## Contents (2004)

### BibTeX

@MISC{Dacorogna04contents,

author = {Bernard Dacorogna and Wilfrid Gangbo},

title = {Contents},

year = {2004}

}

### OpenURL

### Abstract

1.1 Statement of the problem...................... 1 1.2 A motivation for studying extension maps............. 2

### Citations

633 | User’s guide to viscosity solutions of second order partial differential equations
- Crandall, Ishii, et al.
- 1992
(Show Context)
Citation Context ...d here we just mention recent books that deal with this subject: Bardi-Capuzzo Dolcetta [1], Barles [2], Benton [3], Fleming-Soner [16], Subbotin [23] or the reference article of Crandall-Ishii-Lions =-=[7]-=-. However the most appropriate reference for the theorems either in the preceding section or in the present one is Lions [18]. We start with the de…nition of viscosity solution for the equation F (Du ... |

211 |
Solutions de viscosité des équations de Hamilton-Jacobi
- Barles
- 1994
(Show Context)
Citation Context ...ndall and Lions. In [12] we have a large bibliography (as well as an introduction on this method) and here we just mention recent books that deal with this subject: Bardi-Capuzzo Dolcetta [1], Barles =-=[2]-=-, Benton [3], Fleming-Soner [16], Subbotin [23] or the reference article of Crandall-Ishii-Lions [7]. However the most appropriate reference for the theorems either in the preceding section or in the ... |

207 |
Direct Methods in the Calculus of Variations
- Dacorogna
(Show Context)
Citation Context ... = 2 R m n : Fi ( ) = 0; i = 1; :::; N : No simple analogue to Theorem 11 exists and in order explain some results that can be applied to vectorial problems we need to introduce some terminology (cf. =-=[8]-=- or [12]). De…nition 13 (1) A function f : Rm one convex if n ! R = R[ f+1g is said to be rank f (tA + (1 t) B) tf (A) + (1 t) f (B) for every t 2 [0; 1] and every A; B 2 R m n with rank fA Bg = 1. (2... |

204 |
Generalized Solutions of Hamilton-Jacobi Equations
- Lions
(Show Context)
Citation Context ...-Soner [16], Subbotin [23] or the reference article of Crandall-Ishii-Lions [7]. However the most appropriate reference for the theorems either in the preceding section or in the present one is Lions =-=[18]-=-. We start with the de…nition of viscosity solution for the equation F (Du (x)) = 0, a.e. x 2 : (4) 4De…nition 6 A function u 2 C ( ) is said to be a viscosity solution of (4) if the following two co... |

193 |
M.: Controlled Markov processes and viscosity solutions
- Fleming, Soner
- 2006
(Show Context)
Citation Context ...e a large bibliography (as well as an introduction on this method) and here we just mention recent books that deal with this subject: Bardi-Capuzzo Dolcetta [1], Barles [2], Benton [3], Fleming-Soner =-=[16]-=-, Subbotin [23] or the reference article of Crandall-Ishii-Lions [7]. However the most appropriate reference for the theorems either in the preceding section or in the present one is Lions [18]. We st... |

147 |
Proximité et dualité dans un espace hilbertien
- Moreau
- 1965
(Show Context)
Citation Context ...ince the unit sphere S F is strictly convex we obtain as wished. y1 − y = y1 + y ⇒ y = 0 182.3 Extension from a convex subset of E to E In many applications, such as in Browder-Petryshyn [8], Moreau =-=[28]-=-, LionsStampacchia [23], Zabreiko-Kachurovsky-Krasnoselsky [40] –to cite few of them– it is important to know if for every closed convex set Ω ⊂ E, every 1–Lipschitz map u : Ω → F admits a 1–Lipschitz... |

115 |
Capuzzo Dolcetta, Optimal control and viscosity solutions of HamiltonJacobi-Bellman equations, Birkhäuser
- Bardi, I
- 1997
(Show Context)
Citation Context ...ework by Crandall and Lions. In [12] we have a large bibliography (as well as an introduction on this method) and here we just mention recent books that deal with this subject: Bardi-Capuzzo Dolcetta =-=[1]-=-, Barles [2], Benton [3], Fleming-Soner [16], Subbotin [23] or the reference article of Crandall-Ishii-Lions [7]. However the most appropriate reference for the theorems either in the preceding sectio... |

94 |
Shape optimization by the homogenization method
- Allaire, Bonnetier, et al.
- 1997
(Show Context)
Citation Context ...vation for studying extension maps We now motivate these two questions by considerations of optimal design. One of the basic problem in optimal design, which has received a lot of attention (see [2], =-=[3]-=-, [35], [36]), is the study of the variational problem ∫ inf{I[σ] := ˆρ(σ) : σ ∈ ΣF (Ω)}. (3) σ Ω Here, ˆρ : Rd×d → [0, +∞] is a prescribed function, homogeneous of degree 1, so that ∫ Ω ˆρ(σ) is well... |

48 |
Variational inequalities
- Lions, Stampacchia
- 1972
(Show Context)
Citation Context ...F is strictly convex we obtain as wished. y1 − y = y1 + y ⇒ y = 0 182.3 Extension from a convex subset of E to E In many applications, such as in Browder-Petryshyn [8], Moreau [28], LionsStampacchia =-=[23]-=-, Zabreiko-Kachurovsky-Krasnoselsky [40] –to cite few of them– it is important to know if for every closed convex set Ω ⊂ E, every 1–Lipschitz map u : Ω → F admits a 1–Lipschitz extension over E. Thes... |

41 | A geometric approach to monotone functions in Rn
- Alberti, Ambrosio
- 1999
(Show Context)
Citation Context ... ∗ Ω ≤ 1, then the map u + id (as well as −u + id) is monotone over Ω. Moreover if Ω ⊂ Rd is open and connected then u is differentiable everywhere, except on a (d − 1)-dimensional Hausdorff set (see =-=[1]-=-), and ρ(e(u)) ≤ 1. The well-known Korn inequality also ensures that u is continuous and so, is locally bounded (see [29]). - If Ω ⊂ R d is a convex set containing 0 in its interior, we define the Min... |

38 | Optimal design for minimum weight and compliance in plane stress using extremal microstructures - Allaire, Kohn - 1993 |

33 | Normed Linear Spaces - Day - 1973 |

33 |
Nonlinear Functional Analysis
- SCHWARTZ
- 1964
(Show Context)
Citation Context ...me time extended to Hilbert spaces, in several different ways, notably by Valentine [38], [39], Grünbaum [18], Minty [27] and others; one could also consult textbooks such as Federer [16] or Schwartz =-=[34]-=-. When turning to necessary conditions, it was established by Schönbeck [31] that if dimE, dimF ≥ 2 and if the unit sphere S F of F is strictly convex (see below for a precise definition), then [E; F ... |

32 |
General existence theorems for HamiltonJacobi equations in the scalar and vectorial case
- Dacorogna, Marcellini
- 1997
(Show Context)
Citation Context ... this subject is so large that several books would be needed to cover it. We will focus on the work of DacorognaMarcellini that is developed in a recent book [12] (following earlier work [9], [10], 1=-=[11]-=-). We have not touched the very closely related work on convex integration of Gromov [17] (cf. also Spring [22]) as developed by Müller-Sverak [20], [21] and others. We also will speak only little on ... |

30 |
Shape optimization solutions via Monge-Kantorovich equation
- Bouchitté, Buttazzo, et al.
- 1997
(Show Context)
Citation Context ...nd || · ||F be two norms on R d and define ˆρ(ξ) = sup a∈Rd {||ξa||F : ||a||E ≤ 1}. Ψ(a, b) = ||b||F − ||a||E is such that the values in (3) and (6) agree. This case has been intensively studied (see =-=[5]-=-, [6] and [7] for additional references). When the dimension d ≥ 2 and the set { b ∈ Rd : ||b||F = 1 } is strictly convex then Theorem 11 gives a necessary and sufficient condition for LipΨ (Ω) and Li... |

17 | Kirszbraun’s theorem and metric spaces of bounded curvature - Lang, Schroeder - 1997 |

16 |
On inner products in linear metric spaces
- Jordan, Neumann
- 1935
(Show Context)
Citation Context ...oints, if s < d. Iterating the process we have indeed shown that S E ∩ Σmax has at least 2d distinct points. Existence of at least 2d distinct points in S E ∩ Σmin is obtained in a similar manner. In =-=[19]-=- Jordan and von Neumann gave a condition which characterizes the norm induced by an inner product. Lemma 5 (Jordan-von Neumann) Assume that dimE ≥ 2. Then, the norm ‖.‖ E is induced by an inner produc... |

14 |
Construction of fixed points of nonlinear mappings in Hilbert spaces
- Browder, Petryshyn
- 1967
(Show Context)
Citation Context ... y‖ F = 1. Since the unit sphere S F is strictly convex we obtain as wished. y1 − y = y1 + y ⇒ y = 0 182.3 Extension from a convex subset of E to E In many applications, such as in Browder-Petryshyn =-=[8]-=-, Moreau [28], LionsStampacchia [23], Zabreiko-Kachurovsky-Krasnoselsky [40] –to cite few of them– it is important to know if for every closed convex set Ω ⊂ E, every 1–Lipschitz map u : Ω → F admits ... |

14 |
The Hamilton-Jacobi equation: a global approach
- Benton
- 1977
(Show Context)
Citation Context ...ons. In [12] we have a large bibliography (as well as an introduction on this method) and here we just mention recent books that deal with this subject: Bardi-Capuzzo Dolcetta [1], Barles [2], Benton =-=[3]-=-, Fleming-Soner [16], Subbotin [23] or the reference article of Crandall-Ishii-Lions [7]. However the most appropriate reference for the theorems either in the preceding section or in the present one ... |

12 |
Characterizations of Inner Product Spaces
- Amir
- 1986
(Show Context)
Citation Context ... = ρΣE and ρΣF = ‖.‖ min max F = ρΣF . min 2.1 Norms induced by an inner product We start by collecting some well known facts about inner product spaces. One can consult, as a general reference, Amir =-=[4]-=-. Only Lemma 6 and Lemma 8 will be used in the proofs of the next sections, we have however incorporated some other results for the sake of giving a broader panorama. 6Definition 3 An ellipse centere... |

12 | Seppecher: Energies with respect to a measure and applications to low dimensional structures
- Bouchitté, Buttazzo, et al.
- 1997
(Show Context)
Citation Context ... · ||F be two norms on R d and define ˆρ(ξ) = sup a∈Rd {||ξa||F : ||a||E ≤ 1}. Ψ(a, b) = ||b||F − ||a||E is such that the values in (3) and (6) agree. This case has been intensively studied (see [5], =-=[6]-=- and [7] for additional references). When the dimension d ≥ 2 and the set { b ∈ Rd : ||b||F = 1 } is strictly convex then Theorem 11 gives a necessary and sufficient condition for LipΨ (Ω) and LipΨ (R... |

11 |
Partial di¤erential relations
- Gromov
- 1986
(Show Context)
Citation Context ...n the work of DacorognaMarcellini that is developed in a recent book [12] (following earlier work [9], [10], 1[11]). We have not touched the very closely related work on convex integration of Gromov =-=[17]-=- (cf. also Spring [22]) as developed by Müller-Sverak [20], [21] and others. We also will speak only little on the very important tool known as viscosity method introduced by Crandall-Lions following ... |

9 |
l1 and l∞ approximation of vector fields in the plane
- Strang
- 1982
(Show Context)
Citation Context ...n for studying extension maps We now motivate these two questions by considerations of optimal design. One of the basic problem in optimal design, which has received a lot of attention (see [2], [3], =-=[35]-=-, [36]), is the study of the variational problem ∫ inf{I[σ] := ˆρ(σ) : σ ∈ ΣF (Ω)}. (3) σ Ω Here, ˆρ : Rd×d → [0, +∞] is a prescribed function, homogeneous of degree 1, so that ∫ Ω ˆρ(σ) is well defin... |

9 |
On the Dirichlet problem for HamiltonJacobi equations. A Baire category approach
- Blasi, Pianigiani
- 1999
(Show Context)
Citation Context ...na and Friesecke with a very explicit construction that we call ”pyramidal”. The method of proof that we will outline below follows works of Cellina [6], Bressan-Flores [4], De Blasi-Pianigiani [14], =-=[15]-=- and DacorognaMarcellini [12]. Proof. We very roughly outline the idea of the proof only in a very special case that has the advantage to make the procedure transparent and not burdened by too many te... |

8 |
Théorèmes d’existence dans le cas scalaire et vectoriel pour les équations de
- Dacorogna, Marcellini
- 1996
(Show Context)
Citation Context ...ns. Of course this subject is so large that several books would be needed to cover it. We will focus on the work of DacorognaMarcellini that is developed in a recent book [12] (following earlier work =-=[9]-=-, [10], 1[11]). We have not touched the very closely related work on convex integration of Gromov [17] (cf. also Spring [22]) as developed by Müller-Sverak [20], [21] and others. We also will speak o... |

7 |
Some characterization of inner-product spaces
- Day
(Show Context)
Citation Context ...Σmax and in SE . Consequently, Σmax = SE and thus ‖.‖E is induced by an inner product, which is the desired contradiction. We immediately obtain as a corollary the following result established by Day =-=[9]-=-, which is a refinement of the lemma of Jordan-von Neumann. Corollary 7 Assume that dimE ≥ 2. Then the norm ‖.‖E inner product if and only if is induced by an ‖x + y‖ 2 E + ‖x − y‖2 E = 4 (24) for all... |

7 |
Geometric restrictions for the existence of viscosity solutions; Annales Institut Henri Poincaré, Analyse Non Linéaire 16
- Cardaliaguet, Dacorogna, et al.
- 1999
(Show Context)
Citation Context ...of the Hamiltonian F . In absence of convexity it may well happen that Lipschitz solutions do exist but no viscosity ones. The following example has been given by Cardaliaguet-Dacorogna-Gangbo-Georgy =-=[5]-=-. In this article a general theorem relates the existence of viscosity solutions, the zeroes set of F and the geometry of the domain . Theorem 10 Let R2 be convex. Then (writing for u = u (x; y), ux =... |

6 |
On the extension of Lipschitz, Lipschitz-Hölder continuous, and monotone functions
- Minty
- 1970
(Show Context)
Citation Context ...tractions. This result, known as Kirszbraun theorem, has been proved, and at the same time extended to Hilbert spaces, in several different ways, notably by Valentine [38], [39], Grünbaum [18], Minty =-=[27]-=- and others; one could also consult textbooks such as Federer [16] or Schwartz [34]. When turning to necessary conditions, it was established by Schönbeck [31] that if dimE, dimF ≥ 2 and if the unit s... |

6 |
Implicit partial di¤erential equations and the constraints of non linear elasticity; Commun
- Dacorogna, Tanteri
(Show Context)
Citation Context ... recover the classical eikonal equation. In the terminology introduced before we have n E = A = ( ; ) 2 R n R n : j j 2 = j j 2 + f 2 o ; h ; i = 0 Rco E = co E = R 2 n : We can then prove (cf. [12], =-=[13]-=-) Theorem 16 Let ' 2 W 1;1 ; R2 ; then there exists w = (u; v) 2 W 1;1 ; R2 satisfying (14). 3.4 The second order case The method can also be applied to problems with constraints [13] or to second ord... |

5 |
Implicit partial di¤erential equations; Birkhäuser
- Dacorogna, Marcellini
- 1999
(Show Context)
Citation Context ...r partial di¤erential equations. Of course this subject is so large that several books would be needed to cover it. We will focus on the work of DacorognaMarcellini that is developed in a recent book =-=[12]-=- (following earlier work [9], [10], 1[11]). We have not touched the very closely related work on convex integration of Gromov [17] (cf. also Spring [22]) as developed by Müller-Sverak [20], [21] and ... |

4 | Ein Erweiterungssatz für monotone - Debrunner, Flor - 1964 |

4 |
On the radial projection in normed spaces
- Figueiredo, Karlovitz
- 1992
(Show Context)
Citation Context ... important to know if for every closed convex set Ω ⊂ E, every 1–Lipschitz map u : Ω → F admits a 1–Lipschitz extension over E. These questions have been investigated by DeFigueiredo and Karlovitz in =-=[12]-=-, [13] and [14] in the case E = F and ‖.‖E = ‖.‖F . The general case which still remains open, is apparently closely related to whether or not projections on convex sets are contractions. In this sect... |

4 |
The modulus of convexity in normed linear spaces
- Nordlander
- 1960
(Show Context)
Citation Context ...ly we proceed by contradiction and assume that the norm ‖.‖ E is not induced by an inner product. By Lemma 6 we have that (24) does not hold and thus the claim. We conclude with Nordlander inequality =-=[30]-=-. Lemma 8 (Nordlander) Assume that dimE ≥ 2 and that 0 < t < 1. Then inf{‖x + y‖ E : (x, y) ∈ St} ≤ 2 √ 1 − t 2 ≤ sup{‖x + y‖ E : (x, y) ∈ St} (25) where St = { (x, y) ∈ S E × S E : ‖x − y‖ E = 2t } .... |

4 |
Non convex valued di¤erential inclusions in Banach spaces
- Blasi, Pianigiani
- 1991
(Show Context)
Citation Context ... Cellina and Friesecke with a very explicit construction that we call ”pyramidal”. The method of proof that we will outline below follows works of Cellina [6], Bressan-Flores [4], De Blasi-Pianigiani =-=[14]-=-, [15] and DacorognaMarcellini [12]. Proof. We very roughly outline the idea of the proof only in a very special case that has the advantage to make the procedure transparent and not burdened by too m... |

3 |
Optimization of light structures: the vanishing mass conjecture
- Bouchitté
(Show Context)
Citation Context ...me that ˆρ(η) = sup {| 〈η; ξ〉 | : ρ(ξ) ≤ 1}, ρ(ξ) = sup {| 〈ξb; b〉 | : |b| = 1} (7) ξ∈Rd×d b∈Rd which is the Michell case [25], referred to as the fictive materials or light structures case (see also =-=[7]-=-). Let 〈·; ·〉 and ‖·‖ be respectively the Euclidean scalar product and the associated Euclidean norm on R d . Then Ψ(a, b) := |〈a; b〉| − ‖a‖ 2 is such that the values in (3) and (6) agree. Then • Case... |

3 |
On the extention of contractions of normed spaces
- Figueiredo, Karlovitz
- 1970
(Show Context)
Citation Context ... 3, 4, every map u ∈ Lip1(D, F ) admits an extension ũ ∈ Lip1(D ′, F ). When E = F, one can prove some stronger results, see Edelstein and Thompson [15], Schönbeck [32] and DeFigueiredo and Karlovitz =-=[13]-=-, [14]. It is one of our goals to give a still different, and somehow more elementary and more self contained, proof of the result of Schönbeck (see Theorem 11). The approach used to obtain this resul... |

3 |
Extension of range of functions
- Shane
- 1934
(Show Context)
Citation Context ...and F , which in most of our analysis will be Banach spaces, ensuring that [E; F ] has the extension property for contractions. The earliest result in this direction is the celebrated Mac Shane lemma =-=[24]-=- asserting that if dimF = 1, then [E; F ] has the extension property for contractions for any E. It turns out that this is also true for any F , if dimE = 1. At the same time Kirszbraun [21] proved th... |

3 |
The limits of economy of material in framed-structures
- Michell
- 1904
(Show Context)
Citation Context ...following two cases which have attracted a lot of attention. • Case 1. We assume that ˆρ(η) = sup {| 〈η; ξ〉 | : ρ(ξ) ≤ 1}, ρ(ξ) = sup {| 〈ξb; b〉 | : |b| = 1} (7) ξ∈Rd×d b∈Rd which is the Michell case =-=[25]-=-, referred to as the fictive materials or light structures case (see also [7]). Let 〈·; ·〉 and ‖·‖ be respectively the Euclidean scalar product and the associated Euclidean norm on R d . Then Ψ(a, b) ... |

3 |
Weak and Measured-valued Solutions to Evolutionary PDEs
- Malek, Necas, et al.
- 1996
(Show Context)
Citation Context ...s differentiable everywhere, except on a (d − 1)-dimensional Hausdorff set (see [1]), and ρ(e(u)) ≤ 1. The well-known Korn inequality also ensures that u is continuous and so, is locally bounded (see =-=[29]-=-). - If Ω ⊂ R d is a convex set containing 0 in its interior, we define the Minkowski function (or the gauge) associated to Ω to be ρΩ(x) = inf {t : x/t ∈ Ω}. t>0 Acknowledgements. It is a pleasure to... |

3 |
Sur le problème de Cauchy-Dirichlet pour les systèmes d’équations non linéaires du premier ordre
- Dacorogna, Marcellini
- 1996
(Show Context)
Citation Context ...f course this subject is so large that several books would be needed to cover it. We will focus on the work of DacorognaMarcellini that is developed in a recent book [12] (following earlier work [9], =-=[10]-=-, 1[11]). We have not touched the very closely related work on convex integration of Gromov [17] (cf. also Spring [22]) as developed by Müller-Sverak [20], [21] and others. We also will speak only li... |

2 |
The extension of contractions and the intersection of balls in Banach spaces
- DeFigueiredo, Karlovitz
- 1972
(Show Context)
Citation Context ... every map u ∈ Lip1(D, F ) admits an extension ũ ∈ Lip1(D ′, F ). When E = F, one can prove some stronger results, see Edelstein and Thompson [15], Schönbeck [32] and DeFigueiredo and Karlovitz [13], =-=[14]-=-. It is one of our goals to give a still different, and somehow more elementary and more self contained, proof of the result of Schönbeck (see Theorem 11). The approach used to obtain this result invo... |

2 |
Contractions, isometries and some properties of inner product spaces
- Edelstein, Thompson
- 1967
(Show Context)
Citation Context ...y if for every set D ⊂ D ′ of respective cardinality 3, 4, every map u ∈ Lip1(D, F ) admits an extension ũ ∈ Lip1(D ′, F ). When E = F, one can prove some stronger results, see Edelstein and Thompson =-=[15]-=-, Schönbeck [32] and DeFigueiredo and Karlovitz [13], [14]. It is one of our goals to give a still different, and somehow more elementary and more self contained, proof of the result of Schönbeck (see... |

2 |
Michell trusses and existence of lines of principal actions
- Gangbo
(Show Context)
Citation Context ... of bounded strains. It suffices to show this result in R2 and this is achieved in Theorem 22. Our proof does not exhibit an explicit counter example. It exploits the study of Michell trusses done in =-=[17]-=-. Throughout this section, we set e1 = (1, 0), ⃗0 = (0, 0), and e2 = (0, 1). Theorem 20 (i) Assume that X = {a, b} ⊂ R2 , c ∈ R2 , and u : X → R2 satisfies ||u|| ∗ X = 1. Then, u admits an extension ū... |

2 |
A generalization of theorems of Kirszbraun and
- Grünbaum
- 1962
(Show Context)
Citation Context ...erty for contractions. This result, known as Kirszbraun theorem, has been proved, and at the same time extended to Hilbert spaces, in several different ways, notably by Valentine [38], [39], Grünbaum =-=[18]-=-, Minty [27] and others; one could also consult textbooks such as Federer [16] or Schwartz [34]. When turning to necessary conditions, it was established by Schönbeck [31] that if dimE, dimF ≥ 2 and i... |

2 | A further generalization of Kirszbraun theorem - Kamardian - 1972 |

2 |
die zusammenziehende und Lipschitzsche Transformationen
- Uber
- 1934
(Show Context)
Citation Context ...ane lemma [24] asserting that if dimF = 1, then [E; F ] has the extension property for contractions for any E. It turns out that this is also true for any F , if dimE = 1. At the same time Kirszbraun =-=[21]-=- proved that if E and F are both finite dimensional spaces whose norms are induced by a scalar product, then [E; F ] has the extension property for contractions. This result, known as Kirszbraun theor... |

2 | On the simultaneous solution of certain system of linear inequalities - Minty - 1962 |

2 |
Extension of nonlinear contractions
- Schönbeck
- 1966
(Show Context)
Citation Context ...lentine [38], [39], Grünbaum [18], Minty [27] and others; one could also consult textbooks such as Federer [16] or Schwartz [34]. When turning to necessary conditions, it was established by Schönbeck =-=[31]-=- that if dimE, dimF ≥ 2 and if the unit sphere S F of F is strictly convex (see below for a precise definition), then [E; F ] has the extension property for contractions if and only if both E and F ar... |

2 |
On the extension of Lipschitzian maps
- Schönbeck
- 1967
(Show Context)
Citation Context ...et D ⊂ D ′ of respective cardinality 3, 4, every map u ∈ Lip1(D, F ) admits an extension ũ ∈ Lip1(D ′, F ). When E = F, one can prove some stronger results, see Edelstein and Thompson [15], Schönbeck =-=[32]-=- and DeFigueiredo and Karlovitz [13], [14]. It is one of our goals to give a still different, and somehow more elementary and more self contained, proof of the result of Schönbeck (see Theorem 11). Th... |

2 |
On the di¤erential inclusion x 0 2 f 1; 1g ; Atti
- Cellina
- 1980
(Show Context)
Citation Context ...atum is a¢ ne, this result was established by Cellina and Friesecke with a very explicit construction that we call ”pyramidal”. The method of proof that we will outline below follows works of Cellina =-=[6]-=-, Bressan-Flores [4], De Blasi-Pianigiani [14], [15] and DacorognaMarcellini [12]. Proof. We very roughly outline the idea of the proof only in a very special case that has the advantage to make the p... |