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Application to non convex variational problems
We now apply the framework developed in Section 2.2 to the following situation. Let Ω be a bounded Lipschitz domain of RN . We consider the embedding of X = L1(Ω) into Y = L∞(Ω×R) defined by: ϕ : u ∈ X 7→1u ∈ Y, 1u(x, t) := Let us consider K := {v ∈ L∞(Ω × R) : v(x, t) ∈ [0, 1] a.e.(x, t) ∈ Ω × R} .
It is a compact subset of L∞(Ω × R) equipped with its weak-star topology (we are in the case where Y = Z∗ if we set Z = L1(Ω × R)). It is easy to check that 1u is an extreme point of K as it takes values in {0, 1}. Moreover the map u 7→1u is continuous from L1(Ω) to L∞(Ω × R) (embedded with its weak star topology). Let F : u ∈ L1(Ω) → R ∪ +∞ be a possibly non convex functional. We simply assume that F is l.s.c. (with respect to the strong convergence in L1(Ω)) and that the following coercivity assumption holds: ( F (u) ≥ k kuk − 1 , for suitable constant k > 0. (2.11) k (Ω). For every R > 0, the set { u : F (u) ≤ R } is a compact subset of L1 We consider the minimization problem (P) inf nF (u) : u ∈ L1(Ω)o .
Under the assumption (2.11), this problem has at least one solution and the set of solutions Argmin P is a non void compact subset of L1(Ω) (since F is not convex, we expect à priori multiple solutions).
Following the construction developed in Section 2.2, we define for every pair (v, g) ∈ L∞(Ω × R) × L1(Ω × R) ∈ Z × g(x, t) F (u) , G(v) = ∈ sup × Z × ) F ∗ g ) = u sup 1u dxdt − g v dxdt − F ∗ g 0 ( L1(Ω) Ω R g L∞(Ω R) Ω R 0 ( (2.12).
Since F coincides with its l.s.c. envelope, it is a consequence of Theorem 2.1 that it holds G(1u) = F (u) for all u ∈ L1(Ω) (2.13) Our convexification recipe leads to the following convex optimization problem (Q) inf nG(v) : v ∈ L∞(Ω × R; [0, 1])o, whose set of solutions Argmin(Q) is a non empty weakly star compact subset of L∞(Ω ×R; [0, 1]). Lemma 2.6. It holds inf(P) = inf(Q) and the following equivalence holds: u ∈ Argmin(P) ⇐⇒ 1u ∈ Argmin(Q) .
Proof. Applying (2.12) with g = 0, we get inf(Q) = −G∗(0) = −(F0)∗(0) = − sup{−F (u) : u ∈ X} = inf(P) . The equivalence statement follows by using the identity (2.13).
The next step is twofold: first we have to identify the convexified energy in practice in order to settle a duality scheme for Q; then, as some solutions v for (Q) may take intermediate values in (0, 1) (i.e. v is not of the form 1u), we have to specify how solutions to (P) can be recovered. A complete answer to these two requirements will be obtained under an additional assumption on functional F . We will use the following slicing argument on the class A := {v ∈ L∞(Ω × R) : v(x, ·) non increasing (a.e. x ∈ Ω) , v(x, −∞) = 1 , v(x, +∞) = 0} .
Dual problem in Ω × R
Let us decribe the dual problem in the simpler case where f is of the form f(t, z) = g(t) + ϕ(z) being ϕ : RN → R+ convex continuous with ϕ(0) = 0 and g : R → R ∪ {+∞} a lower semicontinuous function with possibly countably many discontinuities. The primal problem reads (Pλ) inf ZΩ (ϕ(ru) + g(u)) dx − λ ZΩ p(x)u dx : u ∈ H01(Ω) and its convexified version.
(Qλ) inf ZΩ×R hf (t, Dv) − λ ZΩ×R p(x)(v − v0) dxdt : v ∈ A , v − v0 ∈ BV0(Ω × R) where BV0(Ω × R) denotes the set of integrable functions with bounded variations on Ω × R and whose trace vanishes on the lateral boundary ∂Ω × R (see [103]).
The dual problem to our non convex problem (Pλ) is then recovered be applying classical duality to problem (Qλ). The competitors of this dual problem (Pλ∗) are vector fields σ = (σx, σt) : Ω × R → RN × R we take in the class n o X1(Ω × R) = σ ∈ L∞(Ω × R; RN+1) : div σ ∈ L1loc(Ω × R) , and (Pλ∗) consists in the following maximal flux problem: Z (Pλ∗) sup − σt(x, 0) dx : σ ∈ K , − div σ = λ p in Ω × R , Ω where σ ∈ K means that the vector field σ ∈ X1(Ω × R) satisfies the pointwise (convex).
An example of -convergence
We revisit here the celebrated asymptotic analysis of the Modica-Mortola functional which arises in the sharp interface model for Cahn-Hilliard fluids, showing how the duality approach developed in Section 2.2 can be used efficiently. In fact we can treat a sligthly more general model where we consider a family of functionals (F ε)ε>0, indexed with a (small) scale parameter ε > 0, of the following form (see [13] ) F ε(u) := 1 Z f(u(x), εru(x)) dx, ε Ω.
where Ω is bounded domain of RN with Lipschitz boundary and f : R × RN → [0, +∞) the following assumptions on :
i) f is continuous in the first variable and convex in the second.
ii) there exist two real numbers 0 < α < β such that f(t, 0) > 0 if t =6 α, β, f(α, 0) = f(β, 0) = 0, and for every z =6 0 and every t, f(t, z) > f(t, 0).
iii) there exists M > β such that f(t, ·) is locally bounded, uniformly in t ∈ [0, M].
iv) there exists a function ψ with superlinear growth at ∞, such that f(t, p) ≥ ψ(p) for every t ∈ R and p ∈ RN . Under such assumptions, it is not difficult to show that the family {F ε, ε > 0} is equicoercive in X = L1(Ω; [0, M]) (that is satisfies the condition (H2) in Section 2.2). Our aim is to compute the -limit of F ε as ε → 0. For every t ∈ R and z ∈ RN , we define fε(t, z) = f(t, εz) ε c( ) = ε>0 ε Zα c ( ) inf f (t, z) , h(z) = β , f t, z f∗∗ t, z dt .
By construction the conical enveloppe of fc is one–homogeneous in z. It follows that h is a convex and one homogeneous function of z. An easy computation involving Moreau-Fenchel conjugates, for fixed t and with respect to the variable z, shows that fε∗(t, z∗) = 1 f∗(t, z∗) , fc∗∗(z) = supN {z · z∗ : f∗(t, z∗) ≤ 0} . (2.26) ε z∗∈R Under these assumptions, we can show the following result: Theorem 2.15. As ε goes to zero, F ε –converges in L1(Ω; [0, M]) to the functional F given by F (u) = Z Su Ω h(νu) dHN−1 if u ∈ BV (Ω; {α, β}) ∩ ∞ otherwise .
Table of contents :
Introduction
A Mathematical background and notations
1 Preliminaries
1.1 Direct method in calculus of variations. -convergence
1.1.1 Lower semicontinuity
1.1.2 Coerciveness conditions
1.1.3 Relaxation
1.1.4 -Convergence
1.2 Convex analysis
1.2.1 Moreau-Fenchel conjugate
1.2.2 Convex optimization
1.3 Integral functionals in Sobolev spaces
1.3.1 Integral functionals with integrand f(x, z)
1.3.2 Integral functionals with integrand f(x, u, z)
1.4 Relaxation in BV
1.4.1 The space BV
1.4.2 Integral representation of relaxed functionals
1.4.3 Relaxed variational problems
1.5 Convex duality. Primal-dual formulations
1.5.1 Duality by perturbation
1.5.2 Duality by min-max
1.5.3 Identification of duality methods
1.6 Classical saddle-point algorithms
1.6.1 Uzawa’s algorithm
1.6.2 Arrow-Hurwicz’ Algorithm
B Non convex variational problems
2 A general duality principle for non convex variational problems
2.1 Introduction
2.2 General framework
2.3 Application to non convex variational problems
2.4 Duality schemes and examples
2.4.1 Dual problem in × R
2.4.2 Saddle point characterization
2.4.3 An example of -convergence
2.5 Extension of the duality principle to the case of linear-growth functionals .
2.5.1 Relaxation in BV
2.5.2 Dual problem
2.5.3 Optimality conditions and saddle point characterization
3 A convex relaxation method for free boundary problems
3.1 Introduction
3.2 Exclusion principle and main results
3.3 Application to free boundary and multiphase problems
3.3.1 A three phases problem
3.3.2 A 4 phases problem
3.4 Min-max approach. A primal-dual algorithm for non differentiable Lagrangians .
3.4.1 Explicit scheme with projection on epi(g)
3.4.2 Semi-implicit scheme with projection on epi(g)
3.5 Counterexamples
4 Calibrating fields for minimal surfaces with free boundary and Cheeger-type problem
4.1 Presentation of two free boundary problems
4.2 Dual problems and calibration method
4.2.1 Dual problems in dimension N + 1
4.2.2 Geometric problem m(,D)
4.2.3 Dual problem associated with m(,D)
4.2.4 -calibrable bodies
4.3 Cut-locus potential and an explicit construction of 2-dimensional calibrating fields
4.3.1 Cheeger sets in R2
4.3.2 Characterization of -calibrable sets in R2. The set
4.3.3 Cut-locus potential
4.3.4 An explicit construction for calibrating fields on D
4.4 Comparison results
C Numerical methods
5 A new semi-implicit scheme based on Arrow-Hurwicz method for saddle point problems
5.1 Introduction
5.2 Saddle point problem and explicit scheme
5.3 Semi-implicit scheme
5.4 Application to the shape optimization of thin torsion rods
5.5 Conclusion
6 Application of primal-dual algorithms to free boundary problems
6.1 An elliptic-type free boundary problem
6.1.1 Discretization
6.1.2 Some simulations in case N = 1
6.1.3 Some simulations in case N = 2
6.2 Free boundary problems with linear growth
Perspectives and open problems
Bibliography
