## 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

3685 | Convex analysis
- Rockafellar
- 1970
(Show Context)
Citation Context ...p 1. Since (3.1) holds we can find a convex and compact set LcRsn such thatsD~(x) e L C int co{rs9 Rn: F(~) ---- 0}. (3.12)sWe can then find a polytope P (cf. the proof of Theorem 20.4 in Rockafellar =-=[35]-=-) withsthe following property:sP -- co{~/1, ..., ~TN}, (3.13)sL C int P C P C int co{~s9 Rn: F(~) = 0}.s12 B. DACOROGNA AND P. MARCELLINIsWe then use the Carath~odory theorem (cf. Theorem 17.1 in Rock... |

2086 |
Elliptic partial differential equations of second order
- Gilbarg, Trudinger
- 1983
(Show Context)
Citation Context ...12 | 0* |D.* | 2 dx&| 0* h.* dx | 0* |D.* | dx&&h& | 0* |.* | dx. (9.8) Note that .* # W 1, 20 (0*). Hence by the Ho lder inequality, followed by a standard Sobolev inequality in W 1, 10 (0*), see =-=[6]-=-, one has | 0* |.* | dx|0* | 1n \|0* . n(n&1)* dx+ (n&1)n |0* | 1n C | 0* |D.* | dx, (9.9) where C is an absolute constant. Combining (9.8) and (9.9) and using the important fact that D.*0, one... |

771 | User's guide to viscosity solutions of second order partial dierential 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 ... |

499 |
Hyperbolic systems of conservation laws
- LAX
- 1957
(Show Context)
Citation Context ...- l ,swe say that u is a scalar function). As usual Du denotes the gradient of u.sThis problem (1.1) has been intensively studied, essentially in the scalar case insmany relevant articles such as Lax =-=[28]-=-, Douglis [23], Kru2kov [27], Crandall-Lions [16],sCrandall-Evans-Lions [14], Capuzzo Dolcetta-Evans [8], Capuzzo Dolcetta-Lions [9],sCrandall-Ishii-Lions [15]. For a more complete bibliography we ref... |

432 |
Minimal surfaces and functions of bounded variation
- Giusti
- 1984
(Show Context)
Citation Context ...(0) & BV(Rn) then u has an internal boundary trace u& in L1(0), and that | 0 |Du|=| 0 |Du| dx+| 0 |u&&0| dHn&1 , (2.4) where dHn&1 denotes the (n&1)-dimensional Hausdorff measure on 0 (see Giusti =-=[8]-=-, Remark 2.14). We now define similar expressions for the area of the graph of a BV(Rn) function which is #0 outside 0. First we define the area of the graph on 0 by | 0 - 1+|Du| 2#sup {|0 g0+u : n i=... |

384 |
Viscosity solutions of HamiltonJacobi equations
- Crandall, Lions
- 1983
(Show Context)
Citation Context ...sentially in the scalar case insmany relevant articles such as Lax [28], Douglis [23], Kru2kov [27], Crandall-Lions [16],sCrandall-Evans-Lions [14], Capuzzo Dolcetta-Evans [8], Capuzzo Dolcetta-Lions =-=[9]-=-,sCrandall-Ishii-Lions [15]. For a more complete bibliography we refer to the main recentsmonographs of Benton [7], Lions [29], Fleming-Soner [25], Barles [6] and Bardi-CapuzzosDolcetta [5].sB. DACORO... |

313 |
Optimal control and viscosity solutions of Hamilton-JacobiBellman equations
- Bardi, Capuzzo-Dolcetta
- 1997
(Show Context)
Citation Context ...etta-Lions [9],sCrandall-Ishii-Lions [15]. For a more complete bibliography we refer to the main recentsmonographs of Benton [7], Lions [29], Fleming-Soner [25], Barles [6] and Bardi-CapuzzosDolcetta =-=[5]-=-.sB. DACOROGNA AND P. MARCELL IN IsOur motivation to study this equation, besides its intrinsic interest, comes from thescalculus of variations. In this context first order partial differential equati... |

256 |
Solutions de viscosité des équations de Hamilton-Jacobi, volume 17
- 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 ... |

255 |
Direct methods in the calculus of variations
- Dacorogna
- 1989
(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... |

240 |
Controlled Markov processes and viscosity solutions
- Fleming, Soner
- 1993
(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... |

233 |
Generalized solution of Hamilton-Jacobi equations
- LIONS
- 1982
(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... |

229 |
Convexity conditions and existence theorems in nonlinear elasticity
- Ball
- 1977
(Show Context)
Citation Context ... [21], [221, [34], in the context of Cauchy problems for ordinary differentialsinclusions.s(ii) The weak lower semicontinuity and the quasiconvexity condition introduced bysMorrey [33] (see also Ball =-=[3]-=- and [17]), that is the appropriate xtension of convexity tosvector valued problems.sWe very roughly outline the idea of the proof following the above scheme. We firstsconstruct a quasiconvex function... |

216 |
Multiple Integrals in the Calculus of Variations
- Morrey
- 1966
(Show Context)
Citation Context ...De Blasi-sPianigiani [21], [221, [34], in the context of Cauchy problems for ordinary differentialsinclusions.s(ii) The weak lower semicontinuity and the quasiconvexity condition introduced bysMorrey =-=[33]-=- (see also Ball [3] and [17]), that is the appropriate xtension of convexity tosvector valued problems.sWe very roughly outline the idea of the proof following the above scheme. We firstsconstruct a q... |

179 |
Some properties of viscosity solutions of Hamilton-Jacobi equations
- Crandall, Evans, et al.
- 1984
(Show Context)
Citation Context ...of u.sThis problem (1.1) has been intensively studied, essentially in the scalar case insmany relevant articles such as Lax [28], Douglis [23], Kru2kov [27], Crandall-Lions [16],sCrandall-Evans-Lions =-=[14]-=-, Capuzzo Dolcetta-Evans [8], Capuzzo Dolcetta-Lions [9],sCrandall-Ishii-Lions [15]. For a more complete bibliography we refer to the main recentsmonographs of Benton [7], Lions [29], Fleming-Soner [2... |

162 |
Fine phase mixtures as minimizers of energy
- Ball, James
(Show Context)
Citation Context ...,swhere SO(n) denotes the set of orthogonal matrices with positive determinant, I is thesidentity matrix andsI _= 1 "s-1sThe problem of potential wells finds its origins in elasticity (cf. Ball-James =-=[4]-=-, for ex-sample). Problem (1.8) has been solved by Cellina-Perrotta [13] if n=3 and rsThe existence results stated in the above examples are a consequence of generalstheorems established in w The main... |

161 |
Proximité et dualité dans un espace hilbertien,” Bulletin de la Sociéte
- 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... |

135 |
Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations
- 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... |

122 | A transmission problem in the calculus of variations
- Acerbi, Fusco
- 1994
(Show Context)
Citation Context ...Tksmeas(12- ~0)s-- k+ lsL 1 = +oc,stETkswhich contradicts the fact that ~ is bounded. It follows that the setso~s{t>O:meast2t>o} C U Tksk=lsis countable. Therefore the set {t>0:measf~t=0} is dense in =-=[0, 1]-=-, and thus (2.16). []s3. The nonconvexssca larscasesandssys tems o fsequat ionssWe now turn to an application of the results of w The main theorem of this section issTHEOREM 3.1 (the nonconvex scalar ... |

118 |
Shape Optimization by the Homogenization Method
- ALLAIRE
(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... |

76 |
The Hamilton-Jacobi theory in the Calculus of Variations
- Rund
- 1973
(Show Context)
Citation Context ...ic interest, comes from thescalculus of variations. In this context first order partial differential equations have beensintensively used, cf. for example the monographs of Carath@odory [10] and Rund =-=[36]-=- (forsmore recent developments on the vectorial case, see [19]).sIn this paper we propose some new hypotheses on the function F in (1.1) that allowsus to treat systems of equations as well as vectoria... |

72 |
Variational inequalities
- Lions, Stampacchia
- 1967
(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... |

69 |
Optimal design and relaxation of variational problems
- Kohn, Strang
(Show Context)
Citation Context ... Rg to concludesat (7.44), since we do not, a priori, know that Qg is rank-one convex all over K (we knowsit only in int K).sRecall that Rg can be obtained by the following procedure (cf. Kohn-Strang =-=[26]-=- orsDacorogna [17]). Let for kENsRog=g,sRa+lg(~) =inf{ARkg(A)+(1-•)Rkg(B): A E [0, 1], A, U E K,srank{A-B} ~< 1, AA+(1-A)B = ~}, (7.45)slimk_.~ Rkg = Rg.sSo in order to prove (7.44) it will be suffici... |

68 |
On a partial differential equation involving the Jacobian determinant
- Dacarogna, Moser
- 1990
(Show Context)
Citation Context ...et Du[=ala> Both equations have beensseparately studied in the literature. For the first one, see for example Kru~kov [27] andsLions [29]. For the second one (without he modulus), cf. Dacorogna-Moser =-=[20]-=-.sThe Dirichlet problem (1.6) can also be rewritten in terms of "potential wells";snamely, if a~=l for i=1,2, ...,n, then (1.6) and (1.7) take the formsDu(x) e SO(n)IUSO(n)I_, a.e. xs9 a, (1.8)sU = ~ ... |

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

43 | A geometrical approach to monotone functions
- 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... |

43 |
General existence theorems for hamiltonjacobi equations in the scalar and vectorial case, Acta Mathematica 178
- 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 ... |

41 | Normed Linear Spaces - Day - 1962 |

40 |
Attainment results for the two-well problem by convex integration. “Geometric Analysis and the Calculus of Variations
- Müller, Šverák
- 1996
(Show Context)
Citation Context ...orial case has been investigated for some special examples notably by Allaire-Francfort [1], Cellina-Zagatti [4], Dacorogna-Ribeiro [10], DacorognaTanteri [11], Mascolo-Schianchi [17], Müller-Sverak =-=[18]-=- and Raymond [20]. A more systematic study was achieved by Dacorogna-Marcellini in [6], [7] and [8]. Building on [6] and owing to the recent developments in the treatment of implicit partial different... |

39 |
Construction of fixed points of nonlinear mappings in Hilbert space
- 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 ... |

39 | Implicit partial differential equations
- Dacorogna, Marcellini
- 1999
(Show Context)
Citation Context ...tions are not really of the same nature. The first one is what is called an implicit partial differential equation, which has recently received a lot of attention and we refer to Dacorogna-Marcellini =-=[7]-=- for some bibliographical and historical comments. The second one is more geometric in nature and has to do with some ”quasiaffinity” of the quasiconvex envelope Qf . The scalar case (n = 1 or m = 1) ... |

36 |
Nonlinear Functional Analysis
- Schwartz
- 1969
(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 ... |

35 | Design and application of a gradient-weighted moving finite element code I: In one dimension - Carlson, Miller - 1998 |

31 |
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... |

29 |
Metodi Diretti nel Calcolo delle Variazioni
- Giusti
- 1994
(Show Context)
Citation Context ...that c1 |!| p&c2F(x, s, !)c3 |!| p+c4 (9.6) for every x in 0, every s in R with |s|&w& , and every ! in Rn. (For our purposes p=2.) Then it is established in Theorem 7.6 and Theorem 7.8 of Giusti =-=[9]-=- that there exists a Holder coefficient, 0<:<1, such that w is in C0, :(0 ). Here F is continuous with respect to (s, !) but it is not required that F be differentiable with respect to its arguments ... |

26 |
Quasi-convexity and lower semicontinuity of multiple variational integrals of any order
- MEYERS
- 1965
(Show Context)
Citation Context ...rtain ~ > 0,sf.y is rank-one convex ca I~1 ~< 2/v~.sThe main theorem which justifies the notion of quasiconvexity is the following es-stablished by Morrey [33] and refined by many authors, cf. Meyers =-=[32]-=-, Acerbi-Fusco [1]sand Marcellini [30].sGENERAL EXISTENCE THEOREMS FOR HAMILTON-JACOBI EQUATIONS 27sTHEOREM 7.1. Let gt be a bounded open set of R n. Let f :Rm• be quasi-sconvex. If u~ converges weak-... |

21 |
Optimal switching for ordinary differential equations
- Dolcetta, Evans
- 1988
(Show Context)
Citation Context ...been intensively studied, essentially in the scalar case insmany relevant articles such as Lax [28], Douglis [23], Kru2kov [27], Crandall-Lions [16],sCrandall-Evans-Lions [14], Capuzzo Dolcetta-Evans =-=[8]-=-, Capuzzo Dolcetta-Lions [9],sCrandall-Ishii-Lions [15]. For a more complete bibliography we refer to the main recentsmonographs of Benton [7], Lions [29], Fleming-Soner [25], Barles [6] and Bardi-Cap... |

21 |
On the regularity of boundaries of sets minimizing perimeter with a volume constraint
- Gonzalez, Massari, et al.
- 1983
(Show Context)
Citation Context ...mizes the ratio P(G) |G| over all subsets G of 0 of positive measure. This is a variational problem which has been studied previously, for example, see Keller [12] and Gonzales, Massari and Tamanini =-=[10]-=-. Remark. Note that (8.18) and (8.19) establish that, if **>0, then the maximum set 0* is almost a Giusti extremal set for the function h(x)&**. We have the desired equality in (8.19) for the set 0* i... |

20 |
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... |

19 |
Rank-one convexity does not imply quasiconvexity
- Šverák
- 1992
(Show Context)
Citation Context ...amples we recall the well-known fact thatsf convex ~ f polyconvex :=> f quasiconvex => f rank-one convex. (7.4)sAll the counter implications are false (for the last one at least when m>~3; cf.sSverAk =-=[37]-=-).sExamples. (i) Let m=n.sFor ~ER n• denote bys0 ~< AI(~) ~< A2(~) < ... < A,(~)sthe singular values of ~ (i.e. eigenvalues of (~t~)1/2). It is well known that (cf. Proposi-stion 1.2 in the appendix i... |

18 | Kirszbrauns theorem and metric spaces of bounded curvature - Lang, Schroeder - 1997 |

17 |
Deformations with finitely many gradients and stability of quasiconvex hulls
- Kirchheim
(Show Context)
Citation Context ... ξ ∈ Rm×n : Fi (ξ) = 0, i = 1, 2, ..., I } where Fi : Rm×n → R, i = 1, 2, ..., I, are quasiconvex. This hypothesis was later removed by Sychev in [22] (see also Müller and Sychev [19]). Kirchheim in =-=[14]-=- pointed out that using a classical result of function theory then the proof of Dacorogna-Marcellini was still valid without the extra hypothesis on E; it is this idea combined with the original proof... |

16 |
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 ... |

15 |
Calculus of Variations
- Caratheodory
- 1989
(Show Context)
Citation Context ...es its intrinsic interest, comes from thescalculus of variations. In this context first order partial differential equations have beensintensively used, cf. for example the monographs of Carath@odory =-=[10]-=- and Rund [36] (forsmore recent developments on the vectorial case, see [19]).sIn this paper we propose some new hypotheses on the function F in (1.1) that allowsus to treat systems of equations as we... |

14 |
Characterizations of Inner Product Spaces, in
- 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... |

14 |
Optimal existence theorems for nonhomogeneous differential inclusions
- Müller, Sychev
(Show Context)
Citation Context ...ypothesis that E = { ξ ∈ Rm×n : Fi (ξ) = 0, i = 1, 2, ..., I } where Fi : Rm×n → R, i = 1, 2, ..., I, are quasiconvex. This hypothesis was later removed by Sychev in [22] (see also Müller and Sychev =-=[19]-=-). Kirchheim in [14] pointed out that using a classical result of function theory then the proof of Dacorogna-Marcellini was still valid without the extra hypothesis on E; it is this idea combined wit... |

14 |
On the equation of surfaces of prescribed mean curvature, Existence and uniqueness without boundary conditions. Invent. math
- Giusti
- 1978
(Show Context)
Citation Context ... in the sense of BV(0), and it is a classical C2, :(0) solution of the PDE (1.1a) in the interior. However, it may ‘‘detach from the desired boundary values’’ on some portions of 0. Moreover, Giusti =-=[7]-=- showed that if 0 is an extremal set for h(x) (i.e. (1.5) holds with strict < for proper subsets, but with = in (1.5a) for 0 itself ) then there exists an extremal solution U(x) on 0 for the PDE (1.1a... |

13 |
Théorème 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... |

13 |
Partial dierential 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 ... |

13 |
On minima of a functional of the gradient: necessary conditions, Nonlinear Anal
- Cellina
- 1993
(Show Context)
Citation Context ... semicontinuous, ξ0 ∈ Rn with Cf (ξ0) < f (ξ0) and Cf not affine at ξ0. Then (P ) has no solution. Remark 24 In the scalar case this result has been obtained by several authors, in particular Cellina =-=[3]-=-, Friesecke [12] and Dacorogna-Marcellini [6]. It also gives, combined with the result of the preceding section, that, provided some appropriate boundedness is assumed, a necessary and sufficient cond... |

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... |