Minimizing movements for dislocation dynamics with a mean curvature term

DOI:10.1051/cocv:2008027 www.esaim-cocv.org

MINIMIZING MOVEMENTS FOR DISLOCATION DYNAMICS WITH A MEAN CURVATURE TERM

Nicolas Forcadel

and Aur´ elien Monteillet

Abstract. We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.

Mathematics Subject Classification.53C44, 49Q15, 49L25, 28A75, 58A25.

Received May 25, 2007. Revised December 20, 2007.

Published online March 28, 2008.

1. Introduction

In this paper, we investigate the existence of minimizing movements (see Almgren, Taylor and Wang [1], Ambrosio [5], and the book by Ambrosio, Gigli and Savar´e [6]) for a non-local geometric law governing the movement of a family {K(t)}0≤t≤T of compact subsets ofR^N:

V_x,t=H_x,t+c₀(·, t)1_K(t)(x) +c₁(x, t), (1.1) whereV_x,t denotes the normal velocity at timetof a pointxof∂K(t),H_x,tthe mean curvature of∂K(t) atx (with negative sign for convex sets),is the convolution in space,1_K(t)is the indicator function of the setK(t) andc₀, c₁:R^N×[0, T]→Rare given functions.

The non-local dependencec₀(·, t)1_K(t)in the expression ofV_x,tis typical of models for dislocation dynamics (see Alvarez et al. [4]). Moreover we think of the term c₁ as a prescribed driving force. Equation (1.1) with only these two terms (and without a mean curvature term) is currently also a center of interest: in the context

Keywords and phrases.Front propagation, non-local equations, dislocation dynamics, mean curvature motion, viscosity solutions, minimizing movements, sets of finite perimeter, currents.

1CERMICS, ´Ecole des Ponts, Paris Tech, 6 et 8 avenue Blaise Pascal, Cit´e Descartes, Champs-sur-Marne, 77455 Marne-la-Vall´ee Cedex 2, France. [email protected]

2 Universit´e de Bretagne Occidentale, UFR Sciences et Techniques, 6 av. Le Gorgeu, BP 809, 29285 Brest, France.

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2008

of viscosity solutions, its level-set formulation, namely

u_t(x, t) = [c₀(·, t)1_{{u(·,t)≥0}}(x) +c₁(x, t)]|Du(x, t)|, (1.2) was first investigated by Alvarezet al.[4], who proved short time existence and uniqueness of a viscosity solution to (1.2), and then by Alvarez, Cardaliaguet and Monneau [2], and by Barles and Ley [9], who proved, by different methods, long time existence and uniqueness under suitable monotonicity assumptions. In (1.2) and throughout the paper, u_t denotes the time derivative ofu, Du denotes the space gradient of u, and | · | is the standard Euclidean norm. The mean curvature term in (1.1) corresponds to an additional line tension term in the elastic energy of the dislocation which better approximates what happens near the dislocation (see the introduction of [18] for a discussion on the model). The level-set formulation of the geometric law (1.1),

u_t(x, t) =

div Du

|Du|

(x, t) +c₀(·, t)1_{{u(·,t)≥0}}(x) +c₁(x, t)

|Du(x, t)|, (1.3) was studied by the first author in [18]. He proved short time existence and uniqueness of a viscosity solution to (1.3).

In both cases, the source of major difficulties is the non-local dependence in the expression of the velocity, c₀(·, t)1_K(t), which prevents comparison principle to hold. Indeed, c₀ is not necessarily non-negative, and physical models show that this situation can not be avoided. The problem of existence and uniqueness of a viscosity solution to the level-set equations (1.2) and (1.3) for general kernels c₀ is therefore still open. For example, the long time existence and uniqueness results mentioned above were obtained under the assumption thatc₀(·, t)1_E+c₁(x, t)≥0 for any setE, which guarantees that the dislocation is expanding, and a regularity assumption on the initial shapeK(0). The short time existence and uniqueness for (1.3) was obtained in the case where the initial shape is a graph or a Lipschitz curve, without assumption on the sign of the non-local term. It is worth mentioning however that this equation benefits from the regularizing effect of the mean curvature term.

To overcome this difficulty, Barleset al. defined in [7] a notion of weak solution for (1.2), and proved existence of such weak solutions under general assumptions onc₀ andc₁. A similar concept of solution already appears in [28] for FitzHugh-Nagumo systems. In this work, we wish to provide such weak solutions for (1.1). We will work with set-valued mappings E : [0, T] → P(R^N) with uniformly bounded images which are continuous in theL¹topology, that is to say,t→1_E(t)belongs toC⁰([0, T], L¹(R^N)). We assume thatc₀andc₁satisfy some regularity assumptions which guarantee that (x, t) →c₀(·, t)1_E(t)(x) +c₁(x, t) is smooth enough for such a mapping E. Let us now explain what we call a weak solution of (1.1):

Definition 1.1 (weak solutions). Assume that c₀ ∈ Lip

[0, T], L¹(R^N)

, c₁ ∈Lip

[0, T], L^∞(R^N)

, that c₀ andc₁are continuous onR^N×[0, T] and Lipschitz continuous in space. LetE: [0, T]→ P(R^N) be a set-valued mapping with uniformly bounded images such thatt→1_E(t)belongs toC⁰([0, T], L¹(R^N)).

Letube the unique uniformly continuous viscosity solution of

⎧⎪

⎨

⎪⎩

u_t(x, t) =

div Du

|Du|

(x, t) +c₀(·, t)1_E(t)(x) +c₁(x, t)

|Du(x, t)| for (x, t)∈R^N×(0, T) u(x,0) =u₀(x) forx∈R^N,

(1.4)

whereu₀ is a uniformly continuous function such thatE₀={u₀≥0},E^◦₀={u₀>0}.

We say thatE is a weak solution of (1.1) if we have, for allt∈[0, T], and almost everywhere in R^N, {u(·, t)>0} ⊂E(t)⊂ {u(·, t)≥0}.

The goal of this paper is to construct a weak solution to the geometric law (1.1). To do this, we wish to adapt the approach of Almgren, Taylor and Wang [1] (also discovered independently by Luckhaus and Sturzenhecker [22]) – initially proposed for the mean curvature motion – to the geometric law (1.1) with its additional non-local term and driving force. The idea of minimizing movements is, for a given initial setE₀, to select a sequence of setsE_h(k) associated with time-steps of sizehby minimizing a suitable functional, so that the corresponding Euler equation is a discretization of our evolution law. A compactness result for sets of finite perimeter guarantees the existence of a subsequence (h_n) and a set-valued mapping E : [0, T]→ P(R^N) such that E_hn([t/h_n]) converges to E(t) inL¹(R^N) for allt, where [·] denotes the integer part. Such a E is called a minimizing movement (or generalized minimizing movement) associated to the geometric law. Moreover, we provea prioriestimates for the discrete evolutionE_h, which imply the H¨older continuity of the limitE in the appropriate metric. This guarantees that the setsE(t) cannot vary in a wildly discontinuous way.

Let us now explain the interest of this approach in the perspective of proving existence of weak solutions.

For any sequence (h_n) going to 0 and such thatE_hn([·/h_n]) converges to a minimizing movement E, we are able, thanks to the Euler equation corresponding to our minimization procedure, to compute the velocity (in the viscosity sense) of the upper and lower limit of theE_hn(k)’s asn→ ∞,E^∗ andE_∗, in function ofE. This enables us to compare E_∗ and E^∗ with the 0 level set of the viscosity solutionuappearing in Definition 1.1.

Since E_∗ ⊂E ⊂E^∗, we will deduce that E is a weak solution of (1.1). In case no fattening occurs for u, we remark thatuis a viscosity solution of (1.3).

Of course it is a natural request that this construction be consistent with smooth flows if they exist. To verify this, we further show that if ∂E₀ is a smooth hypersurface, then there is a unique smooth solution for small times of the evolution law (1.1), and that any minimizing movementE coincides with this smooth evolution as long as the latter exists. This uses the notions of lower/upper limits mentioned above and of sub/super pairs of solutions of Cardaliaguet and Pasquignon [14].

To state our results in more details below, we first need to fix some notation and assumptions that will be used throughout the paper.

Notation

• For k ∈ N, B_r^k(x) (resp. B^k_r(x)) denotes the open (resp. closed) ball of radius r centered at x∈ R^k, and L^k is the Lebesgue measure onR^k. If k is not specified, we mean thatk=N. We setω_k =L^k(B₁^k(0)). The Hausdorff measure of dimensionk onR^N is denoted by H^k.

• The notation Sym_N represents the set of real square symmetric matrices of size N.

•We say that a sequence (E_n) of subsets ofR^N converges toE inL¹(R^N) if1_En→1_EinL¹(R^N) asn→+∞.

•LetPbe the set of all bounded subsets ofR^N having finite perimeter (see [15] for the definition and properties of sets of finite perimeter). We denote byP(E) the perimeter ofE∈ P, byP(E, U) the perimeter of E in U subset ofR^N, and we endowP with the metric

δ(E, F) =1_E−1_FL¹_(R^N₎=L^N(EΔF),

whereEΔF is the symmetric difference of E andF, i.e.,EΔF = (E∪F)\(E∩F).

In particular we callequivalent two setsE andF such thatδ(E, F) = 0, and we also say thatE=F almost everywhere (a.e.). Similarly, we say thatE⊂F almost everywhere ifL^N(E\F) = 0.

Moreover∂^∗E denotes the reduced boundary ofE∈ P. We also define a notion of boundary forE∈ P that is invariant in the class ofE formed by the sets that are equivalent to E:

∂E={x∈R^N; 0<L^N(E∩B_r(x))<L^N(B_r(x)) for allr >0}.

Then∂E is closed, and in fact∂E=∂^∗E.

Definitions of tubes (see [12])

• For any subsetE of R^N ×[0, T], we set E(t) = {x∈ R^N; (x, t)∈ E}. Conversely a mapping t ∈ [0, T] → E(t)∈ P(R^N) can be seen as a subset ofR^N ×[0, T] by identifyingE with its graph∪_t∈[0,T]E(t)× {t}.

•We calltubea bounded subsetEofR^N×[0, T]. We callregular tubea tubeEwithC¹boundary inR^N×[0, T] such that for any regular point (x, t)∈∂E, the unit outer normal (ν_x, ν_t) toE at (x, t) satisfiesν_x= 0. In this case, the normal velocity of Eat (x, t) is−ν_t/|ν_x|.

•Finally a mappingt∈[0, T]→E_r(t) is said to be a smooth evolution withC^3+αboundary ifE_ris a compact regular tube such thatE_r(t) hasC^3+αboundary for allt∈[0, T].

Assumptions on c₀ and c₁

Throughout the paper,c₀ andc₁are assumed to satisfy the following regularity assumption:

(A) c₀∈Lip

[0, T], L¹(R^N)

, c₁∈Lip

[0, T], L^∞(R^N) .

In particular, we setK₀=Lip(c₀), andK₁=Lip(c₁), so that for allt, s∈[0, T],

c0(·, t)−c₀(·, s)1≤K₀|t−s| and c1(·, t)−c₁(·, s)∞≤K₁|t−s|.

We finally set

L₀=c_0L^∞_([0,T],L¹_(R^N₎₎, L₁=c_1L^∞_([0,T],L^∞_(R^N₎₎ andL=L₀+L₁. (1.5) We will sometimes need additional regularity for c₀ and c₁. When this happens, we will specify which assumptions are made in each of the statements of theorems. In particular we will sometimes need to require that c₀ be symmetric, so that the gradient flow of our functional is, at least formally, a solution of (1.1):

(Symmetry of c₀)We say thatc₀ is symmetric if c₀(−(·), t) =c₀(·, t) for allt∈[0, T].

Main results

Forh >0 (the time step),k∈Nsuch thatkh≤T,E andF inP, we define, following the original idea of Almgren, Taylor and Wang [1], the functional

F(h, k, E, F) =P(E) + 1 h

EΔFd_∂F(x) dx−

E

1

2c₀(·, kh)1_E(x) +c₁(x, kh)

dx, (1.6)

whered_C is the distance function to a closed setC.

Let us now define a minimizing movement:

Definition 1.2(minimizing movement [1]). LetT >0 andE₀∈ P. We say thatE: [0, T]→ P is a minimizing movement associated toFwith initial conditionE₀if there exist a sequence (h_n), h_n→0⁺, and setsE_hn(k)∈ P for allk∈Nverifyingkh_n≤T, such that:

(1) E_hn(0) =E₀.

(2) For anyk, n∈Nwith (k+ 1)h_n ≤T,

E_hn(k+ 1) minimizes the functional E→ F(h_n, k+ 1, E, E_hn(k)) (1.7) among allEsin P.

(3) For anyt∈[0, T],E_hn([t/h_n])→E(t) inL¹(R^N) asn→+∞, where [·] denotes the integer part.

The first result of the paper is the existence of minimizing movements associated to our functionalF:

Theorem 1.3 (existence of minimizing movements). Assume that c₀ and c₁ satisfy (A). Let E₀ ∈ P with L^N(∂E₀) = 0. Then, there exists a minimizing movementE associated toF with initial condition E₀such that for allt, s verifyingt≤T and0≤s≤t < s+ 1, we have

δ(E(t), E(s))≤γ(t−s)^N+11 , (1.8)

whereγ=γ(N, T, E₀, K₀, K₁, L₀, L₁)is a constant.

We then prove that any such minimizing movement is a weak solution of (1.1):

Theorem 1.4 (minimizing movements are weak solutions). Assume that c₀ is symmetric, that c₀ and c₁ satisfy (A), are continuous onR^N ×[0, T] and Lipschitz continuous in space. LetE₀ ∈ P with L^N(∂E₀) = 0.

Let E be any minimizing movement associated toF with initial condition E₀.

Then E is a weak solution of (1.1)in the sense of Definition 1.1. In particular if no fattening occurs, i.e.

if the corresponding solution uof (1.4) is such that {u(·, t) = 0} has zero L^N measure, then u is a viscosity solution of (1.3)with initial datum u₀.

Let us already point out that even in the absence of fattening (a favorable situation which is not known to be generic), uniqueness for (1.3) is, to our knowledge, an open problem. The approach we use here provides one particular solution.

Our third result states that any minimizing movementE coincides with the smooth evolutionE_r as long as the latter exists:

Theorem 1.5(agreement with the smooth flow). Assume that c₀ is symmetric, thatc₀ andc₁satisfy (A), are continuous onR^N×[0, T]and Lipschitz continuous in space. LetE₀be a compact subset of R^N with uniformly C^3+α boundary. Let E_r be a smooth evolution with C^3+α boundary defined on [0, T], starting from E₀, with normal velocity given by

V_x,t=H_x,t+c₀(·, t)1_Er(t)(x) +c₁(x, t), (1.9) whereH_x,t is the mean curvature of∂E_r(t)atx.

Then any minimizing movement E associated to F with initial condition E₀ verifies E(t) = E_r(t) almost everywhere, for allt∈[0, T].

In relation with this, we finally prove short time existence and uniqueness of a smooth solutionE_r to (1.1), whenE₀ is sufficiently smooth. The regularity assumptions onc₀ andc₁are the following ones:

c₀∈L^∞([0, T], W^2,∞(R^N))∩W^1,∞([0, T], L^∞(R^N)) (1.10) and

c₁∈W^2,1;∞(R^N ×[0, T]), (1.11)

where f ∈ W^1,∞([0, T], L^∞(R^N)) means that f is Lipschitz continuous with respect to t ∈ [0, T], uniformly with respect to x∈R^N, and forn∈N^∗,

W^n,1;∞(R^N ×(0, T)) =

f ∈L^∞(R^N×(0, T))

f_t, ^∂_∂x^αfα ∈L^∞(R^N×(0, T)) forα∈N^N s.t. _N

i=0α_i≤n

.

Theorem 1.6 (existence and uniqueness of a smooth solution). Assume the regularity (1.10)–(1.11). LetE₀ be a compact subset of R^N with uniformly C^3+α boundary. Then there exists a small timet₀>0 and a unique smooth evolutionE_rwithC^3+αboundary defined on[0, t₀],starting fromE₀, with normal velocity given by (1.9).

Let us now explain how this paper is organized. First, in Section 2, we prove the existence of minimizing movements and the H¨older estimate Theorem 1.3. Section 3 is devoted to proving a regularity result for F-minimizers that we use in Section 4 to prepare the proofs of Theorems 1.4 and 1.5, respectively given in Sections 5 and 6. Finally, in Section 7, we prove Theorem1.6.

2. Existence of minimizing movements

This section is concerned with the existence of minimizing movements associated toF(Th.1.3). Let us start with existence and basic properties of F-minimizers.

2.1. F -minimizers

The first point to check is the existence of F-minimizers:

Proposition 2.1 (existence of F-minimizers). For all h > 0, k∈N with kh≤T, and F ∈ P, there exists a minimizer of E→ F(h, k, E, F)onP. Moreover, ifLis defined by (1.5), then

F ⊂B_R(0)a.e.⇒E⊂B_R+Lh(0)a.e.

wheneverE is a minimizer.

Proof. Let us fixF ∈ P withF ⊂B_R(0) a.e., and setB =B_R+Lh(0). Let (E_n) be a minimizing sequence for F(h, k,·, F). We want to prove that for alln∈N,

F(h, k, E_n∩B, F)≤ F(h, k, E_n, F). (2.1) First, sinceB is open and convex, we know that

P(E_n∩B)≤P(E_n). (2.2)

Let us compare

Enc₀(·, kh)1_En(x) dxand

En∩Bc₀(·, kh)1_En∩B(x) dx: for allx∈R^N, c₀(·, kh)1_En(x) =

En

c₀(x−y, kh) dy

=c₀(·, kh)1_En∩B(x) +

En\Bc₀(x−y, kh) dy.

Therefore

En

c₀(·, kh)1_En(x) dx=

En∩Bc₀(·, kh)1_En∩B(x) dx +

En\B

c₀(·, kh)1_En∩B(x) dx+

En

En\B

c₀(x−y, kh) dydx.

Sincec₀(·, kh)1_AL^∞_(R^N₎≤L₀for any measurable setA, it follows that

En

1

2c₀(·, kh)1_En(x) +c₁(x, kh)

dx≥

En∩B

1

2c₀(·, kh)1_En∩B(x) +c₁(x, kh)

dx−LL^N(E_n\B) (2.3)

thanks to the definition ofL(see (1.5)). MoreoverF ⊂B, so that E_nΔF = (E_n∩B)ΔF ∪ (E_n\B), whence

1 h

EnΔFd_∂F(x) dx= 1 h

(En∩B)ΔFd_∂F(x) dx+1 h

En\Bd_∂F(x) dx

≥ 1 h

(En∩B)ΔF

d_∂F(x) dx+LL^N(E_n\B),

(2.4)

sinced_∂F(x)≥Lhfor all x∈E_n\B by definition of B. Putting (2.2), (2.3) and (2.4) together proves (2.1).

Therefore we can replace (E_n) by (E_n∩B) as a minimizing sequence, and in particular we can assume that E_n⊂B for alln. Then

F(h, k, En, F)≥ −

En

1

2c₀(·, kh)1_En(x) +c₁(x, kh)

dx

≥ − 1

2L₀+L₁

L^N(B), so that inf

E∈PF(h, k, E, F)>−∞. Besides, fornlarge enough, F(h, k, En, F)≤ inf

E∈PF(h, k, E, F) + 1.

This implies that

P(E_n)≤ inf

E∈PF(h, k, E, F) + 1 + 1

2L₀+L₁

L^N(B),

and gives a uniform bound on the perimeter of theE_n’s. Since they are also uniformly bounded byB, it follows from the compactness theorem for sets of finite perimeter [15], Section 5.2.3, that we can extract a converging subsequence (E_nk) of (E_n) in the sense that there existsE_∞ ∈ P,E_∞ ⊂B, such thatE_nk →E_∞ inL¹(R^N).

Therefore

F(h, k, E_∞, F)≤lim inf

k→∞ F(h, k, E_nk, F) = inf

E∈PF(h, k, E, F),

because all terms in the expression ofF are at least lower semi-continuous in theEvariable for theL¹topology.

Thus E_∞ is a minimizer of E → F(h, k, E, F) onP. Finally, if E is any other minimizer, then the previous comparisons show that P(E∩B) =P(E), whenceE⊂B almost everywhere (see the comparison theorem [5],

p. 216).

Remark 2.2. This proposition shows that theE_h(k)’s are uniformly bounded for allhandk, ifE₀∈ P: more precisely, if E₀ ⊂ B_R(0) a.e., then since kh ≤ T, we can choose E_h(k) ⊂ B_R+LT(0) independently of h, k.

Therefore we can choose Ω =B_R+LT+1(0) so that E_h(k)Ω for allk,h. We will always do so in the sequel, and set D=R+LT+ 1.

Remark2.2gives a uniform bound Ω forE_h(k), independently ofh, k, provided thatE₀is bounded. In order to have compactness inP, so as to construct our minimizing movement, we also want a uniform bound on the perimeter ofE_h(k).

Proposition 2.3(uniform bound on the perimeter).LetE₀∈ PwithE₀⊂B_R(0). Then, there exists a constant c=c(T, E₀, D, K₀, K₁, L₀, L₁)>0 independent ofhandksuch that if E_h is defined by the procedure (1.7), we have

P(E_h(k))≤c ∀h, ksuch thatkh≤T.

Proof. By definition ofE_h, we have for allj such thatjh≤T,

F(h, j, E_h(j), E_h(j−1))≤ F(h, j, Eh(j−1), E_h(j−1)), and in particular,

P(E_h(j))−

Eh(j)

1

2c₀(·, jh)1_Eh(j)(x) +c₁(x, jh)

dx≤ P(E_h(j−1))−

Eh(j−1)

1

2c₀(·, jh)1_Eh(j−1)(x) +c₁(x, jh)

dx.

Adding these inequalities for j= 1, . . . , kwithkh≤T, we find:

P(E_h(k))−P(E₀)≤^k

j=1

J_h(j, j)−J_h(j−1, j)

= k j=1

Ωc₁(·, jh)1_Eh(j)−c₁(·, jh)1_Eh(j−1) +1

2 k j=1

Ω(c₀(·, jh)1_Eh(j))1_Eh(j)−(c₀(·, jh)1_Eh(j−1))1_Eh(j−1) (2.5) where we have set

J_h(i, j) =

Eh(i)

1

2c₀(·, jh)1_Eh(i)(x) +c₁(x, jh)

dx. (2.6)

Doing an Abel transformation on the first sum of the last member of (2.5) yields k

j=1

Ωc₁(·, jh)1_Eh(j)−c₁(·, jh)1_Eh(j−1)=

Ωc₁(·, kh)1_Eh(k)−

Ωc₁(·, h)1E0

+

k−1

j=1

Ω[c₁(·, jh)−c₁(·,(j+ 1)h)]1_Eh(j)

≤ 2L₁L^N(Ω) + (k−1)K₁hL^N(Ω)

≤ (2L₁+K₁T)L^N(Ω).

The same manipulation with the second sum gives k

j=1

Ω(c₀(·, jh)1_Eh(j))1_Eh(j)−(c₀(·, jh)1_Eh(j−1))1_Eh(j−1)≤(2L₀+K₀T)L^N(Ω).

This proves that for allk such thatkh≤T, k

j=1

J_h(j, j)−J_h(j−1, j)≤

L₀+ 2L₁+1

2K₀T+K₁T

L^N(Ω) (2.7)

and gives the result, withc=P(E₀) + (L₀+ 2L₁+¹₂K₀T+K₁T)L^N(Ω).

2.2. Minimizing movements

We are now ready to address the problem of existence of minimizing movements. Proofs in this section closely follow the ideas of Almgren, Taylor and Wang [1], and are adaptations of Ambrosio’s simplified presentation of these ideas (see [5]).

The main result in the perspective of the proof of existence of minimizing movements is the following theorem on the behaviour of the solutions of procedure (1.7):

Theorem 2.4(discrete H¨older estimate). LetE₀∈ PwithE₀⊂B_R(0). There exists a constantγ=γ(N, D)>

0 (where D is defined in Rem. 2.2) and h₀>0 such that for all h∈(0, h₀), for all m, l∈Nverifying mh≤T and0< l < m < l+_h¹, we have:

δ(E_h(m), E_h(l))≤γ c[h(m−l)]^N+11 , (2.8) wherec is the uniform bound on P(E_h(k))given by Proposition 2.3.

Theorem 1.3is a corollary of this result, as proved in [5], pp. 231–232. However the arguments of [1], The- orem 4.4, or [5], Theorem 3.3, for the proof of Theorem 2.4 in the mean curvature motion case need a few adaptations due to the particular form ofF. This is what the rest of this section is devoted to. We begin by giving some preliminary results which will be necessary in the proof of Theorem2.4.

Lower density bound forF-minimizers

Theorem 2.5 (density bound forF-minimizers). There exist two positive constants αandβ (depending only on N) and h₀ > 0 such that if E ∈ P is a minimizer of F(h, k,·, F) with F ∈ P, E∪F ⊂ B_D−1(0), and h∈(0, h₀), then

∀x∈∂E, ∀ρ∈

0,αh D

, P(E, B_ρ(x))≥βρ^N−1. (2.9)

Proof. The proof relies on the following lemma relating the perimeter ofE∈ Pand the perimeter ofEreplaced by a cone in a small ball:

Lemma 2.6 ([5], Lem. 3.5).

Let E∈ P,x∈R^N andf(ρ) =P(E, B_ρ(x)). Set E_ρ= (E∩(R^N \B_ρ(x)))∪

y∈B_ρ(x);x+ρ y−x

|y−x| ∈E

. Then for almost all ρ >0 (allρsuch that f is differentiable atρ), we have

P(E_ρ, B_ρ(x))≤ρf(ρ) N−1·

Let us now prove Theorem2.5. Fixx∈∂^∗Eandρ >0 such thatf is differentiable atρ. By definition ofE, we know that F(h, k, E, F)≤ F(h, k, E_ρ, F), that is to say

P(E)≤P(E_ρ) + 1 h

EρΔFd_∂F(y) dy−

EΔFd_∂F(y) dy

+

E

1

2c₀(·, kh)1_E(y) +c₁(y, kh)

dy−

Eρ

1

2c₀(·, kh)1_Eρ(y) +c₁(y, kh)

dy.

(2.10)

But sinceE coincides with E_ρ inR^N\B_ρ, we have

P(E,R^N \B_ρ(x)) =P(E_ρ,R^N \B_ρ(x)).

Moreoverf is continuous atρ, which together with (2.10) implies that P(E, B_ρ(x)) =P(E, B_ρ(x))≤P(E_ρ, B_ρ(x)) +2D

h ω_Nρ^N + 2L ω_Nρ^N,

due to the fact that d_∂F(y)≤2D for all y∈B_ρ(x), provided ρ <1. Now Lemma 2.6implies that for almost allρ∈(0,1),

f(ρ)≤ρf(ρ) N−1 +

2D h + 2L

ω_Nρ^N. (2.11)

Therefore, the function

g:ρ→ f(ρ) ρ^N−1 +

2D h + 2L

(N−1)ω_Nρ is nondecreasing on (0,1). In particular ifx∈∂^∗E andρ∈(0,1),

g(ρ)≥lim inf

ρ→0⁺ g(ρ)≥ω_N−1 (2.12)

because of [15], Corollary 1 (ii) p. 203. As a consequence, for allρ∈(0,1), f(ρ)≥ω_N−1ρ^N−1−

2D h + 2L

(N−1)ω_Nρ^N. (2.13)

Let us setα=_8(N^ωN−1_−1)ω

N andβ =^ωN−1₂ . Then, providedh <min{^D_L,^D_α}=:h₀, we deduce from (2.13) that for allρ∈(0,^αh_D),

P(E, B_ρ(x)) =f(ρ)≥β ρ^N−1.

Since∂^∗E is dense in∂E, this also holds for allx∈∂E.

Corollary 2.7 ([5], Cor. 3.6). Let E∈ P be a minimizer ofF(h, k,·, F)with F∈ P andh∈(0, h₀). Then H^N−1(∂E\∂^∗E) = 0.

Distance-volume comparison

We recall here a general result which makes it possible to compareL^N(A\C) and

Ad_∂C under conditions of density ofCsimilar to (2.9). Such comparison will be essential to prove Theorem2.4.

Theorem 2.8(distance-volume comparison, [5] p. 230). LetC be a compact subset ofR^N such that there exist β >0,τ >0 with

H^N−1(C∩B_ρ(x))≥βρ^N−1 ∀x∈∂C, ∀ρ∈(0, τ).

Then there exists a constant Γ = Γ(N)>0such that for all R > τ, for all Borel setA⊂R^N, we have

L^N(A\C)≤

2Γ R

τ _N−1

H^N−1(C) ¹₂

Ad_C(x) dx ¹₂

+ 1 R

Ad_C(x) dx. (2.14) We are now able to prove Theorem2.4.

Proof of Theorem 2.4. Let us fix h∈(0, h₀), whereh₀ is given by Theorem 2.5. By definition of E_h, we have for allj such thatjh≤T,

F(h, j, E_h(j), E_h(j−1))≤ F(h, j, Eh(j−1), E_h(j−1)),

that is to say,

Eh(j)ΔEh(j−1)d_∂Eh(j−1)(x) dx≤ h[P(E_h(j−1))−P(E_h(j))] + h[J_h(j, j)−J_h(j−1, j)], whereJ_h(i, j) is defined by (2.6). Let us set

I_h(j) ={[P(E_h(j−1))−P(E_h(j))] + [J_h(j, j)−J_h(j−1, j)]}¹².

We now use Theorem 2.8 with C = ∂E_h(j−1), A =E_h(j)ΔE_h(j−1), τ = ^αh_D, which is justified for j ≥ 2 because of the density estimate (2.9). Thanks to Corollary2.7, we know thatL^N(C) = 0, so that for allR > ^αh_D,

L^N(E_h(j)ΔE_h(j−1))≤

2Γ R

τ _N−1

H^N−1(∂E_h(j−1)) ¹₂

√h I_h(j) + 1

Rh I_h(j)². (2.15) Recall that Proposition2.3gives a uniform boundcon the perimeter ofF-minimizers, so thatH^N−1(∂E_h(j− 1))≤c.

Letm, l∈Nverifymh≤T and 0< l < m < l+_h¹. We choose R= αh

D [h(m−l)]^N+1−1 >αh D ,

and add up inequalities (2.15) forj =l+ 1, . . . , m. Recall that (2.5) and (2.7) show that m

j=l+1

I_h(j)²≤P(E_h(l)) + m j=l+1

J_h(j, j)−J_h(j−1, j)

≤P(E₀) + m j=1

J_h(j, j)−J_h(j−1, j)≤c.

Moreover, the Cauchy-Schwarz inequality shows that m

j=l+1

I_h(j)≤√ m−l

⎧⎨

⎩ m j=l+1

I_h(j)²

⎫⎬

⎭

12

≤√

m−l√ c.

Finally, we find that

L^N(E_h(m)ΔE_h(l))≤

2Γ[h(m−l)]^−N−1N+1c ¹₂

h(m−l)√ c+ D

αh[h(m−l)]^N+11 h c

= √

2Γ +D α

c[h(m−l)]^N+11 ,

which concludes the proof.

3. Regularity for F -minimizers

One of the main interests of the variational approach used in [1] is that it enables to use the regularity theory for area-minimizing currents described for instance in [11,17,25,27]. This is the idea we follow in this section.

We use the notation of [1]. In particular, the notation M and S stand respectively for the mass and size of

an integral current: ifT is akintegral current associated to akrectifiable setS⊂R^N and a density functionθ, then M(T) =

SθdH^k, while S(T) = H^k(S) (see [1], Sect. 3.1.3). Besides, ifE ∈ P, [E] denotes the solid associated toE,i.e. the canonicalN-dimensional Euclidean current restricted toE. We use the notationTC for the restriction of a currentT to a setC.

3.1. Existence of tangent cones

A fundamental notion in regularity theory is that of tangent cones defined as follows:

Definition 3.1. Letf_p,R:x→R(x−p), forp∈R^N,R >0. A locally integral current [J] is called a tangent current to ∂E at p ∈ ∂E if there exists a sequence (R_i) → +∞ such that if we set E(R) = f_p,R(E), then [E(R_i)]→[J] locally asi→+∞, in the sense thatL^N((JΔE(R_i))∩B_r(q))→0 for eachq∈R^N andr >0.

Lemma 3.2 (existence of tangent cones). Let F∈ P and let E be a minimizer of F(h, k,·, F)onP. For each p ∈ ∂E, there exists a tangent current [J] to ∂E at p. Each such tangent current [J] is a cone and locally minimizes the perimeterP. Moreover 0∈∂J.

Proof. The proof is inspired by that of [1], Theorem 3.9. We easily check that for allR >0, P(E(R)) =R^N−1P(E),

1 h

E(R)ΔF(R)d_∂F(R)(y) dy=R^N+11 h

EΔFd_∂F(y) dy,

E(R)

1

2c₀(·, kh)1_E(R)(y) dy=R^2N

E

1

2c₀(R(·), kh)1_E(y) dy

E(R)c₁(y, kh) dy=R^N

Ec₁(R(y−p), kh) dy.

By definition ofE we find thatE(R) minimizes E→P(E) + 1

R²h

EΔF(R)d_∂F(R)(y) dy− 1 R^N+1

E

1

2c^R₀(·, kh)1_E(y) dy− 1 R

Ec^R₁(y, kh) dy, (3.1) where we have set c^R₀(y, t) =c₀(y/R, t), c^R₁(y, t) =c₁(p+y/R, t). Let us compareE(R) andE(R)\B_r(q) for fixed q∈R^N andr >0, with respect to this last functional. It follows from manipulations similar to those in the proof of Proposition2.1that for almost all r >0,

P(E(R), B_r(q))≤P(B_r(q)) + 1 R²h

Br(q)d_∂F(R)(x) dx+L

RL^N(B_r(q)), whereL is defined by (1.5). But diamF(R) =RdiamF, so that

R→ 1 R²h

Br(q)d_∂F(R)(x) dx

is bounded as a function of R, and even converges to 0 as R goes to infinity. This provides the sufficient bound on the perimeter ofE(R) in balls to infer the existence of a tangent current [J] (using the compactness result [26], Th. 1.1 p. 225).

Let us prove that [J] locally minimizes the perimeter. This means that for allq∈R^N, allr >0, and all (N−1) integral current X with ∂X = 0 and having support in C =B_r(q), then M(∂[J]C) ≤ M(∂[J]C+X).

We first recall from [1], Section 3.1.6, that there exists anN integral currentQwith compact support inCsuch that ∂Q=X and

S(Q)≤M(Q)≤ r

N M(X). (3.2)

Then according to [17], Section 4.5.17, we can writeQas

Q=

+∞

i=0

[Q_i]−^+∞

i=0

[P_i],

whereQ_i, P_i ∈ P and (Q_i), (P_i) are nested families such that P₁∪Q₁⊂Supp(Q) andP₁∩Q₁=∅. Let us set K = (E(R)\P₁)∪Q₁ and compareE(R) and K with respect to the functional defined by (3.1). We obtain that

P(E(R))≤P(K) + 1 R²h

P1∪Q1

d_∂F(R)(x) dx+ L

RL^N(P₁∪Q₁)

≤M(∂[E(R)] +∂Q) + 1 R²h

Cd_∂F(R)(x) dx+L RS(Q).

SinceQand∂Q=X have compact support inC, and sinceP(E(R), C) =M(∂[E(R)]C), we deduce that M(∂[E(R)]C)≤M(∂[E(R)]C+X) + 1

R²h

Cd_∂F(R)(x) dx+L

RL^N(C).

Knowing this, we can adapt [27], Theorem 34.5, to show that [J] locally minimizes the perimeter and also that if R_i is such that [E(R_i)] →[J] asi → +∞, then for all x∈ R^N and almost all ρ >0, P(E(R_i), B_ρ(x))→ P(J, B_ρ(x)) asi→+∞.

Finally we check that [J] is a cone,i.e. thatJ is invariant under all homothetic expansionsz→λzforλ >0.

To see this we recall from (2.11) and (2.12) that for allx∈∂E, the function g:ρ→ P(E, B_ρ(x))

ρ^N−1 +cρ is nondecreasing on (0,1), wherecis a constant, and that for allρ∈(0,1),

P(E, B_ρ(x))

ρ^N−1 +cρ≥ω_N−1.

It follows that∂Ehas a densityθ(∂E, x) atxwithθ(∂E, x)≥1. For allρ >0, P(E(R), B_ρ(0))

ρ^N−1 = P(E, B_ρ/R(p)) (ρ/R)^N−1 −→

R→+∞θ(∂E, p)ω_N−1. Moreover for almost allρ >0,

P(E(R_i), B_ρ(0))

ρ^N−1 −→

i→+∞

P(J, B_ρ(0)) ρ^N−1 ·

This shows that the ratio ρ^1−NP(J, B_ρ(0)) is independent of ρ, which is known to imply that J is a cone (see [21], proof of Th. 9.3). Moreoverρ^1−NP(J, B_ρ(0)) =θ(∂E, p)ω_N−1>0, so that 0∈∂J. We finally observe

that θ(∂J,0) =θ(∂E, p).

3.2. The regularity results

The existence of tangent cones enables us to prove regularity results forF-minimizers, as in [1], Sections 3.5 and 3.7.

Theorem 3.3 (C¹-regularity forF-minimizers). Let F ∈ P, and let E be a minimizer ofF(h, k,·, F) on P. Then ∂E is aC¹-hypersurface, except for a set of Hausdorff dimension less thanN−8 (empty ifN≤7).

Proof. We verify thatE is an almost minimal current in the sense of Bombieri, that is, for someδ >0, for all (N−1) integral currentX with∂X = 0 and having compact support inCwith diam(C) =r≤δ, then

M(∂[E]C)≤(1 +ω(r))M(∂[E]C+X) (3.3) for some functionωsuch thatω(r)→0 asr→0⁺. To do so we proceed as in the previous proof, writeX =∂Q with

Q=

+∞

i=0

[Q_i]−

+∞

i=0

[P_i], set K= (E\P₁)∪Q₁and compareE andK with respect toF:

P(E, C)≤P(K) + 1 h

P1∪Q1

d_∂F(y) dy+LL^N(P₁∪Q₁).

Let D >0 be such that E∪F ⊂B_D−1(0). If B_D−1(0)∩C =∅ (otherwise (3.3) is obvious), and δ ≤1, the previous comparison yields

M(∂[E]C)≤M(∂[E]C+∂Q) + 2D

h +L

S(Q)

≤M(∂[E]C+X) + 2D

h +L r

NM(X) (using (3.2))

≤M(∂[E]C+X) + 2D

h +L r

N(M(∂[E]C+X) +M(∂[E]C)).

This easily implies the result with ω(r) = 3_2D

h +L _r

N andδ= ^N_{3 2D}

h +L₋₁ .

In addition, at any point pof∂E there exists a tangent cone [J] which minimizes the perimeter (Lem.3.2).

Such a cone must be a hyperplane forN≤7 ([27], Appendix B), so that in particularθ(E, p) =θ(J,0) = 1. We then deduce the result from the final remark in [11]. In caseN≥8, we use the dimension reduction argument

of Federer ([21], Th. 11.8).

Now, we prove that minimizers are smooth at contact points with smooth hypersurfaces:

Theorem 3.4. Let F ∈ P, and letE be a minimizer ofF(h, k,·, F)on P. Assume that there existsK⊂R^N closed such that∂K is aC¹ hypersurface and∂E∩K={p}. Then ∂E is aC¹ hypersurface nearp.

Proof. Let [J] be any tangent cone to ∂E at p. The assumption that ∂E∩K ={p} guarantees that ∂J is contained in the closed half-space orthogonal to the outer unit normal n to K at pand containing n. Since 0 ∈∂J, [21], Theorem 15.5, p. 174, implies that∂J is regular at 0, and therefore is a hyperplane. The result

follows as in the proof of Theorem3.3.

Actually, we can deduce higher regularity for F-minimizers at each point where they areC¹ hypersurfaces:

Theorem 3.5 (higher regularity forF-minimizers). Assume that c₀ is symmetric, that c₀ and c₁ satisfy (A) and are Lipschitz continuous in space. Let F ∈ P, and let E be a minimizer of F(h, k,·, F) on P. Set g(p) =±d∂F(p), where we take the −sign if p∈F, and the+sign otherwise.