Calc. Var. (2018) 57:68 https://doi.org/10.1007/s00526-018-1343-9

Calculus of Variations

Regularity results for an optimal design problem with quasiconvex bulk energies Menita Carozza1 · Irene Fonseca2 · Antonia Passarelli di Napoli3

Received: 11 October 2016 / Accepted: 28 February 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract Regularity results for equilibrium configurations of variational problems involving both bulk and surface energies are established. The bulk energy densities are uniformly strictly quasiconvex functions with quadratic growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u, E), partial Hölder continuity of the gradient of the deformation u is proved, and partial regularity of the boundary of the minimal set E is obtained. Mathematics Subject Classification 49N15 · 49N60 · 49N99

1 Introduction and statements In this paper we study a large class of multidimensional vectorial variational problems involving both bulk and surface energies, relevant to a plethora of problems issuing from material science and imaging science. The regularity of solutions to these problems is a rather subtle issue even in the scalar setting. In [4,24] the authors established existence and regularity of minimal configurations of the model problem

Communicated by L. Ambrosio.

B

Irene Fonseca [email protected] Menita Carozza [email protected] Antonia Passarelli di Napoli [email protected]

1

Dipartimento di Ingegneria, Università del Sannio, Corso Garibaldi, 82100 Benevento, Italy

2

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA

3

Dipartimento di Mat. e Appl. “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy

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σ E (x)|∇u|2 d x + P(E, )

(1.1)

with u = 0 on ∂ and σ E (x) := aχ E + bχ\E for a > b positive constants, where  ⊂ Rn is an open, bounded domain, E ⊂ , and P(E, ) stands for the perimeter of the set E in . In [25] the authors treated more general bulk interfacial energies of the form    I (u, E) := F(x, u, ∇u) + χ E G(x, u, ∇u) d x + P(E, ) , 

subject to the constraints u =  on ∂ and |E| = d, requiring that F and G satisfy restrictive structure assumptions and are convex and with quadratic growth with respect to the gradient variable. Recently in [8] we still dealt with constrained convex scalar problems, without requiring any additional structure assumption on the bulk energies, and considering a general p-growth condition with respect to the gradient. This work is a natural extension of the above mentioned papers to the vectorial setting under the assumption of quasiconvexity on the bulk energies. To be precise, we consider an energy of the type    I (v, A) := (1.2) F(Dv) + χ A G(Dv) d x + P(A, ), 

1, p

where A ⊂  is a set of finite perimeter, u ∈ Wloc (; R N ), χ A is the characteristic function of the set A and P(A, ) denotes the perimeter of A in . We assume that F, G : R N ×n → R are C 2 integrands satisfying, for p > 1 and for positive constants 1 , 2 , L 1 , L 2 > 0 and μ ≥ 0, the following growth and uniformly strict p-quasiconvexity hypotheses, p

0 ≤ F(ξ ) ≤ L 1 (μ2 + |ξ |2 ) 2 ,    p−2 2 2 2 2 d x,  F(ξ + Dϕ) d x ≥  F(ξ ) + 1 |Dϕ| (μ + |Dϕ| )

(F1)p (F2)p

and p

0 ≤ G(ξ ) ≤ L 2 (μ2 + |ξ |2 ) 2 ,    p−2 2 2 2 2 dx  G(ξ + Dϕ) d x ≥  G(ξ ) + 2 |Dϕ| (μ + |Dϕ| )

(G1)p (G2)p

for every ξ ∈ R N ×n and ϕ ∈ C01 (; R N ). We will say that a pair (u, E) is a local minimizer of I in , if for every open set U   and every pair (v, A) where A is a set of finite perimeter with A E  U and v − u ∈ 1, p W0 (U ; R N ), we have       F(∇u) + χ E G(∇u) d x + P(E, U ) ≤ F(∇v) + χ A G(∇v) d x + P(A, U ). U

U

Existence and regularity of local minimizers of integral functionals of the type  F(x, Du), 

with uniformly strict p-quasiconvex integrand F and smooth dependence on the x variable, have been widely investigated ( we refer to [1,2,9–11,18,26] and for an exhaustive treatment to [17,20]). However, as far as we know, neither the existence nor the regularity of the local

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minimizers for functionals involving both bulk and surface energies of the form (1.2), are available in literature. Our results here are a first step to fill this gap. We first establish the existence of minimizers of I . Theorem 1.1 Let p > 1 and assume that (F1)p , (F2)p , (G1)p and (G2)p hold. Then, for 1, p v ∈ Wloc (; R N ) and a set of finite perimeter in , A ⊂ , and for every sequence (vk , Ak ) 1, p such that vk weakly converges to v in Wloc (; R N ) and χ Ak strongly converges to χ A in L 1loc (), we have I (v, A) ≤ lim inf I (vk , Ak ). k→+∞

1, p

In particular, I admits minimal configurations (u, χ E ) ∈ Wloc (; R N ) × BVloc (; [0, 1]). Next we establish a partial regularity result for minimal configurations of the functional I (v, A). Here we focus on the case of quadratic growth (i.e. p = 2). The case of general

p-growth will be treated in a forthcoming paper. We note that when p = 2, (F1)p , (F2)p , (G1)p and (G2)p reduce to 0 ≤ F(ξ ) ≤ L 1 (μ2 + |ξ |2 ),  

F(ξ + Dϕ) d x ≥

  

 F(ξ ) + 1 |Dϕ|2 d x,

(F1) (F2)

and 0 ≤ G(ξ ) ≤ L 2 (μ2 + |ξ |2 ),  

G(ξ + Dϕ) d x ≥

  

 G(ξ ) + 2 |Dϕ|2 d x.

(G1) (G2)

Theorem 1.2 Assume that (F1)–(F2) and (G1)–(G2) hold, and let (u, E) be a local minimizer of I . Then there exist an exponent β ∈ (0, 1) and an open set 0 ⊂  with full 1 measure such that u ∈ C 1,β (0 ). Also, ∂ ∗ E ∩ 0 is a C 1, 2 -hypersurface in 0 , and Hs ((∂ E\∂ ∗ E) ∩ 0 ) = 0 for all s > n − 8. If, in addition, L2 <1 1 + 2

(H )

then there exists an open set 1 ⊂  with full measure such that u ∈ C 1,α (1 ) for every α ∈ (0, 21 ). As it is usual in the vectorial setting, the proof is based on a comparison argument with solutions of a suitable linearized system, aiming at establishing decay estimates of some excess functions. The essential tool here is the use of suitable “hybrid” excess functions U∗ (xo , ρ) and U∗∗ (x0 , ρ) (see (5.1) and (5.50) respectively) that describe the oscillations of the gradient of the minimal deformation u and of the perimeter of the minimal set E in a ball. The decay estimates are achieved by considering points in  at which the excess is small, and using a blow-up argument reducing the problem to the study of convergence of the minimal configurations (u h , E h ) of a suitable rescaled functionals in the unit ball. This argument is hinged on two Caccioppoli type inequalities for minimizers of suitable perturbed rescaled

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functionals. Due to the particular form of our functional, these Caccioppoli type inequalities (see (5.18) and (5.60)) also involve quantities depending on the perimeter of the rescaled minimal set E h . In order to ensure that these terms vanish in the passage to the limit, we need to establish suitable a priori estimates for the perimeter of E h . Remark 1.1 Theorems 1.1 and 1.2 apply, in particular, to energies of the form    χ E F1 (Du) + (1 − χ E )F2 (Du) d x + P(E, ), (u, E)  → 

obtained from I by setting F := F2 and G := F1 − F2 , with F1 and F1 − F2 strict 2quasiconvex functions, and F1 (ξ ) ≥ F2 (ξ ) for all ξ ∈ R N ×n . Note that such assumptions are the natural extension to the vectorial setting of the model case (1.1) treated in [4,24], recalling that there a > b. We end the Introduction by referring to Arroyo-Rabasa [6] where regularity for vectorvalued free interface variational problems is treated within the context of k-th order homogeneous partial differential operators A (for a detailed study of A-quasiconvexification see [15]), and σ E |∇u|2 in (1.1) becomes σ˜ E (x)Au · u, with σ˜ E := σ1 χ E + σ2 χ\E , σ1 and σ2 being two positive symmetric tensors not necessarily well-ordered. Theorem 1.5 in [6] provides C 1,η/2 regularity for some η ∈ [0, 1] while in Theorem 1.2 we achieve C 1,1/2 .

2 Notations and preliminary results We denote by c a generic constant that may vary from expression to expression in the same formula and between formulas. Relevant dependencies on parameters and special constants will be suitably emphasized using parentheses or subscripts. The norms we use on Rn , R N and R N ×n are the standard Euclidean norms, denoted by | · |. In particular, for matrices ξ , η ∈ R N ×n we write ξ, η := trace(ξ T η) for the usual inner product of ξ and η, and 1 |ξ | := ξ, ξ 2 for the corresponding Euclidean norm. When a ∈ R N and b ∈ Rn we write a ⊗ b ∈ R N ×n for the tensor product defined as the matrix that has the element ar bs in its r-th row and s-th column. Observe that (a ⊗ b)x = (b · x)a for x ∈ Rn , and |a ⊗ b| = |a||b|. Let Br (x0 ) be the ball centered at x0 with radius r , and set  (u)x0 ,r = u(x) d x. Br (x0 )

We omit the dependence on the center when it is clear from the context. When F : R N ×n → R is sufficiently differentiable, we write d  d2  Dξ F(ξ )[η] :=  F(ξ + tη) and Dξ ξ F(ξ )[η, η] := 2  F(ξ + tη) dt t=0 dt t=0 for ξ , η ∈ R N ×n . Hereby, F  (ξ ) is interpreted both as a N ×n matrix and as the corresponding linear form on R N ×n , though |F  (ξ )| will always denote the Euclidean norm of the matrix F  (ξ ). The second derivative, F  (ξ ), is a real bilinear form on R N ×n . It is well-known that for quasiconvex C 1 integrands, the assumptions (F1) and (G1) yield the upper bounds 1

1

|Dξ F(ξ )| ≤ c1 L 1 (μ2 + |ξ |2 ) 2 and |Dξ G(ξ )| ≤ c2 L 2 (μ2 + |ξ |2 ) 2

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(2.1)

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for all ξ ∈ R N ×n , with c1 and c2 constants (see [26] or Lemma 5.2 in [20] ). Further, if F and G are C 2 , then (F2) and (G2) imply the following strong Legendre–Hadamard conditions D 2 F(Q)λi λ j μα μβ ≥ c3 |λ|2 |μ|2 ,

D 2 G(Q)λi λ j μα μβ ≥ c4 |λ|2 |μ|2

for all Q ∈ R N ×n , λ ∈ Rn , μ ∈ R N , where c3 = c3 (1 ) and c4 = c4 (2 ) are positive constants (see Proposition 5.2 in [20]). We will need the following regularity result (see [17,20]). Proposition 2.1 Let v ∈ W 1,2 (; R N ) be such that  ij Q αβ Dα vi Dβ ϕ j d x = 0 

αβ

for every ϕ ∈ C ∞ (; R N ), where Q i j are real valued numbers such that |Q αβ | ≤ L and the strong Legendre–Hadamard condition ij

ij

Q αβ λi λ j μα μβ ≥ |λ|2 |μ|2 is satisfied for all λ ∈ Rn , μ ∈ R N , for some , L > 0. Then v ∈ C ∞ , and for any ball B R (x0 ) ⊂  the following estimate holds   |Dv − (Dv)x0 , R |2 d x ≤ c R 2 |Dv − (Dv)x0 ,R |2 d x, B R (x0 )

2

2

B R (x0 )

where c = c(n, N , , L) . The next iteration lemma has important applications in regularity theory (for the proof we refer to [20], pp. 191–192). Lemma 2.1 Let 0 < ρ < R and let  : [ρ, R] → R be a bounded nonnegative function. Assume that for all ρ ≤ s < t ≤ R we have (s) ≤ ϑ(t) + A +

B (s − r )α

where ϑ ∈ (0, 1), α, A, B ≥ 0 are constants. Then there exists a constant c = c(ϑ, α) such that

  B  ρ ≤c A+ (R − ρ)α Given a Borel set E in Rn , P(E, ) denotes the perimeter of E in , defined as  1 N P(E, ) := sup divφ d x : φ ∈ C0 (; R ), |φ| ≤ 1 . E

It is known that, if E is a set of finite perimeter, then P(E, ) = Hn−1 (∂ ∗ E), where

P(E, Bρ (x)) >0 ∂ E := x ∈  : lim sup ρ n−1 + ρ→0



∗

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is the reduced boundary of E (for more details we refer to [5]). Given a set E ⊂  of finite perimeter in , for every ball Br (x)   we measure how far E is from being an area minimizer in the ball by setting   ψ(E, Br (x)) := P(E, Br (x)) − inf P(A, Br (x)) : A E  Br (x), χ A ∈ BV (Rn ) . The following regularity result, due to Tamanini (see [28]), asserts that if the excess ψ(E, Br (x)) decays fast enough when r → 0, then E has essentially the same regularity properties of an area minimizing set. Theorem 2.1 Let  be an open subset of Rn and let E be a set of finite perimeter satisfying, for some σ ∈ (0, 1), ψ(E, Br (x)) ≤ cr n−1+2σ for every x ∈  and every r ∈ (0, r0 ), with c = c(x), r0 = r0 (x) local positive constants. Then ∂ ∗ E is a C 1,σ -hypersurface in  and Hs ((∂ E\∂ ∗ E) ∩ )) = 0 for all s > n − 8. In order to perform the blow up procedure, it will be convenient to introduce suitable translations of the integrands F and G. To be precise, given a C 1 function f : R N ×n → R, Q ∈ R N ×n and λ > 0, set f Q,λ (ξ ) :=

f (Q + λξ ) − f (Q) − Dξ f (Q)λξ . λ2

(2.2)

Lemma 2.2 Let f be a C 2 (R N ×n ) function such that | f (ξ )| ≤ C|ξ |2

and

|Dξ f (ξ )| ≤ C|ξ |

and let f Q,λ (ξ ) be the function defined in (2.2). Then | f Q,λ (ξ )| ≤ c|ξ |2

and

|Dξ f Q,λ (ξ )| ≤ c|ξ |

(2.3)

for all ξ ∈ R N ×n and for some positive constant depending on |Q|. The proof can be found in [1, Lemma II.3, p. 264].

3 Lower semicontinuity This section is devoted to the proof of Theorem 1.1. We recall that a weakly convergent sequence can be truncated in order essentially to obtain an equi-integrable sequence still weakly converging to the same limit. This result is the decomposition lemma proved by Fonseca et al. (see Lemma 2.3 in [16], see also [1,12,22]) . Lemma 3.1 Let (vk ) ⊂ W 1, p (; R N ) be weakly converging to u. Then, there exists a subsequence (vk j ) and a sequence (u j ) ⊂ W 1,∞ (; R N ) such that (i) Ln ({vk j  = u j }) = o(1) and u j → u weakly in W 1, p (; R N ); (ii) (|Du j | p ) is equi-integrable. We now prove Theorem 1.1.

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1, p

Proof of Theorem 1.1 Fix v ∈ Wloc (; R N), A ⊂  a set of finite perimeter in  and 1, p consider (vk ) weakly converging to v ∈ Wloc (; R N), and (χ Ak) strongly converging in L 1loc () to χ A , with lim inf k→∞ I (vk , Ak ) < +∞. Without loss of generality ( and up to the extraction of a subsequence not relabeled), assume that the limits below exist,   lim F(Dvk ) d x, lim χ Ak G(Dvk ) d x and lim P(Ak , ). k→∞ 

k→∞ 

k→∞

In view of the lower semicontinuity property of the perimeter, and by the quasiconvexity and growth assumption on F (see (F1)p , (F2)p ), for all    P(A,  ) ≤ lim inf P(Ak ,  ) ≤ lim P(Ak , ) k→+∞

and



 

F(Dv) d x ≤ lim inf

 

k→∞

(3.1)

k→+∞

F(Dvk ) d x ≤ lim

k→∞ 

F(Dvk ) d x.

(3.2)

Moreover, up to the extraction of a further sequence (not relabeled), there exists (u k ) ∈ W 1,∞ (; R N) such that (i)–(ii) in Lemma 3.1 hold. Hence   χ Ak G(Dvk ) d x ≥ lim sup χ Ak G(Du k ) d x lim k→∞  {u k =vk }∩ k→∞   ≥ lim sup χ Ak G(Du k ) d x − lim sup χ Ak G(Du k ) d x  {u k =vk }∩ k→∞ k→∞   p χ Ak G(Du k ) d x − L 2 lim sup (μ2 + |Du k |2 ) 2 d x ≥ lim sup    {u k =vk }∩ k→∞ k→∞  χ Ak G(Du k ) d x, (3.3) = lim sup k→∞



where we used (G1)p in the second inequality, and the equi-integrability of (|Du k | p ) and condition (i) in Lemma 3.1 to obtain the last equality. Now,    lim sup χ Ak G(Du k ) d x ≥ lim sup χ A G(Du k ) d x − lim sup |χ Ak −χ A |G(Du k ) d x, k→∞



k→∞



k→∞



(3.4) with   |χ Ak − χ A |G(Du k ) d x ≤ L 2 

p

{|χ A −χ A k

since by Chebyshev’s inequality   {|χ − χ | ≥ 1} ∩   ≤ Ak A

|≥1}∩

(μ2 + |Du k |2 ) 2 d x → 0 as k → ∞,

 

|χ Ak − χ A | → 0 as k → ∞.

By the quasiconvexity and growth properties of G we have    lim sup χ A G(Du k ) d x = lim sup G(Du k ) d x ≥ k→∞



k→∞

 ∩A

 ∩A

G(Du) d x,

and the conclusion now follows from (3.1)–(3.5), and by letting   .

(3.5)  

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4 A higher integrability result This section is devoted to the proof of a higher integrability result for the gradient of the function u of the minimal configuration (u, E). Theorem 4.1 Assume that (F1)–(F2) and (G1)–(G2) hold, and let (u, E) be a local minimizer of I . Then there exists δ = δ(n, N , 1 , 2 , L 1 , L 2 ) > 0 such that for every ball B2r (x0 )   it holds

1   1+δ |Du|2(1+δ) d x ≤C |Du|2 d x + Cμ2 Br (x0 )

B2r (x0 )

where C = C(n, N , 1 , 2 , L 1 , L 2 ) > 0. Proof Consider 0 < r < s < t < 2r and let η ∈ C0∞ (Bt ) be a cut-off function between Bs c and Bt , i.e., 0 ≤ η ≤ 1, η ≡ 1 in Bs and |∇η| ≤ t−s . Set ψ1 := η(u − (u)x0 ,2r ) ψ2 := (1 − η)(u − (u)x0 ,2r ). By the uniformly strict quasiconvexity of F in (F2), we have    |Dψ1 (x)|2 d x ≤ F(Dψ1 ) d x = F(Du − Dψ2 ) d x. 1 Bt

Bt

We write   F(Du − Dψ2 ) d x ≤ Bt







F(Du) d x + Bt

F(Du − Dψ2 ) d x − 

Bt



Bt

≤ Bt



Bt

F(Du) d x Bt



1

F(Du) d x −

=

(4.1)

Bt

D F(Du − θ Dψ2 )Dψ2 dθ d x

0

 F(Du) + χ E G(Du) d x

  1 D F(Du − θ Dψ2 )Dψ2 dθ d x − Bt 0    F(Du − Dψ1 ) + χ E G(Du − Dψ1 ) d x ≤ Bt





1

− Bt

D F(Du − θ Dψ2 )Dψ2 dθ d x

(4.2)

0

where we used the fact that G(ξ ) ≥ 0 and the minimality of (u, E) with respect to (u−ψ1 , E). Inserting estimate (4.2) in (4.1), and using the upper bound on D F in (2.1), we obtain       |Du|2 d x = 1 |Dψ1 |2 d x ≤ F(Dψ2 ) d x + χ E G Dψ2 d x 1 Bs Bs Bt Bt  1 (μ2 + |Du|2 + |Dψ2 |2 ) 2 |Dψ2 | d x +c Bt \Bs   2 ≤c |Dψ2 | dy + c |Du|2 d x + cμ2 |Bt |  ≤c

123

Bt \Bs

Bt \Bs

 |Du|2 d x + c

Bt \Bs

   (u − (u)x0 ,2r ) 2  d x + cμ2 |Bt |,    t −s Bt \Bs

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where weused assumptions (F1) and (G1), Young’s inequality and the definition of ψ2 . Adding c Bs |Du|2 d x to both sides of the previous estimate we get 





|u − (u)x0 ,2r |2 d x + cμ2 |Bt | (t − s)2 Bs Bt Bt \Bs

  |u − (u)x0 ,2r |2 2 2 μ + dx ≤c |Du| d x + c (t − s)2 Bt B2r  and by the iteration Lemma 2.1 with (z) := Bz |Du|2 d x for z ∈ [r, 2r ], θ := 1c+c ,   A := B2r μ2 and B := B2r |u − (u)x0 ,2r |2 d x, we deduce that        u − u 2r 2 2 2  d x. |Du| d x ≤ c μ +   r Br B2r (1 + c)

|Du| d x ≤ c 2

|Du| d x + c 2

The Sobolev–Poincaré inequality ([20], p.102) implies that 

 |Du| d x ≤ c

|Du|

2

Br

2n n+2

n+2 n

dx

+ cμ2 ,

B2r

with the constant c depending only on n and not on r , and the conclusion follows by virtue of Giaquinta–Modica Theorem ([20], p. 203).  

5 The decay estimates Consider the excess function defined as  U (x0 , r ) :=

Br (x0 )

|Du(x) − (Du)x0 ,r |2 d x,

for Br (x0 ) ⊂ , and let the “hybrid” excess be given by  P(E, Br (x0 )) |Du(x) − (Du)x0 ,r |2 d x + + r. U∗ (x0 , r ) := r n−1 Br (x0 )

(5.1)

Proposition 5.1 Let (u, E) be a local minimizer of I under the assumptions (F1), (F2), (G1), (G2) and (H). For every M > 0 and every 0 < τ < 41 there exist ε0 = ε0 (τ, M) and c∗ = c∗ (M, 1 , L 1 , 2 , L 2 , n, N ) such that whenever Br (x0 )   verifies |(Du)x0 ,r | ≤ M and U∗ (x0 , r ) ≤ ε0 , then U∗ (x0 , τr ) ≤ c∗ τ U∗ (x0 , r ).

(5.2)

Proof In order to prove (5.2), we argue by contradiction. Let M > 0 and τ ∈ (0, 1/4) be such that for every h ∈ N, C∗ > 0, there exists a ball Brh (x h )   such that |(Du)xh ,rh | ≤ M, U∗ (x h , rh ) → 0

(5.3)

U∗ (x h , τrh ) ≥ C∗ τ U∗ (x h , rh ).

(5.4)

and

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The constant C∗ will be determined later. Remark that we can confine ourselves to the case in which E ∩ Brh (x h )  = ∅, since the case in which Brh (x h ) ⊂ \E is easier because then U = U∗ . Step 1 Blow-up. Set λ2h := U∗ (x h , rh ), Ah := (Du)xh ,rh , ah := (u)xh ,rh , and define vh (y) :=

u(x h + rh y) − ah − rh Ah y λh r h

(5.5)

for y ∈ B1 := B1 (0). One can easily check that (Dvh )0,1 = 0 and (vh )0,1 = 0. Set E − xh E − xh , E h∗ := ∩ B1 . E h := rh rh Note that

 U∗ (x h , rh ) =

|Du(x h + rh y) − Ah |2 dy + 

P(E, B(x h , rh ))

B1

=

rhn−1

+ rh

|λh Dvh |2 dy + P(E h , B1 ) + rh .

(5.6)

B1

By the definition of λh and by (5.6), it follows that  rh ≤ 1, |Dvh |2 ≤ 1, rh → 0, P(E h , B1 ) → 0, λ2h B1 (0)

P(E h , B1 ) ≤ 1. (5.7) λ2h

Therefore, by (5.3) and (5.7), there exist a subsequence of {vh } (not relabeled), A ∈ R N ×n and v ∈ W 1,2 (B1 ; R N ), such that vh  v weakly in W 1,2 (B1 ; R N ), vh → v strongly in L 2 (B1 ; R N ), Ah → A, λh Dvh  0 in L 2 (B1 ) and pointwise a.e.,

(5.8)

where we used the fact that (vh )0,1 = 0. Moreover, by (5.7) and (5.3), we also deduce that   n n−1   1 P(E h , B1 ) P(E h , B1 ) n−1 lim = lim P(E h , B1 ) lim sup = 0. (5.9) 2 h h λh λ2h h Therefore, by the relative isoperimetric inequality in a ball (see [5]),

 n |E h∗ | |B1 \E h | (P(E h , B1 )) n−1 lim min , = 0. ≤ c lim h h λ2h λ2h λ2h

(5.10)

We expand F and G around Ah as follows: F(Ah + λh ξ ) − F(Ah ) − Dξ F(Ah )λh ξ , λ2h G(Ah + λh ξ ) − G(Ah ) − Dξ G(Ah )λh ξ , G h (ξ ) := λ2h Fh (ξ ) :=

and we consider the corresponding rescaled functionals    Fh (Dw)dy + χ E ∗ G h (Dw) dy + P(E h , B1 ). Ih (w) := B1 (0)

123

h

(5.11)

(5.12)

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We claim that vh satisfies the minimality inequality Ih (vh ) ≤ Ih (vh + ψ) +

1 λh

 B1

χ E ∗ Dξ G(Ah )Dψ(y) dy,

(5.13)

h

for ψ ∈ W01,2 (B1 ). Indeed, using the change of variable x = x h + rh y, the minimality of h y) yields (u, E) with respect to (u +ϕ, E), for ϕ ∈ W01,2 (B(x h , rh )), setting ψ(y) := ϕ(xhr+r h 



 Fh (Dvh (y)) + χ E ∗ G h (Dvh (y)) dy h B1    Fh (Dvh (y) + Dψ(y)) + χ E ∗ G h (Dvh (y) + Dψ(y)) dy ≤ h B1  1 χ ∗ Dξ G(Ah )Dψ(y) dy + λh B1 Eh

and (5.13) follows by the definition of Ih in (5.12). Next we claim that    Fh (Dvh (y)) + G h (Dvh (y)) dy B1    Fh (Dvh (y) + Dψ(y)) + G h (Dvh (y) + Dψ(y)) dy ≤ B1  L2 + 2 (μ2 + |Ah + λh Dvh |2 ) dy, λh (B1 \E h )∩suppψ

(5.14)

(5.15)

for all ψ ∈ W01,2 (B1 ). In fact, the minimality of (u, E) with respect to (u + ϕ, E) for ϕ ∈ W01,2 (B(x h , rh )), implies that 

 B(x h ,rh )

(F + G)(Du) d x =  ≤

 B(x h ,rh ) B(x h ,rh )



  F(Du) + χ E G(Du) d x +

B(x h ,rh )\E

G(Du)d x

 F(Du + Dϕ) + χ E G(Du + Dϕ) d x



+  =

B(x h ,rh )\E

B(x h ,rh )

(F + G)(Du + Dϕ)d x



+  ≤



B(x h ,rh )\E

B(x h ,rh )

G(Du)d x

 G(Du) − G(Du + Dϕ) d x

(F + G)(Du + Dϕ)d x



+

(B(x h ,rh )\E)∩suppϕ

G(Du)d x,

(5.16)

where we used that last integral vanishes outside the support of ϕ and that G(ξ ) ≥ 0. Using the change of variable x = x h + rh y in (5.16), we get

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  (F + G)(Du(x h + rh y) + Dϕ(x h + rh y)) dy

(F + G)(Du(x h + rh y))dy ≤ B1

B1



+

(B1 \E h )∩suppψ

G(Du(x h + rh y))dy

h y) where, we recall, ψ(y) := ϕ(xλhh+r , or, equivalently, using the definitions of vh , rh   (F + G)(Ah + λh Dvh )dy ≤ (F + G)(Ah + λh (Dvh + Dψ))dy B1 B1  G(Ah + λh Dvh )dy +

(B1 \E h )∩suppψ

for all ψ ∈ W01,2 (B1 ). Therefore, setting Hh := Fh + G h by the definition of Fh and G h in (5.11) and using the assumption (G1), we have that    1 Hh (Dvh )dy ≤ Hh (Dvh + Dψ)dy + 2 G(Ah + λh Dvh ) dy λh (B1 \E h )∩suppψ B1 B1   L2 Hh (Dvh + Dψ)dy + 2 (μ2 + |Ah + λh Dvh |2 ) dy, ≤ λ B1 h (B1 \E h )∩suppψ (5.17) i.e. (5.15). Step 2. A Caccioppoli type inequality. We claim that there exists a constant c = c(M, μ, 1 , 2 , L 1 , L 2 , n, N ) such that for every 0 < ρ < 1 there exists h 0 ∈ N such that for all h > h 0 we have n   v − (v ) − (Dv ) ρ y 2  h ρ h 2  P(E h , B1 ) n−1  h 2 |Dvh − (Dvh ) ρ | dy ≤ c dy + c .   2  ρ λ2h Bρ Bρ  2

(5.18) We divide the proof into two substeps. Substep 2.a The case min{|E h∗ |, |B1 \E h |} = |E h∗ |. Consider 0 < ρ2 < s < t < ρ < 1 and let η ∈ C0∞ (Bt ) be a cut off function between Bs and c Bt , i.e., 0 ≤ η ≤ 1, η ≡ 1 on Bs and |∇η| ≤ t−s . Set bh := (vh ) Bρ , Bh := (Dvh ) B ρ , and 2 set wh (y) := vh (y) − bh − Bh y. Define h (ξ ) := F(Ah + λh Bh + λh ξ ) − F(Ah + λh Bh ) − Dξ F(Ah + λh Bh )λh ξ , F λ2h h (ξ ) := G(Ah + λh Bh + λh ξ ) − G(Ah + λh Bh ) − Dξ G(Ah + λh Bh )λh ξ . G λ2h (5.19)

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Regularity results for an optimal design problem with…

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68

h , G h , for some constants that could It is easy to check that Lemma 2.2 applies to each F depend on M (see (5.3)) and also on ρ through |λh Bh |. However, given ρ we may choose h large enough to have |λh Bh | < λhn < 1. In fact, by (5.7) we have ρ2

  ⎛ ⎞1 2      1 c |Bh | =  Dvh  ≤ ⎝ |Dvh |2 ⎠ · ≤ n, 1 2 ρ ρ ρ 2 ρ B B   |B | 2 2

(5.20)

2

and so the constant in (2.3) can be taken independently of ρ. Set ψ1,h (y) := ηwh and ψ2,h (y) := (1 − η)wh . h we have By the uniformly strict quasiconvexity of F    h (Dψ1,h ) dy = F |Dψ1,h (y)|2 dy ≤ 1 Bt

Bt

h (Dwh − Dψ2,h ) dy. F

(5.21)

Bt

Using the change of variable x = x h + rh y, the fact that G(ξ ) ≥ 0 and the minimality of (u, E) with respect to (u + ϕ, E) for ϕ ∈ W01,2 (B(x h , rh )), we have     F(Du(x h + rh y)) + χ E ∗ G(Du(x h + rh y)) dy F(Du(x h + rh y))dy ≤ h B1 B1    ≤ F(Du(x h + rh y) + Dϕ(x h + rh y)) + χ E ∗ G(Du(x h + rh y) + Dϕ(x h + rh y)) dy, h

B1

i.e., by the definitions of vh and wh ,  F(Ah + λh Bh + λh Dwh )dy B1    ≤ F(Ah + λh Bh + λh (Dwh + Dψ))+χ E ∗ G(Ah + λh Bh + λh (Dwh + Dψ)) dy h

B1

h y) h and G h in (5.19), for ψ := ϕ(xλhh+r ∈ W01,2 (B1 ). Therefore, recalling the definitions of F rh we have that     h (Dwh )dy ≤ h (Dwh + Dψ) dy h (Dwh + Dψ) + χ ∗ G F F Eh B1 B1    1 χ E ∗ G(Ah + λh Bh ) + Dξ G(Ah + λh Bh )λh (Dwh + Dψ) dy. + 2 λh B1 h (5.22)

Choosing −ψ1,h (y) as test function in (5.22), we get  Bt



    h Dwh − Dψ1,h dy h Dwh − Dψ1,h dy + χ ∗ G F Eh Bt    1 χ E ∗ G(Ah + λh Bh ) + Dξ G(Ah + λh Bh )λh (Dwh − Dψ1,h ) dy + 2 λ B1 h   h   h (Dψ2,h ) dy + h Dψ2,h dy χE ∗ G = F

h (Dwh ) dy ≤ F



Bt \Bs

+

1 λ2h





B1

Bt \Bs

h

 χ E ∗ G(Ah + λh Bh ) + Dξ G(Ah + λh Bh )λh Dψ2,h ) dy h

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 ≤

Bt \Bs

+c

|E h∗ |

Bt \Bs

+c

c + λh

λ2h

 ≤

h (Dψ2,h ) dy + F

E h∗

+

λ2h

c λh

  h Dψ2,h dy χE ∗ G

Bt \Bs



h

|Dψ2,h | dy

h (Dψ2,h ) dy + F

|E h∗ |





  h Dψ2,h dy χE ∗ G

Bt \Bs



h

1 2

E h∗

|Dψ2,h |2 dy

1

|E h∗ | 2 ,

(5.23)

for a constant c = c(M, μ, L 2 ), and where we used the second estimate in (2.1), Hölder’s inequality, and the fact that |Ah + λh Bh | ≤ M + 1 (see (5.20)). We write 

h (Dwh − Dψ2,h ) dy = F Bt

 

h (Dwh ) dy + F Bt

h (Dwh ) dy − F

= Bt

 

h (Dwh − Dψ2,h ) dy − F Bt

Bt





h (Dwh ) dy F Bt

1

h (Dwh − θ Dψ2,h )Dψ2,h dθ dy. DF

0

(5.24) h in Lemma 2.2, we Inserting estimate (5.23) in (5.24), and using the upper bound on D F obtain 

h (Dwh − Dψ2,h ) dy ≤ F

 Bt \Bs

Bt

h (Dψ2,h ) dy + F



+c +c

Bt \Bs

|E h∗ | λ2h

 Bt \Bs

  h Dψ2,h dy χE∗ G h

(|Dwh | + |Dψ2,h |)|Dψ2,h | dy

c + λh

1



2

|Dψ2,h (y)| dy 2

E h∗

1

|E h∗ | 2 . (5.25)

Hence, combining (5.21) with (5.25), using the properties of η and Lemma 2.2, we obtain  1

  |Dwh |2 dy = 1 |Dψ1,h |2 dy ≤ 1 |Dψ1,h |2 dy Bs Bs Bt      h Dψ2,h dy ≤ Fh (Dψ2,h ) dy + χE∗ G h Bt \Bs Bt \Bs  (|Dwh | + |Dψ2,h |)|Dψ2,h | dy +c Bt \Bs

+c  ≤c  ≤c

123

|E h∗ | λ2h

Bt \Bs

Bt \Bs

c + λh

1



2

|Dψ2,h (y)| dy 2

E h∗



|Dψ2,h |2 dy + c  |Dwh |2 dy + c

1

|E h∗ | 2

 Bt \Bs

Bt \Bs

χ E ∗ |Dψ2,h |2 dy + c h

  ∗  w h 2  dy + c |E h | ,  t − s  λ2 h

Bt \Bs

|Dwh |2 dy + c

|E h∗ | λ2h

Regularity results for an optimal design problem with…

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68

 where we used Young’s inequality. Adding to both sides of previous estimate c Bs |Dwh |2 dy we get    |E h∗ | c 2 |Dwh |2 dy ≤ c |Dwh |2 dy + |w | dy + c (1 + c) h (t − s)2 Bt \Bs λ2h Bs Bt   ∗ |E | c ≤c |Dwh |2 dy + |wh |2 dy + c 2h (5.26) 2 (t − s) Bρ λh Bt and by the iteration Lemma 2.1, we deduce that    2 ∗  wh    dy + c |E h | . |Dwh |2 dy ≤ c  ρ  2 λh Bρ Bρ 2

Therefore, by the definition of wh , we conclude that   v − (v ) − (Dv ) ρ y 2  h ρ h 2  |E ∗ |  h 2 |Dvh − (Dvh ) ρ | dy ≤ c   dy + c 2h , 2  ρ λh Bρ Bρ 

(5.27)

2

which, by the relative isoperimetric inequality and using the hypothesis of this substep that min{|E h∗ |, |B1 \E h |} = |E h∗ | , yields the estimate (5.18). Substep 2.b The case min{|E h∗ |, |B1 \E h |} = |B1 \E h |. As in the previous substep, we fix 0 < ρ2 < s < t < ρ < 1 and let η ∈ C0∞ (Bt ) be a cut c off function between Bs and Bt , i.e., 0 ≤ η ≤ 1, η ≡ 1 on Bs and |∇η| ≤ t−s . Also, we set bh := (vh ) Bρ , Bh := (Dvh ) B ρ and define 2

wh (y) := vh (y) − bh − Bh y and h := F h (ξ ) + G h (ξ ). H h and so Remark that Lemma 2.2 applies to H h (ξ )| ≤ c(M)|ξ |2 , |H and by the uniformly strict quasiconvexity conditions (F2) and (G2)      h (ξ ) +  Hh (ξ + Dψ) d x ≥ H |Dψ|2 d x, B1

(5.28)

B1

for all ψ ∈ W01,2 (B1 ), where   is such that   ≥ 1 + 2 . Set ψ1,h (y) := ηwh and ψ2,h (y) := (1 − η)wh . h (0) = 0, we have By (5.28) and since H     h (Dψ1,h ) dy = |Dψ1,h (y)|2 dy ≤ H  Bt

Bt

h (Dwh − Dψ2,h ) dy. H

(5.29)

Bt

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By virtue of the minimality inequality in (5.17) and since Dvh = Dwh + Bh , we get 

 Hh (Dwh + Bh )dy ≤ B1

Hh (Dwh + Bh + Dψ) dy  L2 (μ2 + |Ah + λh Bh + λh Dwh |2 ) dy, + 2 λh (B1 \E h )∩suppψ B1

h , or, equivalently, by the definition of H 

h (Dwh )dy ≤ H B1



h (Dwh + Dψ) dy H  L2 (μ2 + |Ah + λh Bh + λh Dwh |2 ) dy. + 2 λh (B1 \E h )∩suppψ B1

(5.30) h (0) = 0, we get Choosing − ψ1,h (y) as test function in (5.30) and using the fact that H 



  h Dwh (y) − Dψ1,h dy H Bt  L2 (μ2 + |Ah + λh Bh + λh Dwh |2 ) dy + 2 λh (Bt \E h )    h Dψ2,h dy = H Bt \Bs  L2 + 2 (μ2 + |Ah + λh Bh + λh Dwh |2 ) dy. λh Bt \E h

h (Dwh ) dy ≤ H Bt

(5.31)

Now we have that 

h (Dwh − Dψ2,h ) dy H    h (Dwh ) dy + h (Dwh − Dψ2,h ) dy − = H H

Bt



Bt

h (Dwh ) dy − H

=



Bt

Bt

Bt



h (Dwh ) dy H Bt

1

h (Dwh − θ Dψ2,h )Dψ2,h dθ dy. DH

(5.32)

0

h given by Hence, inserting the estimate (5.31) in (5.32), by the upper bound on D H Lemma 2.2 , we obtain 

h (Dwh − Dψ2,h ) dy ≤ H Bt



h (Dψ2,h ) dy H  L2 (μ2 + |Ah + λh Bh + λh Dwh |2 ) dy + 2 λh Bt \E h  +c (|Dwh | + |Dψ2,h |)|Dψ2,h | dy. (5.33) Bt \Bs

Bt \Bs

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Regularity results for an optimal design problem with…

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68

Combining (5.29) with (5.33) and using again Lemma 2.2 and (5.20), we have    Bs

 |Dψ1,h |2 dy ≤   |Dψ1,h |2 dy Bt   h (Dψ2,h ) dy + L 2 ≤ (μ2 + |Ah + λh Bh + λh Dwh |2 ) dy H 2 λ Bt \Bs h Bt \E h  +c (|Dwh | + |Dψ2,h |)|Dψ2,h | dy Bt \Bs



1 c(M, L 2 ) |Dψ2,h |2 dy + 1 + |B1 \E h | ε λ2h Bt \Bs   |Dwh |2 dy + c |Dwh |2 dy, + (1 + ε)L 2 

≤c

Bt \E h

Bt \Bs

for every ε > 0, and thus    Bs

 |Dwh |2 dy ≤   |Dψ1,h |2 dy Bt   ≤c |Dwh |2 dy + (1 + ε)L 2 Bt \Bs



+c Bρ

|Dwh |2 dy Bt

   w h 2 c(M, L 2 )   |B1 \E h |.  t − s  dy + λ2h

Using the hole filling technique as in (5.26) , we obtain (c +  )



  |Dwh |2 dy ≤ c + (1 + ε)L 2 Bs

|Dwh |2 dy Bt

 +c Bρ

   wh 2 c(M, L 2 )   |B1 \E h |.  t − s  dy + λ2h

The assumption (H) implies that there exists ε > 0 such that have

(1+ε)L 2 1 +2

< 1. Therefore we

c + (1 + ε)L 2 c + (1 + ε)L 2 <1 ≤ c + 1 + 2 c +  So, by virtue of the iteration Lemma 2.1, from the previous estimate we deduce that 

 Bρ

|Dwh |2 dy ≤ c Bρ

2

 2  wh    dy + c |B1 \E h | ,  ρ  λ2h

where c = c(M, μ, 1 , L 1 , 2 , L 2 , n, N ). Therefore, by the definition of wh , we conclude that 

 |Dvh − (Dvh ) ρ | dy ≤ c 2

Bρ

2

2

Bρ

   vh − (vh )ρ − (Dvh ) ρ y 2 |B1 \E h |  2  ,   dy + c   ρ λ2h

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which, by the relative isoperimetric inequality and since we have |B1 \E h | = min{|E h∗ |, |B1 \E h |}, gives the estimate (5.18). Step 3. We prove that   |Dv − (Dv) τ2 |2 d x ≤ cτ 2 Bτ

2

Bτ

|Dv − (Dv)τ |2 d x,

(5.34)

for Bτ = Bτ (0) with τ < 1. As before, we will divide the proof in two substeps. Let A and v be as introduced in (5.8). Substep 3.a The case min{|E h∗ |, |B1 \E h |} = |E h∗ |. We claim that v solves the linear system  Dξ ξ F(A)Dv Dψ dy = 0 B1

for all ψ ∈

C01 (B1 ).

Since vh satisfies (5.13), we have that  1 0 ≤ Ih (vh + sψ) − Ih (vh ) + χ ∗ Dξ G(Ah )s Dψ(y) dy λh B1 Eh

(5.35)

for every ψ ∈ C01 (B1 (0)) and s ∈ (0, 1). By the definition of Ih we get  1 0 ≤ Ih (vh + sψ) − Ih (vh ) + χ ∗ Dξ G(Ah )s Dψ(y) dy λh B1 Eh ⎛ ⎞  1   1 ⎝ = Dξ F(Ah + λh (Dvh + ts Dψ)) s Dψ dt − Dξ F(Ah )s Dψ ⎠ dy λh 0

B1

1 + λh + 1 = λh +

h



B1

⎛



⎝

h

0

B1

1 λh

χ E ∗ Dξ G(Ah )s Dψ(y) dy

1

h



⎞  Dξ F(Ah + λh (Dvh + ts Dψ)) s Dψ dt − Dξ F(Ah )s Dψ ⎠ dy

0

B1

1 λh

⎛ 1 ⎞    ⎝ χ ∗ Dξ G(Ah + λh (Dvh + ts Dψ)) s Dψ dt − χ ∗ Dξ G(Ah )s Dψ ⎠ dy E E



 1 χ E ∗ Dξ G(Ah + λh (Dvh + ts Dψ))]s Dψ dt dy h

B1 0

Dividing by s and taking the limit as s → 0, we deduce that    1 0≤ Dξ F(Ah + λh Dvh ) − Dξ F(Ah ) Dψdy λh B1  1 χ E ∗ Dξ G(Ah + λh Dvh )Dψ dy. + h λh B1

We partition the unit ball as − B1 = B+ h ∪ Bh = {y ∈ B1 : λh |Dvh | > 1} ∪ {y ∈ B1 : λh |Dvh | ≤ 1}.

123

(5.36)

Regularity results for an optimal design problem with…

By (5.7), we get



|B+ h|

≤

Page 19 of 34

68

 B+ h

λ2h |Dvh |2 dy

≤

λ2h

B+ h

|Dvh |2 dy ≤ cλ2h .

(5.37)

By virtue of the first estimate in (2.1) and Hölder’s inequality, we get    1   [Dξ F(Ah + λh Dvh ) − Dξ F(Ah )]Dψ dy    λh  B+h  c + ≤ |B | + c |Dvh | dy λh h B+ h  1 2

≤ cλh + c

B+ h

|Dvh |2 dy

2 |B+ h | ≤ cλh 1

(5.38)

for a constant c = c(L 1 , M), thanks to (5.4) (to bound |Ah | ≤ M), (5.7) and (5.37), and therefore    1   (5.39) [Dξ F(Ah + λh Dvh ) − Dξ F(Ah )]Dψ dy  = 0. lim   h→∞ λh  B+ h

On

B− h

we have 

1 λh

B− h

[Dξ F(Ah + λh Dvh ) − Dξ F(Ah )]Dψ dy

 

=

1

B− h 0

  =

B− h

1

0

 

+

Dξ ξ F(Ah + tλh Dvh ) dt Dvh Dψ dy

B− h 0

1

 Dξ ξ F(Ah + tλh Dvh ) − Dξ ξ f (A) dt Dvh Dψ dy Dξ ξ f (A) dt Dvh Dψ dy.

(5.40)

− By (5.3) and the definition of B− h we have that |A h + λh Dvh | ≤ M + 1 on Bh . Hence the uniform continuity of Dξ ξ F on bounded sets implies

     1    Dξ ξ F(Ah + tλh Dvh ) − Dξ ξ f (A) dt Dvh Dψ dy  lim   h  B− 0 h    1      ≤ lim Dξ ξ F(Ah + tλh Dvh ) − Dξ ξ f (A) dt  |Dvh ||Dψ| dy  B− h

h

 ≤ lim h

0

   − 

0

Bh

 ≤ c lim h

1

   − 

Bh

Dξ ξ F(Ah + tλh Dvh ) − Dξ ξ

1 0

 21   2  f (A) dt  dy ||Dvh || L 2 (B1 ) ||Dψ|| L ∞ (B1 )

Dξ ξ F(Ah + tλh Dvh ) − Dξ ξ

 21   2 f (A) dt  dy = 0,

(5.41)

where we used (5.7) and the fact that by (5.8) λh Dvh → 0 a.e. in B1 .

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Note that (5.37) yields that χB− → χ B1 in L r for every r < ∞. Therefore by (5.7) h

     1   1   lim  Dξ ξ f (A) dt Dvh Dψ dy − Dξ ξ f (A) dt Dvh Dψ dy   h  B− 0 0 B 1 h  1     ≤ lim |χB− − χ B1 |  Dξ ξ f (A) dt  |Dvh ||Dψ| dy h

h

B1

0

≤ c lim ||χB− − χ B1 || L 2 (B1 ) ||Dvh || L 2 (B1 ) = 0. h

(5.42)

h

Hence using (5.41) and (5.42) in (5.40), we have that   1 lim [Dξ F(Ah + λh Dvh ) − Dξ F(Ah )]Dψ dy = Dξ ξ F(A)Dv Dψ dy. h λh B− B1 h (5.43) By the second estimate in (2.1), we deduce that     1  1  2 2 2 ≤ 1 μ χ [D G(A + λ Dv )Dψ dy χ + |A + λ Dv | |Dψ| dy ∗ ∗ ξ h h h h h h  λh  B1 Eh λh B1 Eh  c ∗ |E | + c |Dvh | dy ≤ λh h E h∗ 

1 2 1 c ∗ 2 ≤ |E | + c |Dvh | dy |E h∗ | 2 λh h B1 c ∗ ∗ 21 ≤ |E | + c|E h | , λh h for a constant c = c(L 2 , M), thanks to (5.3) and (5.7). Since min{|E h∗ |, |B1 \E h |} = |E h∗ |, by (5.10) we have lim h

and so

|E h∗ | =0 λh

   1   = 0. χ D G(A + λ Dv )Dψ dy ξ h h h   E h∗ h→∞ λh B1 lim

(5.44)

By (5.39), (5.43) and (5.44), passing to the limit as h → ∞ in (5.36) yields  0≤ Dξ ξ F(A)Dv Dψ dy, B1

and with −ψ in place of ψ we get  Dξ ξ F(A)Dv Dψ dy = 0, B1

i.e., v solves a linear system with constant coefficients. By Proposition 2.1 we deduce that v ∈ C ∞ , and for every 0 < τ < 1, we have   |Dv − (Dv) τ2 |2 ≤ cτ 2 |Dv − (Dv)τ |2 d x ≤ cτ 2 , Bτ

2

123

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Regularity results for an optimal design problem with…

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68

since ||Dv|| L 2 (B1 ) ≤ lim sup ||Dvh || L 2 (B1 ) ≤ 1. h

min{|E h∗ |, |B1 \E h |}

= |B1 \E h |. Substep 3.b The case We claim that v solves the linear system  Dξ ξ (F + G)(A)Dv Dψ dy = 0, B1

for all ψ ∈ C01 (B1 ). Arguing as (5.16) and rescaling, we have that    1 Hh (Dvh )dy ≤ Hh (Dvh + s Dψ)dy + c s|Dψ|dy λh B1 \E h B1 B1  1  |Dvh + ts Dψ||s Dψ| dt dy, +c B1 \E h

0

for every ψ ∈ C01 (B1 (0)) and for every s ∈ (0, 1), and so   0≤ Hh (Dvh + s Dψ)dy − Hh (Dvh )dy B1

1 +c λh

B1



 B1 \E h

s|Dψ|dy + c



B1 \E h

1

|Dvh + ts Dψ||s Dψ| dt dy.

0

Therefore 

 0≤ B1

1

0

1 +c λh

Dξ Hh (Dvh + sθ Dψ)dθ s Dψdy 



 B1 \E h

s|Dψ|dy +

B1 \E h

1

|Dvh + ts Dψ||s Dψ| dt dy.

0

Dividing the previous inequality by s and taking the limit as s → 0, we obtain that    1 0≤ Dξ Hh (Dvh )Dψdy + c |Dψ|dy + |Dvh ||Dψ| dy. λh B1 \E h B1 B1 \E h By the definition of Hh , we conclude that    1 Dξ (F + G)(Ah + λh Dvh )Dψ − Dξ (F + G)(Ah )Dψ dy 0≤ λh B1   1 |Dψ|dy + c |Dvh ||Dψ| dy. +c λh B1 \E h B1 \E h Just as before, we partition B1 as − B1 (0) = B+ h ∪ Bh = {y ∈ B1 : λh |Dvh | > 1} ∪ {y ∈ B1 : λh |Dvh | ≤ 1},

and arguing as in (5.39) , we obtain that    1   lim [Dξ (F + G)(Ah + λh Dvh ) − Dξ (F + G)(Ah )]Dψ dy  = 0, (5.45)   h→∞ λh  B+ h

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and as in (5.43), lim h

1 λh

 B− h



= B1

[Dξ (F + G)(Ah + λh Dvh ) − Dξ (F + G)(Ah )]Dψ dy

Dξ ξ (F + G)(A)Dv Dψ dy.

(5.46)

Moreover, we have that 1 λh





1 2 1 c |B1 \E h | + c|B1 \E h | 2 |Dvh |2 dy λh B1 \E h 1 c ≤ |B1 \E h | + c|B1 \E h | 2 , λh

 B1 \E h

|Dψ|dy +

B1 \E h

|Dvh ||Dψ| dy ≤

where we used (5.7). Since min{|E h∗ |, |B1 \E h |} = |B1 \E h |, by (5.10), we have lim h

and we obtain

lim h

1 λh

|B1 \E h | = 0, λh



 B1 \E h

|Dψ|dy +

B1 \E h

|Dvh ||Dψ| dy

= 0.

(5.47)

By (5.45), (5.46) and (5.47), passing to the limit as h → ∞ in (5.45) we conclude that  Dξ ξ (F + G)(A)Dv Dψ dy 0≤ B1

and with −ψ in place of ψ we finally get  Dξ ξ (F + G)(A)Dv Dψ dy = 0, B1

asserting the claim. By Proposition 2.1, we deduce also in this case that v ∈ C ∞ and for every 0 < τ < 1 satisfies estimate (5.34). Step 4. An estimate for the perimeters.  where E  is a set of finite perimeter such By the minimality of (u, E) with respect to (u, E),  that E E  Brh (x h ), where Brh (x h ) are the balls of the contradiction argument, we get    Brh (x h )). χ E G(Du)d x + P(E, Brh (x h )) ≤ χ EG(Du)d x + P( E, Brh (x h )

Brh (x h )

Using the change of variable x = x h + rh y we have  χ E h G(Du(x h + rh y))dy + rhn−1 P(E h , B1 ) rhn B1  n h , B1 ), χ Eh G(Du(x h + rh y))dy + rhn−1 P( E ≤ rh B1

and so

 rh

χ E h G(Ah + λh Dvh )dy + P(E h , B1 )  h , B1 ). χ Eh G(Ah + λh Dvh )dy + P( E ≤ rh B1

B1

123

(5.48)

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Assume first that min{|B1 \E h |, |E h∗ |} = |B1 \E h |. Then by the relative isoperimetric inequality, we have n

|B1 \E h | ≤ c(n)P(E h , B1 ) n−1 . By Fubini’s Theorem and choosing as a representative of E h the set of points of density one, we get  2θ |B1 \E h | ≥ Hn−1 (∂ Bρ \E h )dρ, θ

for every θ ∈ (0, 1/4), therefore we may choose ρ ∈ (θ, 2θ ) such that Hn−1 (∂ Bρ \E h ) ≤

n c P(E h , B1 ) n−1 . θ

h := E h ∪ Bρ and observe that Set E h , B1 ) ≤ P(E h , B1 \ B¯ ρ ) + Hn−1 (∂ Bρ \E h ). P( E h , (5.48) yields With such a choice of E  χ Bρ G(Ah + λh Dvh )dy + P(E h , B1 \ B¯ ρ ) + Hn−1 (∂ Bρ \E h ) P(E h , B1 ) ≤ rh B1

and so by (5.3) and (5.8)

 n c P(E h , B1 ) n−1 + c(L 2 )rh (μ2 + |Ah + λh Dvh |2 ) d x θ B2θ n c ≤ P(E h , B1 ) n−1 + c(L 2 , μ, M)rh θ n + c(L 2 , μ, M)rh λ2h . θ

P(E h , Bθ ) ≤

(5.49)

We arrive at the same conclusion (5.49) if min{|B1 \E h |, |E h∗ |} = |E h∗ |, choosing as a comh = E h \Bρ . petiting set E Step 5. Conclusion. Using the change of variable x = x h + rh y and the Caccioppoli inequality in (5.18), for every 0 < τ < 41 we have  U∗ (x h , τrh ) 1 lim sup ≤ lim sup |Du(x) − (Du)xh ,τrh |2 d x 2 λ2h h→∞ h→∞ λh Bτrh (x h ) + lim sup h→∞

P(E, B(x h , τrh )) λ2h τ n−1 rhn−1



≤ c lim sup h→∞



Bτ

≤ c lim sup h→∞

B2τ

+ lim sup h→∞

τrh λ2h

|Dvh − (Dvh )τ |2 dy + lim sup |vh − (vh )2τ − (Dvh )τ τ2

h→∞ y|2

P(E h , Bτ ) +τ λ2h τ n−1

dy

  n rh τ n P(E h , B1 ) n−1 c c + n lim sup + n−1 lim sup + rh + τ, τ h→∞ τ λ2h λ2h h→∞ where we used (5.7) and estimate (5.49). By virtue of the strong convergence of vh → v in L 2 (B1 ), since (Dvh )τ → (Dv)τ in Rn N , by (5.8), (5.9), (5.10) and by the Poincaré–Wirtinger inequality, we get

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 U∗ (x h , τrh ) |v − (v)2τ − (Dv)τ y|2 ≤ c dy + cτ 2 τ2 λh B2τ  ≤ c lim |Dv − (Dv)τ |2 dy + cτ h→∞ B2τ 2

≤ cτ + cτ ≤ Cτ, where we used the estimate (5.34), and where C = C(M, μ, 1 , L 1 , 2 , L 2 , n, N ). The contradiction follows by choosing C∗ such that C∗ > C, since by (5.4) lim inf h

U∗ (x h , τrh ) ≥ C∗ τ. λ2h  

Next, we obtain a suitable decay estimate that allow us to prove Theorem 1.2 without the assumption (H). To this aim, we introduce a new “hybrid” excess as

δ  P(E, Br (x0 )) 1+δ U∗∗ (x0 , r ) := |Du(x) − (Du)x0 ,r |2 d x + + r β , (5.50) r n−1 Br (x0 ) δ . where δ has been determined in Theorem 4.1 and 0 < β < 1+δ In the proof of Proposition 5.2 we will only elaborate on the steps that substantially differ from the corresponding ones in the proof of Proposition 5.1.

Proposition 5.2 Let (u, E) be a local minimizer of I under the assumptions (F1), (F2), (G1) and (G2). For every M > 0 and every 0 < τ < 41 there exist ε0 = ε0 (τ, M) and c∗∗ = c∗∗ (M, 1 , L 1 , 2 , L 2 , n, N ) for which whenever Br (x0 )   verifies |(Du)x0 ,r | ≤ M and U∗∗ (x0 , r ) ≤ 0 , then U∗∗ (x0 , τr ) ≤ c∗∗ τ β U∗∗ (x0 , r ).

(5.51)

Proof In order to prove (5.51), we argue by contradiction. Let M > 0 and τ ∈ (0, 1/4) be such that that for every h ∈ N, C∗∗ > 0, there exists a ball Brh (x h )   such that |(Du)xh ,rh | ≤ M, U∗∗ (x h , rh ) → 0

(5.52)

U∗∗ (x h , τrh ) ≥ C∗∗ τ β U∗∗ (x h , rh ).

(5.53)

but

The constant C∗∗ will be determined later. Remark that we can confine ourselves to the case in which E ∩ Brh (x h )  = ∅, since the case in which Brh (x h ) ⊂ \E is easier because U = U∗∗ − r β where, we recall ,  U (x0 , r ) := |Du(x) − (Du)x0 ,r |2 d x Br (x0 )

for Br (x0 ) ⊂ . Step 1. Blow-up. Set λ2h := U∗∗ (x h , rh ), Ah := (Du)xh ,rh , ah := (u)xh ,rh , and define as before vh (y) :=

123

u(x h + rh y) − ah − rh Ah y λh r h

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68

for y ∈ B1 (0). One can easily check that (Dvh )0,1 = 0 and (vh )0,1 = 0. Again, as before, we set E h :=

E − xh , rh

E h∗ :=

E − xh ∩ B1 . rh

Note that 

 U∗∗ (x h , rh ) =  =

|Du(x h + rh y) − Ah | dy + 2

B1 (0)



δ 1+δ

rhn−1 δ

B1 (0)

P(E, B(x h , rh ))

β

+ rh

β

|λh Dvh |2 dy + (P(E h , B1 )) 1+δ + rh .

(5.54)

By the definition of λh , it follows that 

β

rh → 0,

P(E h , B1 ) → 0,

rh

λ2h

≤ 1,

δ

|Dvh | dy ≤ 1, 2

B1 (0)

(P(E h , B1 )) 1+δ ≤ 1. λ2h (5.55)

Therefore, by virtue of (5.52), (5.54) and (5.55), there exist a subsequence {vh } (not relabeled), A ∈ R N ×n and v ∈ W 1,2 (B1 (0); R N ), such that vh  v weakly in W 1,2 (B1 (0); R N ), vh → v strongly in L 2 (B1 (0); R N ), Ah → A, λh Dvh → 0 in L 2 (B1 (0)) and pointwise a.e.,

(5.56)

where we used the fact that (vh )0,1 = 0. We also note that δ

rh1+δ λ2h since 0 < β <  lim h

δ 1+δ .

β −β r h λ2h

→ 0,

(5.57)

Moreover, by (5.55) and (5.52), we deduce that

 n δ n−1 1+δ P(E h , B1 ) λ2h

δ

= rh1+δ

δ δ   (P(E h , B1 )) 1+δ (n−1)(1+δ) = lim P(E h , B1 ) lim sup = 0. h λ2h h (5.58)

Therefore, by the relative isoperimetric inequality in a ball (see [5]),   n δ

n−1 δ δ  1+δ ∗ P(E , B ) 1+δ h 1 1+δ |E h | |B1 \E h | lim min , = 0. ≤ c lim h h λ2h λ2h λ2h

(5.59)

Step 2. A Caccioppoli type inequality We claim that there exists a constant c = c(M, μ, 1 , 2 , L 1 , L 2 ) such that, for every 0 < ρ < 1, there exists h 0 ∈ N such that for all h > h 0 we have   v − (v ) − (Dv ) ρ y 2  h ρ h 2   h 2 ρ |Dvh − (Dvh ) | dy ≤ c   dy 2   ρ Bρ Bρ 2

+

c nδ

ρ 1+δ

nδ

P(E h , B1 ) (n−1)(1+δ) . λ2h

(5.60)

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We divide the proof into two substeps. Substep 2.a The case min{|E h∗ |, |B1 \E h |} = |E h∗ |. The proof of this substep goes exactly as that of Substep 2.a of Proposition 5.1 up to estimate (5.27). Next we observe that 

 |Dvh − (Dvh ) ρ | dy ≤ c 2

Bρ

2

Bρ

2

 ≤c Bρ

 ≤c Bρ

   vh − (vh )ρ − (Dvh ) ρ y 2 |E ∗ |  2    dy + c 2h   ρ λh  2  vh − (vh )ρ − (Dvh ) ρ y  ∗ δ 1 |E | 1+δ  2    dy + c|E h∗ | 1+δ h 2   ρ λh  2 δ  vh − (vh )ρ − (Dvh ) ρ y  |E ∗ | 1+δ  2    dy + c h 2 ,   ρ λh

and this, by the relative isoperimetric inequality and using the hypothesis of this substep that min{|E h∗ |, |B1 \E h |} = |E h∗ | , yields the estimate (5.60). Substep 2.b The case min{|E h∗ |, |B1 \E h |} = |B1 \E h | Fix 0 < ρ2 < s < t < ρ < 1 and let η ∈ C0∞ (Bt ) be a cut off function between Bs and Bt , c i.e., 0 ≤ η ≤ 1, η ≡ 1 on Bs and |∇η| ≤ t−s . Also, we set bh := (vh ) Bρ , Bh := (Dvh ) B ρ 2 and define wh (y) := vh (y) − bh − Bh y, ψ1,h (y) := ηwh and ψ2,h (y) := (1 − η)wh . We recall that 



  h Dwh (y) − Dψ1,h dy H Bt  L2 + 2 (μ2 + |Ah + λh Dvh |2 ) dy λh (Bt \E h )     L2  (μ2 + |Ah + λh Dvh |2 ) dy. = Hh Dψ2,h dy + 2 λh Bt \E h Bt \Bs (5.61)

h (Dwh ) dy ≤ H Bt

We remark that the higher integrability result of Theorem 4.1, through the change of variable x = x h + rh y, translates into the following

 |Du(x h + rh y)|2(1+δ) dy

1 1+δ

Bt

 ≤c

|Du(x h + rh y)|2 dy + cμ2 , B2t

or, equivalently,

 |Ah + λh Dvh |2(1+δ) dy Bt

123

1 1+δ

 ≤c

|Ah + λh Dvh |2 dy + cμ2 . B2t

(5.62)

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68

Using Hölder’s inequality and inequality (5.62) in the estimate (5.61), we get 

h (Dwh ) dy ≤ H Bt

 Bt \Bs

  h Dψ2,h dy H

1    1+δ δ L2 2(1+δ) + c(μ) 2 1 + |Ah + λh Dvh | dy |Bt \E h | 1+δ λh Bt \E h    h Dψ2,h dy ≤ H

Bt \Bs

1    1+δ δ L2 n 2(1+δ) 1+δ t + λ Dv | |B1 \E h | 1+δ 1 + |A dy h h h λ2h Bt   h Dψ2,h dy H

+ c(μ)  ≤

Bt \Bs

 δ L2 n 2 1+δ t |A + λ Dv | dy |B1 \E h | 1+δ 1 + h h h 2 λh B2t   n δ h Dψ2,h dy + c(μ, M) L 2 t 1+δ −n |B1 \E h | 1+δ . H 2 λh

+ c(μ)  ≤

Bt \Bs

Therefore we have 

h (Dwh ) dy ≤ H Bt

 Bt \Bs

  h Dψ2,h dy + H

δ

c nδ

ρ 1+δ

|B1 \E h | 1+δ , λ2h

(5.63)

where we used the fact that t > ρ2 . Now we observe that 

h (Dwh − Dψ2,h ) dy = H Bt



h (Dwh ) dy H   h (Dwh − Dψ2,h ) dy − + H Bt

Bt



h (Dwh ) dy − H

= Bt



 Bt

h (Dwh ) dy H Bt

1

h (Dwh DH

0

− θ Dψ2,h )Dψ2,h dθ dy.

(5.64)

h given by Hence, inserting the estimate (5.63) in (5.64), by the upper bound on D H Lemma 2.2 , we obtain 

h (Dwh − Dψ2,h ) dy ≤ H Bt

 Bt \Bs

h (Dψ2,h ) dy + H



+c

Bt \Bs

c nδ

ρ 1+δ

δ

|B1 \E h | 1+δ λ2h

(|Dwh | + |Dψ2,h |)|Dψ2,h | dy.

(5.65)

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Combining (5.29) with (5.65), using Lemma 2.2 and Young’s inequality, we have  |Dψ1,h |2 dy ≤  

   Bs

 |Dψ1,h |2 dy ≤

Bt \Bs

Bt



+c

Bt \Bs

 ≤c

Bt \Bs

|Dψ2,h |2 dy +

Bt \Bs

nδ

ρ 1+δ

(|Dwh | + |Dψ2,h |)|Dψ2,h | dy



+c

δ

|B1 \E h | 1+δ λ2h

c

h (Dψ2,h ) dy + H

δ

|B1 \E h | 1+δ λ2h

c nδ

ρ 1+δ

|Dwh |2 dy.

(5.66)

By the properties of η, we obtain by (5.66)  |Dwh |2 dy ≤  

   Bs

 |Dψ1,h |2 dy ≤ c Bt

+

 |Dwh |2 dy + c

Bt \Bs

c ρ

nδ 1+δ

|B1 \E h | λ2h

δ 1+δ

Bt \Bs

   w h 2    t − s  dy

.

Using the hole filling technique as in (5.26) , we get

(c +  )







|Dwh |2 dy ≤ c

|Dwh |2 dy + c

Bs

Bt \Bs

Bt





≤c

|Dwh | dy + c 2

Bt

Bρ

   w h 2    t − s  dy +

   w h 2    t − s  dy +

c nδ

ρ 1+δ c nδ

ρ 1+δ

δ

|B1 \E h | 1+δ λ2h δ

|B1 \E h | 1+δ . λ2h

By virtue of the iteration Lemma 2.1, from previous estimate we deduce that 

 |Dwh | dy ≤ c 2

Bρ

Bρ

2

 2  wh    dy +  ρ 

c nδ

ρ 1+δ

δ

|B1 \E h | 1+δ , λ2h

where c = c(M, μ, 1 , L 1 , 2 , L 2 , n, N ). Therefore, by the definition of wh , we conclude that 

 |Dvh − (Dvh ) ρ | dy ≤ c 2

Bρ

2

2

Bρ

  δ  vh − (vh )ρ − (Dvh ) ρ y 2 c |B1 \E h | 1+δ  2  ,   dy + nδ   ρ λ2h ρ 1+δ

which, by the relative isoperimetric inequality and since we have |B1 \E h | = min{|E h∗ |, |B1 \E h |}, gives the estimate (5.60). The proofs of the Step 3 and 4 of Proposition 5.1 hold true also in this case.

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Step 5. Conclusion. Using the change of variable x = x h + rh y and the Caccioppoli inequality in (5.60), for every 0 < τ < 41 we have lim sup h→∞

U∗∗ (x h , τrh ) 1 ≤ lim sup 2 λ2h h→∞ λh + lim sup h→∞



 Bτrh (x h )

≤ c lim sup h→∞





1 λ2h

Bτ

P(E, B(x h , τrh )



δ 1+δ

τ n−1 rhn−1

β

+ lim sup

τ β rh

h→∞

|Dvh − (Dvh )τ |2 dy + lim sup h→∞

1 λ2h



λ2h P(E h , Bτ ) τ n−1



δ 1+δ

+ τβ

|vh − (vh )2τ − (Dvh )τ y|2 dy τ2

≤ c lim sup h→∞

|Du(x) − (Du)xh ,τrh |2 d x

B2τ

nδ

δ P(E h , B1 ) (n−1)(1+δ) + c(τ, δ) lim sup + crh τ 1+δ + τ β , λ2h h→∞

where we used (5.55) and estimate (5.49). By virtue of the the strong convergence of vh → v in L 2 (B1 ), since (Dvh )τ → (Dv)τ in Rn N , by (5.55), (5.56), (5.57) (5.58), (5.59) and by the use of Poincaré–Wirtinger inequality, we get  U∗ (x h , τrh ) |v − (v)2τ − (Dv)τ y|2 lim sup ≤ c dy + τ β τ2 λ2h B2τ h→∞  ≤ c lim |Dv − (Dv)τ |2 dy + τ β h→∞ B2τ 2 β

≤ cτ + τ ≤ Cτ β , where we used the estimate (5.34) and where C = C(M, μ, 1 , L 1 , 2 , L 2 , n, N ). The contradiction follows by choosing C∗∗ such that C∗∗ > C, since by (5.4) lim inf h

U∗ (x h , τrh ) ≥ C∗∗ τ β . λ2h  

6 Proof of the main theorem Here we give the proof of Theorem 1.2 through a suitable iteration procedure.

6.1 An iteration lemma In the next Lemma the constant c∗ is that introduced in (5.2). Lemma 6.1 Let (u, E) be a minimizer of the functional  I .1 For every M > 0, for every − α ∈ (0, 1) and for every ϑ ∈ (0, ϑ0 ), with ϑ0 := min c∗ 1−α , 41 , there exist ε1 > 0 and R > 0 such that, if r < R and x0 ∈  satisfy Br (x0 )  , |Du|x0 ,r < M and U∗ (x0 , r ) < ε1 ,

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then U∗ (x0 , ϑ k r ) ≤ ϑ kα U∗ (x0 , r )

(Dk )

for all k ∈ N. Proof Fix M > 0 and ϑ ∈ (0, ϑ0 ). Let

  1 ε1 := min ε0 , (1 − ϑ 2 )2 ϑ n+1

where ε0 is determined in Proposition 5.1 corresponding to M + 1 and to ϑ. We will argue by induction. First note that (D1 ) holds true by virtue of Proposition 5.1 and since ε1 < ε0 and c∗ ϑ ≤ ϑ α . Assume that (Dk ) hold true for k ≤ h and we prove that (Dh+1 ) is satisfied. Observe that |Du|x0 ,ϑ h r ≤ |Du|x0 ,r +

h 

||Du|x0 ,ϑ j r − |Du|x0 ,ϑ j−1 r |

j=1

≤ |Du|x0 ,r +

h 

|(Du)x0 ,ϑ j r − (Du)x0 ,ϑ j−1 r |

j=1

≤ |Du|x0 ,r +

≤ |Du|x0 ,r +

h  

1

 h 

2

|Du − (Du)x0 ,ϑ j−1 r | d x 2

Bϑ j r

j=1

≤ |Du|x0 ,r +

|Du − (Du)x0 ,ϑ j−1 r |d x

j=1 Bϑ j r

h  |B

ϑ j−1 r |

|Bϑ j r |

j=1 n

≤ |Du|x0 ,r + ϑ − 2

h 

1

1  2

2

|Du − (Du)x0 ,ϑ j−1 r | d x 2

Bϑ j−1 r 1

U (x0 , ϑ j−1 r ) 2

j=1 h 

n

≤ M + ϑ− 2

1

U∗ (x0 , ϑ j−1 r ) 2

j=1 1

n

1

≤ M + (c∗ ) 2 ϑ − 2 U∗ (x0 , r ) 2

h 

ϑ

j−1 2

j=1

≤ M + ϑ−

n+1 2

≤ M + ϑ−

n+1 2

≤ M + 1,

1 2

ε1

1

ε12

1 2

(c∗ ) ϑ

1 2

1

1−ϑ2 1 1

1−ϑ2

(6.1)

because, by our choice of ε1 , we have ϑ−

123

n+1 2

1

ε12

1 1

1−ϑ2

≤ 1.

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Moreover, since (Dh ) holds true, we have that U∗ (x0 , ϑ h r ) ≤ ϑ hα U∗ (x0 , r ) < ε1 ,

(6.2)

by our choice of ϑ and ε1 , and so by (6.1) we can apply Proposition 5.1 with ϑ h r in place of r to deduce U∗ (x0 , ϑ h+1 r ) ≤ ϑ α U∗ (x0 , ϑ h r ) ≤ ϑ (h+1)α U∗ (x0 , r ), by (6.2). Therefore (Dk ) holds true for every k ∈ N.

 

Arguing exactly in the same way and by using Proposition 5.2 instead of Proposition 5.1, we have the following

Lemma 6.2 Let (u, E) be a minimizer of the functional I and let β be the exponent of Lemma 5.2. For every M > 0 and for every ϑ ∈ (0, ϑ0 ), with ϑ0 < min c∗∗ , 41 , there exist ε1 > 0 and R > 0 such that, if r < R and x0 ∈  satisfy Br (x0 )  , |Du|x0 ,r < M and U∗∗ (x0 , r ) < ε1 , then U∗∗ (x0 , ϑ k r ) ≤ ϑ kβ U∗∗ (x0 , r ) for all k ∈ N.

6.2 Proof of Theorem 1.2 Proof Assume first that (F1), (F2), (G1), (G2) and (H) hold, consider the set 0 := {x ∈  : lim sup |(Du)x,ρ | < +∞ and lim sup U∗ (x, ρ) = 0}, ρ→0

ρ→0

and let x0 ∈ 0 . For every M > 0 and for ε1 determined in Lemma 6.1 there exists a radius R M,ε1 such that |Du|x0 ,r < M and U∗∗ (x0 , r ) < ε1 ,

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for every 0 < r < R M,ε1 . If 0 < ρ < ϑ20 r < R, let h ∈ N be such that ϑ h+1 r < ρ < ϑ h r , and let ϑ = ϑ20 where ϑ0 is as in Lemma 6.1. By Lemma 6.1, we obtain  P(E, Bρ (x0 )) |Du − (Du)x0 ,ρ |2 d x + +ρ U∗ (x0 , ρ) = ρ n−1 Bρ  P(E, Bρ (x0 )) = |Du − (Du)ϑ h r + (Du)ϑ h r − (Du)x0 ,ρ |2 d x + +ρ ρ n−1 Bρ   P(E, Bρ (x0 )) |Du −(Du)ϑ h r |2 d x +2 |(Du)ϑ h r − (Du)x0 ,ρ |2 + +ρ ≤2 ρ n−1 Bρ Bρ  P(E, Bρ (x0 )) ≤c |Du − (Du)ϑ h r |2 d x + +ρ ρ n−1 Bρ h n  P(E, Bρ (x0 )) ϑ r ≤c |Du − (Du)ϑ h r |2 d x + +ρ ρ ρ n−1 Bϑ h r h+1 n  P(E, Bρ (x0 )) ϑ r |Du − (Du)ϑ h r |2 d x + +ρ =c ϑρ ρ n−1 Bϑ h r n  P(E, Bρ (x0 )) ρ |Du − (Du)ϑ h r |2 d x + +ρ ≤c ϑρ (ϑ h+1 r )n−1 Bϑ h r  1 P(E, Bϑ h r (x0 )) c |Du − (Du)ϑ h r |2 d x + n−1 + ϑ hr ≤ n ϑ Bϑ h r ϑ (ϑ h r )n−1  2n−1 P(E, Bϑ h r (x0 )) c |Du − (Du)ϑ h r |2 d x + n−1 + ϑ hr = n ϑ0 Bϑ h r (ϑ h r )n−1 ϑ0  ρ α ≤ CU∗ (x0 , ϑ h r ) ≤ C∗ ϑ hα U∗ (x0 , r ) ≤ C∗ U∗ (x0 , r ), (6.3) r  1 − 1−α 1 , 4 , ϑ = ϑ20 and ϑ h+1 r < ρ < ϑ h r . The where we used the fact that ϑ0 := min c∗ previous estimate implies that   ρ α U (x0 , ρ) = |Du − (Du)ρ |2 d x ≤ C∗ U∗ (x0 , r ). r Bρ Since U∗ (y, r ) is continuous in y, we have that U∗ (y, r ) < ε1 for all y in a suitable neighborhood I of x0 . For every y ∈ I we then have that  ρ α U∗ (y, r ). U (y, ρ) ≤ C∗ r The last inequality implies, by the Campanato characterization of Hölder continuous functions [20, Theorem 2.9], that u is C 1,α in I for every 0 < α < 21 , and we conclude that the function u has Hölder continuous derivatives in an open set 0 that contains all points y such that lim sup U∗ (y, r ) = 0. r →0

Next we prove a suitable decay estimate for the perimeter of the minimal set. For every 0 < ρ < ϑ20 r , let h ∈ N be such that ϑ h+1 r < ρ < ϑ h r , where ϑ = ϑ20 as before. We observe that

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|(Du)ρ | ≤ |Du|ρ ≤

ϑ hr ρ

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n  |Du| ≤ Bϑ h r

68

M +1 c(M) = = c(M, c∗ ), ϑn ϑ0n

(6.4)

where we used (6.1). Consider any set A of finite perimeter such that E A ⊂⊂ Bρ (x0 ). From the minimality of (u, E) we have that    F(Du) + χ E G(Du) d x + P(E, )     F(Du) + χ A G(Du) d x + P(A, ). ≤ 

Using the fact that E A ⊂⊂ Bρ (x), we deduce that    P(E, Bρ (x0 )) − P(A, Bρ (x0 )) ≤ χ A (x) − χ E (x) G(Du) d x Bρ (x0 )



≤ L2 

Bρ (x0 )

(μ2 + |Du|2 ) d x

|Du − (Du)ρ |2 d x + c(μ, Bρ (x0 ) cρ n U (x0 , ρ) + c(μ, M, c∗ )ρ n ρα Cρ n α U∗ (x0 , r ) + c(μ, M, c∗ )ρ n r c(μ, M, c∗ , r )ρ n ,

≤c = ≤ ≤

M, c∗ )ρ n

where we invoked the assumption (G2) and we used estimates (6.4) and (6.3). At this point the result follows from Theorem 2.1. When the assumption (H) is not enforced, the proof goes exactly in the same way provided we use Lemma 6.2 in place of Lemma 6.1, with 1 := {x ∈  : lim sup |(Du)x0 ,ρ | < +∞ and lim sup U∗∗ (x0 , ρ) = 0}. ρ→0

ρ→0

  Acknowledgements The authors warmly thank the Center for Nonlinear Analysis, where part of this research was carried out. The research of I. Fonseca was partially funded by the National Science Foundation under Grant no. DMS-1411646. The research of M. Carozza and A. Passarelli di Napoli was partially supported by MIUR through the Project PRIN (2012) “Calcolo delle Variazioni” and by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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