. Then by the methods in Baldo [10] {u ǫ k } is relatively compact in L 1 (D; R m ) and along a subsequence Moreover the interface ∂D 1 ∩∂D 2 minimizes H n−1 (∂A 1 ∩∂A 2 ) among all partitions of D with Dirichlet conditions B. For two-phase partitions, if n ≤ 7, the interface ∂D 1 ∩ ∂D 2 is locally a real analytic classical minimal surface (see [21]).
Proof. (Blow-up, cfr. Theorem 2 in [14]) Suppose that the convergence is not uniform over a compact set K ⊂⊂ D. Then there are sequences ǫ k → 0 + , y k ∈ S ǫ k ∩ K, k = 1, . . . and r > 0 such that d(y k , ∂D 1 ∩ ∂D 2 ) ≥ r. We can assume that all the points y k are in one of the sets D j , j = 1, 2. For definiteness we suppose y k ∈ D 1 , k = 1, . . .. Actually K ⊂⊂ D implies that we can assume B r (y k ) ⊂ D 1 , k = 1, . . .. Set Since u ǫ k is a minimizer we have ∆v k − W (v k ) = 0, ̺ k (0) = γ and |v k − a| < M . Thus we also have the gradient bound |∇v k | < M which implies Thus we can apply Theorem A that yields the density estimate that holds uniformly over the family {v k }. This estimate is equivalent to In particular, for R = r/ǫ k , we get Since B r (y k ) ⊂ D 1 and ρ 0 = 0 a.e. on D 1 (4.3) implies Cr n which contradicts (4.1). The proof is complete.

A Lower Bound for the Energy
In this section we adopt the following hypothesis (HC) There exists N ≥ 2 and N distinct points a 1 , . . . , a N ∈ R m such that Moreover W : R m → R is as in (HA) for a = a j , j = 1, . . . , N .
From the monotonicity formula (see (1.4) in [2]), which holds for general Lipschitz W ≥ 0, it follows that any solution to ∆u − W u (u) = 0 satisfies the lower bound If W (u) = (1 − |u| 2 ) 2 and, more generally, if the set of the zeros of W is not totally disconnected, the lower bound above is sharp (see (2.4) in Farina [17]). On the other hand for the class of phase transition potentials defined in (HC) above, under the hypothesis of minimality we have Proposition 5.1. Let u : R n → R m be nonconstant and minimal in the sense of (HB), and pointwise bounded uniformly over R n (cfr. (2.5)). Then we have Proof. Since u is continuous and nonconstant there are γ > 0 and ξ ∈ R n such that |u(ξ) − a j | > γ, j = 1, . . . , N . Thus L n (B 1 (ξ) ∩ {|u(ξ) − a j | > γ/2}) ≥ µ 0 , j = 1, . . . , N for some µ 0 > 0 and so by Theorem A This and the same argument as in the proof of Theorem 2.1 imply It follows that, for each R ≥ 1, at least for two distinct a − , a + ∈ {a 1 , . . . , a N } we have Define ϕ : R n → R by setting ϕ(x) = d(a − , u(x)) with d(z 1 , z 2 ) 1 given by.
). (5.5) This and the relative isoperimetric inequality ( see [16] pag. 190) imply via (5.4) the estimate where α = C γ 2 and β = d(a − , a + ) − C γ 2 . From Proposition 2.1 in [10] and (2.5) we have that ϕ is lipschitz and for all bounded smooth open subsets A ⊂ R n . Therefore by the coarea formula, the estimate (5.6), and Young's inequality we obtain that concludes the proof.
We give another proof of Proposition 5.1 via linking with the sharp interface problem in [10].
Proof. (Blow-down) Let ξ ∈ R n as before and set x − ξ = y ǫ , u ǫ (y) = u(ξ + y ǫ ). Then (2.35) implies where r = ǫR is fixed once and for all. From (2.5) and (5.8) it follows (see pages 73, 82 in [10]) that u ǫ BV (Br(0);R m ) < C and so along a subsequence and by passing to the limit for k → ∞ we obtain From this it follows that at least for two distinct values a h = a l the sets A h , A l have full measure: Then the relative isoperimetric inequality implies where ∂A j is the the relative boundary of A j in B r (0) and C > 0 a constant. Finally by lower semicontinuity (see pag.76 in [10]) and (5.9) we have (5.10) Since the right hand side of (5.10) is independent of the particular subsequence {ǫ k } considered, we conclude that there is ǫ 0 > 0 such that and in the original variables To conclude the proof we show that given x 0 ∈ R n there is R(x 0 ) such that where C > 0 is the constant in (5.11). Indeed from (5.11), for R ≥ R 0 + |x 0 − ξ|, we have, with d = |x 0 − ξ|, This completes the proof.

PART II
6 Theorem B

Hypotheses and statement
In this subsection we consider in the class of symmetric solutions where for z ∈ R d , d ≥ 1 we denote byẑ the symmetric of z in the plane {z 1 = 0} that iŝ z = (−z 1 , z 2 , . . . , z d ).
We assume that W : R m → R is a C 3 potential that satisfies where W uu (a + ) is the Hessian matrix of W at a. The connection e is hyperbolic in the class of symmetric perturbations in the sense that the operator T defined by is the subspace of symmetric maps, satisfies for some η > 0. Here , is the inner product in L 2 (R; R m ) and the associated norm and W uu is the Hessian matrix of W .
(hc) u ∈ W 1,2 loc (R n ; R m ) ∩ L ∞ (R n ; R m ), is minimal in the class of symmetric maps in the sense that J Ω (u) ≤ J Ω (u + v), for each symmetric v ∈ W 1,2 0 (Ω; R m ) and for every open symmetric bounded lipschitz set Ω ⊂ R n . Moreover u satisfies the estimates (6.7) |u − a| + |∇u| ≤ Ke −kx 1 , on R n + for some k, K > 0.
Since we have it follows, via (6.7), that We denote E xp ⊂ W 1,2 S,loc (R; R m ) the exponential class of symmetric maps which, as e, satisfy (6.8) with k, K > 0 fixed constants.

Notes
(i) Under hypotheses (ha) by Theorems 3.6, 3.7 in [7] there is a connection e symmetric and global minimizer of A.
(ii) In the proof of Theorem B we utilize minimality only in symmetric cylinders.

Notation
As before by · we denote the Euclidean inner product in R d d ≥ 2. We write the typical x = (x 1 , . . . , x n ) ∈ R n in the form x = (s, y) with s = x 1 ∈ R and y = (x 2 , . . . , x n ) ∈ R n−1 . For r > 0 and y 0 ∈ R n−1 we set B r (y 0 ) = {y ∈ R n−1 : |y − y 0 | < r}. By C r (y 0 ) ⊂ R n we denote the cylinder R × B r (y 0 ).
Theorem B has the following important consequence Theorem 6.1. Assume that W satisfies (ha) and that u : R n → R m is minimal in the sense of (hc). Assume that there are exactly two global minimizers e + = e − of the action A in the symmetric class with the properties of e in (hb) above. Then the condition where C = C(µ 0 , λ, K), is independent of y 0 and independent of u. An analogous statement applies to e − .

1.The Polar Form and the Effective Potential
We will utilize the polar form with respect to e of a vector map u ∈ W 1,2 loc (R n ; R m ) ∩ L ∞ (R n ; R m ). We write u(s, y) = e(s) + q u (y)ν u (s, y), (s, y) ∈ R n where q u (y) = u(·, y) − e(· ) and ν u (·, y) = if q u (y) = 0, 0, otherwise . (6.14) We have (6.15) ∂u ∂y i = ∂q u ∂y i ν u + q u ∂ν u ∂y i and therefore observing that we obtain the following polar representation of the energy of u Cr(y 0 ) ∂ν u ∂y i 2 + W(u) + A(e) dy (6.17) where W : e + W 1,2 (R; R m ) → R the Effective Potential is defined by As it is standard in variational arguments, adding a constant to the integrand in (6.17) does not affect what follows. Therefore we disregard the constant A(e) in (6.17) and define the modified energy J Cr(y 0 ) (u) by setting where we have slightly abused the notation in (2.4). Note that (6.20) J Cr(y 0 ) (e) = 0.
Lemma 6.2. We have Assume that v(s) = e(s) + qν, q ∈ R, ν ∈ S satisfies (6.9). Then there are constants c 0 > 0 andq > 0 such that Proof. (i) follows from (hb). To prove (ii) we begin by differentiating twice W(e + qν) with respect to q. We obtain From the interpolation inequality: (6.23) applied to qν we obtain via the second inequality with M 1 the constant in (6.9), and via the first since qν = q and qν s ≤ M 1 . Therefore we have where W ′′′ is defined by where C 1 > 0 is a constant that depends on M 1 . We now observe that where we have also utilized (6.6). Thus (6.29) and (6.28)  In the following lemma we show that in the definition of minimality in (hc) we can extend the class of sets to include unbounded cylinders aligned to the x 1 axis.
Proof. Assume there are η > 0 and v ∈ u + W 1,2 0S (R × O; R m ) such that The minimality of u implies where we have also used the fact that both u and v belong to W 1,2 S (R × O; R m ) and satisfy (6.7). Taking the limit for l → +∞ in (6.33) yields in contradiction with (6.32).
For q h ∈ W 1,2 (C R ; R m ) ∩ L ∞ (C R ; R m ), q h ≥ 0, let the map σ defined via σ = e + q σ ν u , q σ = min{q h , q u }. We have σ ∈ W 1,2 (C R ; R m ) ∩ L ∞ (C R ; R m ) [9]. The minimality of u and the polar form (6.19) of the energy imply the inequality (6.34) Indeed minimality and Lemma 6.3 imply J C R (u) − J C R (σ) ≤ 0 and the second term is also nonpositive by 0 ≤ q σ ≤ q u .
2.An Upper Bound for the Energy Next we establish the analogous of Lemma 2.2 that is Lemma 6.4. Assume that W satisfies (ha) and assume that u is minimal as defined in (hc) and e a global minimizer of the Action as in (hb) above (hyperbolicity is not required). Then there is a constant C > 0 depending on K, independent of u and independent of y 0 such that 0 ≤ Cr(y 0 ) , for y ∈ B R−1 (y 0 ), e(· ) + (|y − y 0 | − R + 1)q u (y)ν u (·, y), for y ∈ B R (y 0 ) \ B R−1 (y 0 ).
(6.36) From Lemma 6.3 we have and via (6.7) The proof of the lemma is complete.

3.The Isoperimetric Inequality for Minimizers
As in the proof of the case α = 2 in Theorem A we let ϕ : B R ⊂ R n−1 → R be the solution of the problem ∆ϕ = c 1 ϕ, on B R , ϕ = 1, on ∂B R , (6.37) where c 1 < c 0 will be chosen later and c 0 is the constant in Lemma 6.2. We set q M = sup y∈R n−1 u(·, y) and define h = e + q h ν u , q h = ϕq M , and as before σ = e + q σ ν u , q σ = min{q u , q h }, β = min{q u − q σ , λ}, where λ ∈ (0,q) withq as in Lemma 6.2. We also recall the exponential estimate for some c 2 > 0. We remark that the definition of σ in (6.38) implies q σ = q u , on ∂B R and that σ ∈ W 1,2 (C R ; R m ) ∩ L ∞ (C R ; R m ) (see [9]). Proceeding as in the proof of Theorem A by applying the inequality in [16] on B R ⊂ R n−1 to β 2 yields where we have utilized ∇β = 0 a.e. on q u − q σ > λ and Young's inequality. Thus via (6.34) we derive

4.Conclusion
The inequality (6.41), aside from the fact that n is replaced by n − 1, B R is the ball of radius R in R n−1 and W is replaced by W, coincides with (2.19). Moreover, by Lemma 6.2, W has the properties of W in (HA), α = 2 and Lemma 6.4 is the counterpart of Lemma 2.2. The only difference is that the inequality Thus the arguments developed in the proof of Theorem A for the case α = 2 can be repeated verbatim to complete the proof of Theorem B.
The proof of Theorem 6.1 is complete.

On the Product Structure of Solutions
In this subsection we give alternative proofs of some of the results in [8]. Proof. It is sufficient to establish that, given a small number γ > 0, there is d γ > 0 such that (6.43) d(y, ∂O) ≥ d γ ⇒ |u(s, y) − e(s)| < γ.
Therefore arguing as in the proof of Theorem 3.1 we deduce from Theorem B and Lemma 6.4 that there is R(λ) > 0 such that This and (6.44) imply that we can take d γ = R( γ C ) 3 2 ) in (6.43). The proof is complete.
Choosing first p ′ so that p ′ (p − 1) = 2 and finally noting that max p−1 p = 2 3 we arrive at (6.46). The proof of the lemma is complete.
In [8] Theorem 6.5 was established by a different approach which also applies to a larger class of minimizers not necessarily defined on cylinders. We conclude with the following Rigidity result Theorem 6.7. (see Theorem 1.3 in [8]) Assume u : R n → R m and otherwise the hypothesis of Theorem 6.5. Then u(x) = e(x 1 ), for x ∈ R n .
The proof is concluded.