# Volume-constrained minimizers for the prescribed curvature problem in periodic media

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## Study of the Primal-Dual Method

In this section, unless otherwise stated, everything holds for general functionals J of the type (2.1). Before starting the study of the Primal-Dual method, let us remind some facts about pairings between measures and bounded functions.
Following Anzellotti [16], we define R R ­[»,Du] which has to be understood as ­ » · Du, for functions u with bounded variation and bounded functions » with divergence in L2.
Definition 2.3.1. • Let X2 = © » 2 (L1(­))m / div » 2 L2(­) ª .
• For (u, ») 2 BV 2 × X2 we define the distribution [»,Du] by h[»,Du], ‘i = − Z ­ u’ div(») − Z ­ u » · r’ 8′ 2 C1 c (­).
Theorem 2.3.2. [16] The distribution [»,Du] is a bounded Radon measure on ­ and if º is the outward unit normal to ­, we have Green’s formula, Z ­ [»,Du] = − Z ­ u div(») + Z @­ (» · º)u.

### Existence of surfaces with prescribed mean curvature

In this section we shall assume that g has zero average and satisfies Z E g · (1 − ¤)P(E,Q) 8E ½ Q (3.14) for some ¤ > 0. Notice that (3.14) is always satisfied if kgkLm(Q) is small enough, and is precisely the assumption needed in [52] (see also [37]) to prove existence of planelike minimizers of F. Notice also that, if g satisfies (3.14), then the inequality in (3.14) holds for all sets E ½ Rm of finite perimeter.
In particular, this implies the following estimate on the function f: c v m−1 m · f(v) · C v m−1 m for some 0 < c < C. (3.15) In the sequel we will need a representation result for the functional F, due to Bourgain and Brezis [30].
Theorem 3.4.1. Let g be a function verifying (3.14) then there exists a periodic and continuous function ¾ with max ¾(x) < 1 satisfying div ¾ = g.

#### Compact solutions with big volume.

From (3.15), Proposition 3.3.2 and Remark 3.3.4, we immediately obtain the following result.
Proposition 3.4.3. Let g be a periodic C0,® function of zero average satisfying (3.14). Assume that f0(v) · 0 for some v > 0. Then there exists w > 0 such that f0(w) = 0, therefore problem (3.1) admits a compact solution.
Theorem 3.4.4. Let g be a periodic C0,® function with zero average and satisfying (3.14). There exist vn ! +1 and compact minimizers En of (3.4) such that |En| = vn and En solves · = g + ¸n with ¸n ¸ 0 and ¸n ! 0 as n ! +1.

Wiener space and functions of bounded variation

A clear and comprehensive reference on the Wiener space is the book by Bogachev [27] (see also [106]). We follow here closely the notation of [12]. Let X be a separable Banach space and let X¤ be its dual. We say that X is a Wiener space if it is endowed with a non-degenerate centered Gaussian probability measure °. That amounts to say that ° is a probability measure for which x¤]° is a centered Gaussian measure on R for every x¤ 2 X¤. The non-degeneracy hypothesis means that ° is not concentrated on any proper subspace of X.
As a consequence of Fernique’s Theorem [27, Th. 2.8.5], for every x¤ 2 X¤, the function R¤x¤(x) = hx¤, xi is in L2 °(X) = L2(X, °). Let H be the closure of R¤X¤ in L2 °(X); the space H is usually called the reproducing kernel Hilbert space of °. Let R, the operator from H to X, be the adjoint of R¤ that is, for ˆh 2 H, Rˆh = Z X xˆh(x) d° where the integral is to be intended in the Bochner sense. It can be shown that R is a compact and injective operator. We will let Q = RR¤so that for
every x¤, y¤ 2 X¤, hQx¤, y¤i = Z X hx¤, xihy¤, xi d°.

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Table des mati`eres
1 Introduction
1.1 Introduction g´en´erale
1.2 D´efinitions et propri´et´es principales des fonctions
1.3 ¡-convergence et enveloppes semi-continues
1.4 Les fonctions `a variation born´ee en traitement d’images
1.4.1 Coupures minimales et flots maximaux sur les graphes
1.4.2 L’approche d’Appleton et Talbot
1.4.3 Contributions de la th`ese
1.5 Surfaces de courbure moyenne prescrite
1.5.1 Surfaces minimales et probl`eme de Plateau
1.5.2 Surfaces de courbure moyenne constante et probl`eme isop´erim´etrique
1.5.3 Le probl`eme g´en´eral
1.5.4 Contributions de la th`ese
1.6 Fonctions `a variation born´ee dans les espaces de Wiener
1.7 Approximation et relaxation du p´erim`etre dans les espaces de Wiener
1.7.1 Isop´erim´etrie et sym´etrisation de Ehrhard
1.7.2 Contributions de la th`ese
1.8 Convexit´e des solutions de certains probl`emes variationnels en dimension infinie
1.8.1 Le cas Euclidien
1.8.2 Contributions de la th`ese
2 Continuous Primal-Dual methods for Image Processing
2.1 Introduction
2.1.1 Presentation of the problem
2.1.2 Idea of the Primal-Dual method
2.2 Maximal Monotone Operators
2.2.1 Definitions and first properties of maximal monotone operators
2.2.2 Application to Arrow-Hurwicz methods
2.3 Study of the Primal-Dual Method
2.4 Numerical Experiments
2.4.1 The numerical scheme
2.4.2 The experiments
2.5 Conclusion and perspectives
3 Volume-constrained minimizers for the prescribed curvature problem in periodic media
3.1 Introduction
3.2 Existence of minimizers
3.3 Properties of the isovolumetric function
3.4 Existence of surfaces with prescribed mean curvature
3.4.1 Compact solutions with big volume.
3.4.2 Asymptotic behavior of minimizers.
3.5 Conclusion and perspectives
4 Approximation and relaxation of perimeter in the Wiener space
4.1 Introduction
4.2 Wiener space and functions of bounded variation
4.3 The Ehrhard symmetrization
4.4 Relaxation of perimeter
4.5 ¡-limit for the Modica-Mortola functional
4.6 Conclusion and perspectives
5 Convex minimizers for infinite dimensional variational problems
5.1 Introduction
5.2 Representation formula and relaxation of integral functionals
5.3 The finite dimensional case
5.4 The infinite dimensional case
5.5 A Geometric proof for the total variation in Gauss space
5.5.1 Convexity of the minimizer
5.6 Conclusion and perspectives
Bibliographie

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