1622:
2936:
2759:
2041:
2365:
2236:
2717:
2594:
1785:
2931:{\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}({\boldsymbol {u}})=\left(\varepsilon _{ik}({\boldsymbol {u}})\right)=\left({\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{k}}}+{\frac {\partial u_{k}}{\partial x_{i}}}\right)\right)}
599:
3191:
1575:
1942:
3095:
3605:(1964a), "Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno" [Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions],
3271:
533:
2412:
2241:
2129:
333:
2607:
2484:
1675:
1502:
2988:
1159:
638:
of the problem, showing that the solution can be used to represent a physical phenomenon. It is a particularly important step since most of the problems modeled by variational inequalities are of physical
2083:
2752:
1340:
1048:
961:
541:
871:
387:
1926:
2443:
1192:
3106:
2114:
819:
3302:
2036:{\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}}-{\boldsymbol {u}})-F({\boldsymbol {v}}-{\boldsymbol {u}})\geq 0\qquad \forall {\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }}
1510:
458:
409:
355:
290:
268:
243:
1611:
1296:
1244:
1084:
3003:
999:
909:
1852:
753:
2466:
3471:(English translation of the title) is an historical paper describing the beginning of the theory of variational inequalities from the point of view of its founder.
3200:
3830:
726:
3345:
1887:
1818:
1433:
1413:
1389:
1368:
1267:
773:
685:
436:
3717:
470:
2370:
3414:
2360:{\displaystyle F({\boldsymbol {v}})=\int _{A}v_{i}f_{i}\,\mathrm {d} x+\int _{\partial A\setminus \Sigma }\!\!\!\!\!v_{i}g_{i}\,\mathrm {d} \sigma }
2231:{\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}})=-\int _{A}\sigma _{ik}({\boldsymbol {u}})\varepsilon _{ik}({\boldsymbol {v}})\,\mathrm {d} x}
3987:
3297:
3885:
2712:{\displaystyle {\boldsymbol {g}}({\boldsymbol {x}})=\left(g_{1}({\boldsymbol {x}}),g_{2}({\boldsymbol {x}}),g_{3}({\boldsymbol {x}})\right)}
2589:{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=\left(f_{1}({\boldsymbol {x}}),f_{2}({\boldsymbol {x}}),f_{3}({\boldsymbol {x}})\right)}
1780:{\displaystyle {\boldsymbol {u}}({\boldsymbol {x}})=\left(u_{1}({\boldsymbol {x}}),u_{2}({\boldsymbol {x}}),u_{3}({\boldsymbol {x}})\right)}
295:
3634:
Fichera, Gaetano (1964b), "Elastostatic problems with unilateral constraints: the
Signorini problem with ambiguous boundary conditions",
1441:
82:, recalled by the name of the general abstract problem. The applicability of the theory has since been expanded to include problems from
3292:
2947:
3825:
1092:
3785:
3759:
3684:
3549:
3507:
3992:
3462:
2049:
3965:
155:
111:
3915:
3960:
3404:
2726:
594:{\displaystyle \langle \cdot ,\cdot \rangle \colon {\boldsymbol {E}}^{\ast }\times {\boldsymbol {E}}\to \mathbb {R} }
3312:
1304:
1004:
914:
91:
3860:. An important paper, describing the abstract approach of the authors to the theory of variational inequalities.
3389:
3194:
2117:
1198:
824:
461:
412:
363:
163:
44:
40:
36:
3955:
3940:
1900:
143:
3186:{\displaystyle W({\boldsymbol {\varepsilon }})=a_{ikjh}({\boldsymbol {x}})\varepsilon _{ik}\varepsilon _{jh}}
3452:
2420:
1206:
665:
1570:{\displaystyle \langle \cdot ,\cdot \rangle \colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }
1164:
3868:, ISNM. International Series of Numerical Mathematics, vol. 153 (2nd ed.), Basel–Boston–Berlin:
3426:
3287:
2088:
782:
416:
48:
3598:. A short research note announcing and describing (without proofs) the solution of the Signorini problem.
3385:
79:
3440:. A brief research survey describing the field of variational inequalities, precisely the sub-field of
441:
392:
338:
273:
251:
226:
3422:
3090:{\displaystyle \sigma _{ik}=-{\frac {\partial W}{\partial \varepsilon _{ik}}}\qquad \forall i,k=1,2,3}
1587:
1272:
1220:
3869:
2991:
2939:
3400:
1056:
167:
3817:
3777:
3656:
3441:
2469:
1929:
1828:
1670:
1633:
187:
3575:
Rendiconti della
Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
968:
878:
3709:
3627:
3322:
2472:
1861:
1858:
151:
3855:
1834:
1197:
The absolute minimum must be searched between the solutions (if more than one) of the preceding
3607:
Memorie della
Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
731:
3911:
3881:
3821:
3781:
3755:
3713:
3680:
3623:
3545:
3503:
3381:
3317:
3274:
2476:
2121:
1933:
1891:
1791:
1667:
1663:
1630:
1626:
1621:
1299:
139:
107:
75:
67:
63:
3451:(1995), "La nascita della teoria delle disequazioni variazionali ricordata dopo trent'anni",
3266:{\displaystyle {\boldsymbol {a}}({\boldsymbol {x}})=\left(a_{ikjh}({\boldsymbol {x}})\right)}
2451:
3927:
3899:
3873:
3847:
3839:
3797:
3765:
3698:
3652:
3614:
3590:
3555:
3513:
3372:
3354:
3307:
1647:
1393:
171:
3895:
3694:
3586:
3368:
3931:
3903:
3891:
3851:
3801:
3769:
3702:
3690:
3618:
3602:
3594:
3582:
3566:
3559:
3541:
3517:
3499:
3448:
3393:
3376:
3364:
1824:
1659:
688:
115:
71:
693:
17:
3751:
3573:[On the elastostatic problem of Signorini with ambiguous boundary conditions],
1872:
1803:
1418:
1398:
1374:
1353:
1252:
758:
670:
421:
3981:
3731:
3668:
3570:
3537:
3495:
3340:
2720:
1794:
1637:
1578:
3359:
3417:, vol. Mathématiques appliquées (E), Histoire et Enseignement (F) – Volume 3,
1582:
652:
The problem of finding the minimal value of a real-valued function of real variable
620:
613:
528:{\displaystyle \langle F(x),y-x\rangle \geq 0\qquad \forall y\in {\boldsymbol {K}}}
221:
56:
3469:
The birth of the theory of variational inequalities remembered thirty years later
214:, p. 283), the definition of a variational inequality is the following one.
3526:
Finite
Dimensional Variational Inequalities and Complementarity Problems, Vol. 2
3484:
Finite
Dimensional Variational Inequalities and Complementarity Problems, Vol. 1
1202:
95:
28:
2407:{\displaystyle {\boldsymbol {u}},{\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }}
3877:
3533:
3491:
2598:
1855:
1788:
1641:
1343:
623:. The three obvious steps in the study of the problem are the following ones:
609:
602:
358:
52:
3971:
3735:
3664:
3571:"Sul problema elastostatico di Signorini con ambigue condizioni al contorno"
1821:
1371:
1347:
617:
83:
3843:
3972:
Alessio
Figalli, On global homogeneous solutions to the Signorini problem
3661:
Numerical analysis of variational inequalities. Translated from the French
3408:
3676:
1644:
3343:(1983), "The influence of elasticity in analysis: modern developments",
1053:
These necessary conditions can be summarized as the problem of finding
608:
In general, the variational inequality problem can be formulated on any
3936:
3806:
3747:
1890:
of the problem exists and is unique (under precise assumptions) in the
661:
178:
on 1965 where
Fichera presented his study of the Signorini problem, as
174:
to study and expand on
Fichera's work, after attending a conference in
87:
328:{\displaystyle F\colon {\boldsymbol {K}}\to {\boldsymbol {E}}^{\ast }}
182:, p. 283 reports: thus the theory become widely known throughout
3739:
3727:
3723:
3672:
3529:
3487:
1865:
1798:
1247:
246:
183:
175:
70:: in that model problem, the functional involved was obtained as the
59:
3920:
Comptes rendus hebdomadaires des séances de l'Académie des sciences
3790:
Comptes rendus hebdomadaires des séances de l'Académie des sciences
1497:{\displaystyle \langle F(x),y-x\rangle \geq 0\qquad \forall y\in K}
3743:
3719:
An
Introduction to Variational Inequalities and Their Applications
3418:
1620:
634:
Prove the uniqueness of the given solution: this step implies the
2983:{\displaystyle {\boldsymbol {\sigma }}=\left(\sigma _{ik}\right)}
162:
the name "variational inequality" for all the problems involving
62:
of variational inequalities was initially developed to deal with
3639:
3458:
159:
1154:{\displaystyle f^{\prime }(x^{\ast })(y-x^{\ast })\geq 0\quad }
3636:
Seminari dell'istituto
Nazionale di Alta Matematica 1962–1963
198:): full proofs of their results appeared later in the paper (
106:
The first problem involving a variational inequality was the
3663:, Studies in Mathematics and its Applications, vol. 8,
2393:
2022:
1907:
1101:
1013:
923:
833:
3939:. The paper containing Stampacchia's generalization of the
3916:"Formes bilineaires coercitives sur les ensembles convexes"
3866:
Nonlinear Partial Differential Equations with Applications
3454:
Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993
3298:
Extended Mathematical Programming for Equilibrium Problems
631:
of the problem, showing that there is at least a solution.
3405:"Problèmes unilatéraux en mécanique des milieux continus"
627:
Prove the existence of a solution: this step implies the
411:, the variational inequality problem is the problem of
3457:, Atti dei Convegni Lincei (in Italian), vol. 114,
3380:. An historical paper about the fruitful interaction of
3410:
Actes du Congrès international des mathématiciens, 1970
2078:{\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}})}
3303:
Mathematical programming with equilibrium constraints
3203:
3109:
3006:
2950:
2762:
2729:
2610:
2487:
2454:
2423:
2373:
2244:
2132:
2091:
2052:
1945:
1903:
1875:
1837:
1806:
1678:
1590:
1513:
1444:
1421:
1401:
1377:
1356:
1307:
1275:
1255:
1223:
1213:
The general finite-dimensional variational inequality
1167:
1095:
1059:
1007:
971:
917:
881:
827:
785:
761:
734:
696:
673:
544:
473:
444:
424:
395:
366:
341:
298:
276:
254:
229:
1617:
The variational inequality for the Signorini problem
660:, p. 283): consider the problem of finding the
3265:
3185:
3089:
2982:
2930:
2746:
2711:
2588:
2460:
2437:
2406:
2359:
2230:
2108:
2077:
2035:
1920:
1881:
1846:
1812:
1779:
1605:
1569:
1496:
1427:
1407:
1383:
1362:
1334:
1290:
1261:
1238:
1186:
1153:
1078:
1042:
993:
955:
903:
865:
813:
767:
747:
720:
679:
593:
527:
452:
430:
403:
381:
349:
327:
284:
262:
237:
2740:
2736:
2327:
2326:
2325:
2324:
2323:
775:where the minimum occurs. Three cases can occur:
3810:
656:This is a standard example problem, reported by
199:
195:
2747:{\displaystyle \partial A\!\setminus \!\Sigma }
1350:variational inequality problem associated with
3831:Communications on Pure and Applied Mathematics
3722:, Pure and Applied Mathematics, vol. 88,
3524:Facchinei, Francisco; Pang, Jong-Shi (2003),
3482:Facchinei, Francisco; Pang, Jong-Shi (2003),
3346:Bulletin of the American Mathematical Society
1662:describes the genesis of his solution to the
1335:{\displaystyle F\colon K\to \mathbb {R} ^{n}}
1043:{\displaystyle f^{\prime }(x^{\ast })\leq 0.}
956:{\displaystyle f^{\prime }(x^{\ast })\geq 0;}
8:
1526:
1514:
1472:
1445:
557:
545:
501:
474:
3786:"Inéquations variationnelles non coercives"
191:
147:
3528:, Springer Series in Operations Research,
3486:, Springer Series in Operations Research,
642:Find the solution or prove its regularity.
3809:. Announcements of the results of paper (
3358:
3250:
3232:
3212:
3204:
3202:
3174:
3161:
3149:
3131:
3116:
3108:
3044:
3026:
3011:
3005:
2967:
2951:
2949:
2909:
2894:
2884:
2872:
2857:
2847:
2832:
2811:
2799:
2779:
2771:
2763:
2761:
2728:
2696:
2687:
2672:
2663:
2648:
2639:
2619:
2611:
2609:
2573:
2564:
2549:
2540:
2525:
2516:
2496:
2488:
2486:
2453:
2424:
2422:
2398:
2392:
2391:
2382:
2374:
2372:
2349:
2348:
2342:
2332:
2308:
2293:
2292:
2286:
2276:
2266:
2251:
2243:
2220:
2219:
2211:
2199:
2187:
2175:
2165:
2147:
2139:
2131:
2098:
2090:
2067:
2059:
2051:
2027:
2021:
2020:
2011:
1993:
1985:
1968:
1960:
1952:
1944:
1912:
1906:
1905:
1902:
1874:
1836:
1805:
1764:
1755:
1740:
1731:
1716:
1707:
1687:
1679:
1677:
1597:
1593:
1592:
1589:
1563:
1562:
1553:
1549:
1548:
1538:
1534:
1533:
1512:
1443:
1420:
1400:
1376:
1355:
1326:
1322:
1321:
1306:
1282:
1278:
1277:
1274:
1254:
1230:
1226:
1225:
1222:
1166:
1135:
1113:
1100:
1094:
1064:
1058:
1025:
1012:
1006:
976:
970:
935:
922:
916:
886:
880:
866:{\displaystyle f^{\prime }(x^{\ast })=0;}
845:
832:
826:
796:
784:
760:
739:
733:
695:
672:
587:
586:
578:
569:
564:
543:
520:
472:
445:
443:
423:
396:
394:
382:{\displaystyle {\boldsymbol {E}}^{\ast }}
373:
368:
365:
342:
340:
319:
314:
305:
297:
277:
275:
255:
253:
230:
228:
3645:
1921:{\displaystyle {\mathcal {U}}_{\Sigma }}
1217:A formulation of the general problem in
135:
131:
126:): the first papers of the theory were (
3251:
3213:
3205:
3150:
3117:
2952:
2812:
2780:
2772:
2764:
2737:
2697:
2673:
2649:
2620:
2612:
2574:
2550:
2526:
2497:
2489:
2425:
2383:
2375:
2315:
2252:
2212:
2188:
2148:
2140:
2099:
2068:
2060:
2012:
1994:
1986:
1969:
1961:
1953:
1765:
1741:
1717:
1688:
1680:
1655:
579:
565:
521:
446:
397:
369:
343:
315:
306:
278:
256:
231:
127:
123:
3642:: Edizioni Cremonese, pp. 613–679
2438:{\displaystyle {\boldsymbol {x}}\in A}
657:
211:
179:
119:
118:in 1963, according to the references (
3444:problems with unilateral constraints.
1666:: the problem consist in finding the
1187:{\displaystyle \quad \forall y\in I.}
7:
3630:for the Signorini problem is proved.
2109:{\displaystyle F({\boldsymbol {v}})}
814:{\displaystyle a<x^{\ast }<b,}
3293:Differential variational inequality
47:for all possible values of a given
3396:is described in §5, pages 282–284.
3057:
3037:
3029:
2902:
2887:
2865:
2850:
2741:
2730:
2455:
2399:
2350:
2318:
2309:
2294:
2221:
2028:
2008:
1913:
1838:
1482:
1169:
511:
25:
1636:of the orange spherically shaped
453:{\displaystyle {\boldsymbol {K}}}
404:{\displaystyle {\boldsymbol {E}}}
350:{\displaystyle {\boldsymbol {K}}}
285:{\displaystyle {\boldsymbol {E}}}
263:{\displaystyle {\boldsymbol {K}}}
238:{\displaystyle {\boldsymbol {E}}}
194:), announcing them in the paper (
142:proved his generalization to the
3425:, pp. 71–78, archived from
3388:: the creation of the theory of
1606:{\displaystyle \mathbb {R} ^{n}}
1291:{\displaystyle \mathbb {R} ^{n}}
1239:{\displaystyle \mathbb {R} ^{n}}
186:. Also in 1965, Stampacchia and
3954:Panagiotopoulos, P.D. (2001) ,
3659:; Trémolières, Raymond (1981),
3360:10.1090/S0273-0979-1983-15185-6
3056:
2007:
1481:
1168:
1150:
510:
3988:Partial differential equations
3463:Accademia Nazionale dei Lincei
3255:
3247:
3217:
3209:
3154:
3146:
3121:
3113:
2816:
2808:
2784:
2776:
2701:
2693:
2677:
2669:
2653:
2645:
2624:
2616:
2578:
2570:
2554:
2546:
2530:
2522:
2501:
2493:
2256:
2248:
2216:
2208:
2192:
2184:
2152:
2136:
2103:
2095:
2072:
2056:
1998:
1982:
1973:
1949:
1769:
1761:
1745:
1737:
1721:
1713:
1692:
1684:
1559:
1457:
1451:
1317:
1201:: note that the solution is a
1141:
1122:
1119:
1106:
1079:{\displaystyle x^{\ast }\in I}
1031:
1018:
941:
928:
851:
838:
715:
703:
583:
486:
480:
310:
156:partial differential equations
1:
3644:. An English translation of (
2475:(or more generally a contact
1934:ambiguous boundary conditions
1205:, therefore this is a finite
190:extended earlier results of (
3811:Lions & Stampacchia 1967
994:{\displaystyle x^{\ast }=b,}
904:{\displaystyle x^{\ast }=a,}
200:Lions & Stampacchia 1967
196:Lions & Stampacchia 1965
3961:Encyclopedia of Mathematics
3622:. The first paper where an
2940:infinitesimal strain tensor
4009:
3956:"Variational inequalities"
3826:"Variational inequalities"
3313:Projected dynamical system
1847:{\displaystyle \partial A}
1654:In the historical survey (
1246:is the following: given a
3878:10.1007/978-3-0348-0513-1
1932:satisfying the system of
748:{\displaystyle x^{\ast }}
122:, pp. 282–284) and (
51:, belonging usually to a
3864:RoubĂÄŤek, Tomáš (2013),
3390:variational inequalities
3195:elastic potential energy
1896:admissible displacements
1864:and subject only to its
1209:variational inequality.
629:mathematical correctness
150:) in order to study the
66:problems, precisely the
18:Variational inequalities
2461:{\displaystyle \Sigma }
666:differentiable function
3993:Calculus of variations
3844:10.1002/cpa.3160200302
3288:Complementarity theory
3267:
3187:
3091:
2984:
2932:
2748:
2713:
2590:
2462:
2439:
2408:
2361:
2232:
2110:
2079:
2037:
1922:
1883:
1848:
1814:
1781:
1651:
1607:
1571:
1498:
1429:
1409:
1385:
1364:
1336:
1292:
1263:
1240:
1188:
1155:
1080:
1044:
995:
957:
905:
867:
815:
769:
749:
722:
681:
595:
529:
454:
432:
405:
383:
351:
329:
286:
264:
239:
114:in 1959 and solved by
78:. Therefore, it has a
33:variational inequality
3679:, pp. xxix+776,
3386:mathematical analysis
3334:Historical references
3268:
3188:
3092:
2985:
2933:
2749:
2714:
2591:
2463:
2440:
2409:
2367:,
2362:
2238:,
2233:
2120:, written using the
2111:
2080:
2038:
1923:
1884:
1849:
1815:
1782:
1624:
1608:
1572:
1499:
1430:
1410:
1386:
1370:consist of finding a
1365:
1337:
1293:
1264:
1241:
1189:
1156:
1081:
1045:
996:
958:
906:
868:
816:
770:
750:
723:
682:
596:
530:
455:
433:
406:
384:
352:
330:
287:
265:
240:
3818:Lions, Jacques-Louis
3778:Lions, Jacques-Louis
3657:Lions, Jacques-Louis
3201:
3107:
3004:
2992:Cauchy stress tensor
2948:
2760:
2727:
2608:
2602:applied to the body,
2485:
2452:
2421:
2371:
2242:
2130:
2089:
2050:
1943:
1930:displacement vectors
1901:
1873:
1835:
1804:
1676:
1640:resting on the blue
1588:
1511:
1442:
1419:
1399:
1375:
1354:
1305:
1273:
1253:
1221:
1165:
1093:
1057:
1005:
969:
915:
879:
825:
783:
759:
732:
694:
671:
636:physical correctness
542:
471:
442:
422:
393:
364:
339:
296:
274:
252:
227:
3941:Lax–Milgram theorem
3872:, pp. xx+476,
3710:Kinderlehrer, David
3442:continuum mechanics
1668:elastic equilibrium
1629:: what will be the
292:, and a functional
188:Jacques-Louis Lions
144:Lax–Milgram theorem
3912:Stampacchia, Guido
3822:Stampacchia, Guido
3782:Stampacchia, Guido
3714:Stampacchia, Guido
3628:uniqueness theorem
3609:, 8 (in Italian),
3577:, 8 (in Italian),
3323:Unilateral contact
3263:
3183:
3087:
2980:
2928:
2744:
2709:
2586:
2458:
2435:
2404:
2357:
2228:
2116:are the following
2106:
2075:
2033:
1918:
1879:
1844:
1810:
1777:
1652:
1603:
1567:
1494:
1425:
1405:
1381:
1360:
1332:
1288:
1259:
1236:
1184:
1151:
1076:
1040:
991:
953:
901:
863:
811:
765:
745:
721:{\displaystyle I=}
718:
677:
591:
525:
450:
428:
401:
379:
347:
325:
282:
260:
235:
152:regularity problem
80:variational origin
43:, which has to be
3887:978-3-0348-0512-4
3870:Birkhäuser Verlag
3653:Glowinski, Roland
3382:elasticity theory
3318:Signorini problem
3275:elasticity tensor
3054:
2916:
2879:
2840:
2122:Einstein notation
1882:{\displaystyle u}
1813:{\displaystyle A}
1664:Signorini problem
1627:Signorini problem
1428:{\displaystyle K}
1408:{\displaystyle x}
1384:{\displaystyle n}
1363:{\displaystyle K}
1262:{\displaystyle K}
768:{\displaystyle I}
680:{\displaystyle f}
431:{\displaystyle x}
172:graduate students
140:Guido Stampacchia
112:Antonio Signorini
108:Signorini problem
68:Signorini problem
16:(Redirected from
4000:
3968:
3934:
3906:
3859:
3854:, archived from
3804:
3772:
3705:
3643:
3621:
3603:Fichera, Gaetano
3597:
3567:Fichera, Gaetano
3562:
3520:
3476:Scientific works
3466:
3465:, pp. 47–53
3449:Fichera, Gaetano
3439:
3438:
3437:
3431:
3423:Gauthier-Villars
3379:
3362:
3308:Obstacle problem
3272:
3270:
3269:
3264:
3262:
3258:
3254:
3246:
3245:
3216:
3208:
3192:
3190:
3189:
3184:
3182:
3181:
3169:
3168:
3153:
3145:
3144:
3120:
3096:
3094:
3093:
3088:
3055:
3053:
3052:
3051:
3035:
3027:
3019:
3018:
2989:
2987:
2986:
2981:
2979:
2975:
2974:
2955:
2937:
2935:
2934:
2929:
2927:
2923:
2922:
2918:
2917:
2915:
2914:
2913:
2900:
2899:
2898:
2885:
2880:
2878:
2877:
2876:
2863:
2862:
2861:
2848:
2841:
2833:
2823:
2819:
2815:
2807:
2806:
2783:
2775:
2767:
2753:
2751:
2750:
2745:
2718:
2716:
2715:
2710:
2708:
2704:
2700:
2692:
2691:
2676:
2668:
2667:
2652:
2644:
2643:
2623:
2615:
2595:
2593:
2592:
2587:
2585:
2581:
2577:
2569:
2568:
2553:
2545:
2544:
2529:
2521:
2520:
2500:
2492:
2467:
2465:
2464:
2459:
2444:
2442:
2441:
2436:
2428:
2413:
2411:
2410:
2405:
2403:
2402:
2397:
2396:
2386:
2378:
2366:
2364:
2363:
2358:
2353:
2347:
2346:
2337:
2336:
2322:
2321:
2297:
2291:
2290:
2281:
2280:
2271:
2270:
2255:
2237:
2235:
2234:
2229:
2224:
2215:
2207:
2206:
2191:
2183:
2182:
2170:
2169:
2151:
2143:
2115:
2113:
2112:
2107:
2102:
2084:
2082:
2081:
2076:
2071:
2063:
2042:
2040:
2039:
2034:
2032:
2031:
2026:
2025:
2015:
1997:
1989:
1972:
1964:
1956:
1928:i.e. the set of
1927:
1925:
1924:
1919:
1917:
1916:
1911:
1910:
1888:
1886:
1885:
1880:
1868:. The solution
1853:
1851:
1850:
1845:
1819:
1817:
1816:
1811:
1786:
1784:
1783:
1778:
1776:
1772:
1768:
1760:
1759:
1744:
1736:
1735:
1720:
1712:
1711:
1691:
1683:
1612:
1610:
1609:
1604:
1602:
1601:
1596:
1577:is the standard
1576:
1574:
1573:
1568:
1566:
1558:
1557:
1552:
1543:
1542:
1537:
1503:
1501:
1500:
1495:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1390:
1388:
1387:
1382:
1369:
1367:
1366:
1361:
1341:
1339:
1338:
1333:
1331:
1330:
1325:
1297:
1295:
1294:
1289:
1287:
1286:
1281:
1268:
1266:
1265:
1260:
1245:
1243:
1242:
1237:
1235:
1234:
1229:
1193:
1191:
1190:
1185:
1160:
1158:
1157:
1152:
1140:
1139:
1118:
1117:
1105:
1104:
1085:
1083:
1082:
1077:
1069:
1068:
1049:
1047:
1046:
1041:
1030:
1029:
1017:
1016:
1000:
998:
997:
992:
981:
980:
962:
960:
959:
954:
940:
939:
927:
926:
910:
908:
907:
902:
891:
890:
872:
870:
869:
864:
850:
849:
837:
836:
820:
818:
817:
812:
801:
800:
774:
772:
771:
766:
754:
752:
751:
746:
744:
743:
727:
725:
724:
719:
686:
684:
683:
678:
600:
598:
597:
592:
590:
582:
574:
573:
568:
534:
532:
531:
526:
524:
459:
457:
456:
451:
449:
437:
435:
434:
429:
410:
408:
407:
402:
400:
388:
386:
385:
380:
378:
377:
372:
356:
354:
353:
348:
346:
334:
332:
331:
326:
324:
323:
318:
309:
291:
289:
288:
283:
281:
269:
267:
266:
261:
259:
244:
242:
241:
236:
234:
219:
192:Stampacchia 1964
148:Stampacchia 1964
76:potential energy
74:of the involved
21:
4008:
4007:
4003:
4002:
4001:
3999:
3998:
3997:
3978:
3977:
3953:
3950:
3935:, available at
3910:
3888:
3863:
3816:
3805:, available at
3776:
3762:
3708:
3687:
3651:
3633:
3601:
3565:
3552:
3542:Springer-Verlag
3523:
3510:
3500:Springer-Verlag
3481:
3478:
3447:
3435:
3433:
3429:
3415:ICM Proceedings
3401:Duvaut, Georges
3399:
3394:Gaetano Fichera
3339:
3336:
3331:
3284:
3228:
3227:
3223:
3199:
3198:
3170:
3157:
3127:
3105:
3104:
3040:
3036:
3028:
3007:
3002:
3001:
2963:
2959:
2946:
2945:
2905:
2901:
2890:
2886:
2868:
2864:
2853:
2849:
2846:
2842:
2831:
2827:
2795:
2794:
2790:
2758:
2757:
2725:
2724:
2683:
2659:
2635:
2634:
2630:
2606:
2605:
2560:
2536:
2512:
2511:
2507:
2483:
2482:
2450:
2449:
2419:
2418:
2417:where, for all
2390:
2369:
2368:
2338:
2328:
2304:
2282:
2272:
2262:
2240:
2239:
2195:
2171:
2161:
2128:
2127:
2087:
2086:
2048:
2047:
2019:
1941:
1940:
1936:if and only if
1904:
1899:
1898:
1871:
1870:
1854:, resting on a
1833:
1832:
1825:euclidean space
1802:
1801:
1797:that lies in a
1792:non-homogeneous
1751:
1727:
1703:
1702:
1698:
1674:
1673:
1660:Gaetano Fichera
1619:
1591:
1586:
1585:
1547:
1532:
1509:
1508:
1440:
1439:
1417:
1416:
1397:
1396:
1373:
1372:
1352:
1351:
1320:
1303:
1302:
1276:
1271:
1270:
1251:
1250:
1224:
1219:
1218:
1215:
1163:
1162:
1131:
1109:
1096:
1091:
1090:
1060:
1055:
1054:
1021:
1008:
1003:
1002:
972:
967:
966:
931:
918:
913:
912:
882:
877:
876:
841:
828:
823:
822:
792:
781:
780:
757:
756:
735:
730:
729:
692:
691:
689:closed interval
669:
668:
654:
649:
603:duality pairing
563:
540:
539:
469:
468:
440:
439:
420:
419:
391:
390:
367:
362:
361:
337:
336:
313:
294:
293:
272:
271:
250:
249:
225:
224:
217:
208:
170:encouraged his
116:Gaetano Fichera
104:
72:first variation
23:
22:
15:
12:
11:
5:
4006:
4004:
3996:
3995:
3990:
3980:
3979:
3976:
3975:
3969:
3949:
3948:External links
3946:
3945:
3944:
3908:
3886:
3861:
3838:(3): 493–519,
3814:
3774:
3760:
3752:Academic Press
3706:
3685:
3649:
3631:
3599:
3581:(2): 138–142,
3563:
3550:
3521:
3508:
3477:
3474:
3473:
3472:
3445:
3397:
3353:(3): 267–291,
3341:Antman, Stuart
3335:
3332:
3330:
3327:
3326:
3325:
3320:
3315:
3310:
3305:
3300:
3295:
3290:
3283:
3280:
3279:
3278:
3261:
3257:
3253:
3249:
3244:
3241:
3238:
3235:
3231:
3226:
3222:
3219:
3215:
3211:
3207:
3180:
3177:
3173:
3167:
3164:
3160:
3156:
3152:
3148:
3143:
3140:
3137:
3134:
3130:
3126:
3123:
3119:
3115:
3112:
3100:
3099:
3098:
3097:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3050:
3047:
3043:
3039:
3034:
3031:
3025:
3022:
3017:
3014:
3010:
2996:
2995:
2978:
2973:
2970:
2966:
2962:
2958:
2954:
2943:
2926:
2921:
2912:
2908:
2904:
2897:
2893:
2889:
2883:
2875:
2871:
2867:
2860:
2856:
2852:
2845:
2839:
2836:
2830:
2826:
2822:
2818:
2814:
2810:
2805:
2802:
2798:
2793:
2789:
2786:
2782:
2778:
2774:
2770:
2766:
2755:
2743:
2739:
2735:
2732:
2707:
2703:
2699:
2695:
2690:
2686:
2682:
2679:
2675:
2671:
2666:
2662:
2658:
2655:
2651:
2647:
2642:
2638:
2633:
2629:
2626:
2622:
2618:
2614:
2603:
2584:
2580:
2576:
2572:
2567:
2563:
2559:
2556:
2552:
2548:
2543:
2539:
2535:
2532:
2528:
2524:
2519:
2515:
2510:
2506:
2503:
2499:
2495:
2491:
2480:
2457:
2434:
2431:
2427:
2415:
2414:
2401:
2395:
2389:
2385:
2381:
2377:
2356:
2352:
2345:
2341:
2335:
2331:
2320:
2317:
2314:
2311:
2307:
2303:
2300:
2296:
2289:
2285:
2279:
2275:
2269:
2265:
2261:
2258:
2254:
2250:
2247:
2227:
2223:
2218:
2214:
2210:
2205:
2202:
2198:
2194:
2190:
2186:
2181:
2178:
2174:
2168:
2164:
2160:
2157:
2154:
2150:
2146:
2142:
2138:
2135:
2105:
2101:
2097:
2094:
2074:
2070:
2066:
2062:
2058:
2055:
2044:
2043:
2030:
2024:
2018:
2014:
2010:
2006:
2003:
2000:
1996:
1992:
1988:
1984:
1981:
1978:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1948:
1915:
1909:
1878:
1843:
1840:
1809:
1775:
1771:
1767:
1763:
1758:
1754:
1750:
1747:
1743:
1739:
1734:
1730:
1726:
1723:
1719:
1715:
1710:
1706:
1701:
1697:
1694:
1690:
1686:
1682:
1625:The classical
1618:
1615:
1600:
1595:
1565:
1561:
1556:
1551:
1546:
1541:
1536:
1531:
1528:
1525:
1522:
1519:
1516:
1505:
1504:
1493:
1490:
1487:
1484:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1424:
1415:belonging to
1404:
1380:
1359:
1329:
1324:
1319:
1316:
1313:
1310:
1285:
1280:
1258:
1233:
1228:
1214:
1211:
1195:
1194:
1183:
1180:
1177:
1174:
1171:
1149:
1146:
1143:
1138:
1134:
1130:
1127:
1124:
1121:
1116:
1112:
1108:
1103:
1099:
1075:
1072:
1067:
1063:
1051:
1050:
1039:
1036:
1033:
1028:
1024:
1020:
1015:
1011:
990:
987:
984:
979:
975:
963:
952:
949:
946:
943:
938:
934:
930:
925:
921:
900:
897:
894:
889:
885:
873:
862:
859:
856:
853:
848:
844:
840:
835:
831:
810:
807:
804:
799:
795:
791:
788:
764:
755:be a point in
742:
738:
717:
714:
711:
708:
705:
702:
699:
676:
653:
650:
648:
645:
644:
643:
640:
632:
589:
585:
581:
577:
572:
567:
562:
559:
556:
553:
550:
547:
536:
535:
523:
519:
516:
513:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
460:the following
448:
427:
399:
376:
371:
345:
322:
317:
312:
308:
304:
301:
280:
258:
233:
207:
204:
168:Georges Duvaut
166:of this kind.
103:
100:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4005:
3994:
3991:
3989:
3986:
3985:
3983:
3973:
3970:
3967:
3963:
3962:
3957:
3952:
3951:
3947:
3942:
3938:
3933:
3929:
3926:: 4413–4416,
3925:
3921:
3917:
3913:
3909:
3905:
3901:
3897:
3893:
3889:
3883:
3879:
3875:
3871:
3867:
3862:
3858:on 2013-01-05
3857:
3853:
3849:
3845:
3841:
3837:
3833:
3832:
3827:
3823:
3819:
3815:
3812:
3808:
3803:
3799:
3795:
3791:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3761:0-89871-466-4
3757:
3753:
3749:
3745:
3741:
3737:
3733:
3729:
3725:
3721:
3720:
3715:
3711:
3707:
3704:
3700:
3696:
3692:
3688:
3686:0-444-86199-8
3682:
3678:
3677:North-Holland
3674:
3670:
3666:
3662:
3658:
3654:
3650:
3647:
3646:Fichera 1964a
3641:
3637:
3632:
3629:
3625:
3620:
3616:
3613:(2): 91–140,
3612:
3608:
3604:
3600:
3596:
3592:
3588:
3584:
3580:
3576:
3572:
3568:
3564:
3561:
3557:
3553:
3551:0-387-95581-X
3547:
3543:
3539:
3535:
3531:
3527:
3522:
3519:
3515:
3511:
3509:0-387-95580-1
3505:
3501:
3497:
3493:
3489:
3485:
3480:
3479:
3475:
3470:
3464:
3460:
3456:
3455:
3450:
3446:
3443:
3432:on 2015-07-25
3428:
3424:
3420:
3416:
3412:
3411:
3406:
3402:
3398:
3395:
3391:
3387:
3383:
3378:
3374:
3370:
3366:
3361:
3356:
3352:
3348:
3347:
3342:
3338:
3337:
3333:
3328:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3289:
3286:
3285:
3281:
3276:
3259:
3242:
3239:
3236:
3233:
3229:
3224:
3220:
3196:
3178:
3175:
3171:
3165:
3162:
3158:
3141:
3138:
3135:
3132:
3128:
3124:
3110:
3102:
3101:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3048:
3045:
3041:
3032:
3023:
3020:
3015:
3012:
3008:
3000:
2999:
2998:
2997:
2993:
2976:
2971:
2968:
2964:
2960:
2956:
2944:
2941:
2924:
2919:
2910:
2906:
2895:
2891:
2881:
2873:
2869:
2858:
2854:
2843:
2837:
2834:
2828:
2824:
2820:
2803:
2800:
2796:
2791:
2787:
2768:
2756:
2733:
2722:
2721:surface force
2705:
2688:
2684:
2680:
2664:
2660:
2656:
2640:
2636:
2631:
2627:
2604:
2601:
2600:
2582:
2565:
2561:
2557:
2541:
2537:
2533:
2517:
2513:
2508:
2504:
2481:
2478:
2474:
2471:
2448:
2447:
2446:
2432:
2429:
2387:
2379:
2354:
2343:
2339:
2333:
2329:
2312:
2305:
2301:
2298:
2287:
2283:
2277:
2273:
2267:
2263:
2259:
2245:
2225:
2203:
2200:
2196:
2179:
2176:
2172:
2166:
2162:
2158:
2155:
2144:
2133:
2126:
2125:
2124:
2123:
2119:
2092:
2064:
2053:
2016:
2004:
2001:
1990:
1979:
1976:
1965:
1957:
1946:
1939:
1938:
1937:
1935:
1931:
1897:
1893:
1889:
1876:
1867:
1863:
1860:
1857:
1841:
1830:
1826:
1823:
1820:of the three-
1807:
1800:
1796:
1793:
1790:
1773:
1756:
1752:
1748:
1732:
1728:
1724:
1708:
1704:
1699:
1695:
1672:
1671:configuration
1669:
1665:
1661:
1657:
1649:
1646:
1643:
1639:
1635:
1634:configuration
1632:
1628:
1623:
1616:
1614:
1598:
1584:
1580:
1579:inner product
1554:
1544:
1539:
1529:
1523:
1520:
1517:
1491:
1488:
1485:
1478:
1475:
1469:
1466:
1463:
1460:
1454:
1448:
1438:
1437:
1436:
1422:
1402:
1395:
1392:
1378:
1357:
1349:
1345:
1327:
1314:
1311:
1308:
1301:
1283:
1256:
1249:
1231:
1212:
1210:
1208:
1204:
1200:
1181:
1178:
1175:
1172:
1147:
1144:
1136:
1132:
1128:
1125:
1114:
1110:
1097:
1089:
1088:
1087:
1073:
1070:
1065:
1061:
1037:
1034:
1026:
1022:
1009:
988:
985:
982:
977:
973:
964:
950:
947:
944:
936:
932:
919:
898:
895:
892:
887:
883:
874:
860:
857:
854:
846:
842:
829:
808:
805:
802:
797:
793:
789:
786:
778:
777:
776:
762:
740:
736:
712:
709:
706:
700:
697:
690:
674:
667:
663:
662:minimal value
659:
651:
646:
641:
637:
633:
630:
626:
625:
624:
622:
619:
615:
611:
606:
604:
575:
570:
560:
554:
551:
548:
517:
514:
507:
504:
498:
495:
492:
489:
483:
477:
467:
466:
465:
463:
438:belonging to
425:
418:
414:
389:of the space
374:
360:
320:
302:
299:
248:
223:
218:Definition 1.
215:
213:
205:
203:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
157:
153:
149:
145:
141:
138:). Later on,
137:
136:Fichera 1964b
133:
132:Fichera 1964a
129:
125:
121:
117:
113:
109:
101:
99:
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
58:
54:
50:
46:
42:
38:
34:
30:
19:
3959:
3923:
3919:
3865:
3856:the original
3835:
3829:
3793:
3789:
3718:
3660:
3635:
3610:
3606:
3578:
3574:
3525:
3483:
3468:
3453:
3434:, retrieved
3427:the original
3409:
3350:
3344:
2994:, defined as
2597:
2416:
2045:
1895:
1869:
1859:frictionless
1795:elastic body
1656:Fichera 1995
1653:
1645:frictionless
1638:elastic body
1583:vector space
1506:
1391:-dimensional
1216:
1196:
1052:
658:Antman (1983
655:
635:
628:
621:Banach space
607:
537:
222:Banach space
216:
212:Antman (1983
209:
164:inequalities
128:Fichera 1963
124:Fichera 1995
105:
92:optimization
57:mathematical
39:involving a
32:
26:
2723:applied to
2118:functionals
1866:mass forces
1822:dimensional
1789:anisotropic
1631:equilibrium
1348:dimensional
1207:dimensional
1203:real number
618:dimensional
180:Antman 1983
120:Antman 1983
110:, posed by
96:game theory
64:equilibrium
29:mathematics
3982:Categories
3932:0124.06401
3904:1270.35005
3852:0152.34601
3802:0136.11906
3770:0457.35001
3703:0463.65046
3619:0146.21204
3595:0128.18305
3560:1062.90001
3534:Heidelberg
3518:1062.90001
3492:Heidelberg
3436:2015-07-25
3377:0533.73001
3329:References
2599:body force
1435:such that
1199:inequality
1086:such that
462:inequality
359:dual space
210:Following
206:Definition
53:convex set
41:functional
37:inequality
3966:EMS Press
3796:: 25–27,
3736:San Diego
3665:Amsterdam
3624:existence
3172:ε
3159:ε
3118:ε
3058:∀
3042:ε
3038:∂
3030:∂
3024:−
3009:σ
2965:σ
2953:σ
2903:∂
2888:∂
2866:∂
2851:∂
2797:ε
2773:ε
2765:ε
2742:Σ
2738:∖
2731:∂
2456:Σ
2430:∈
2400:Σ
2388:∈
2355:σ
2319:Σ
2316:∖
2310:∂
2306:∫
2264:∫
2197:ε
2173:σ
2163:∫
2159:−
2029:Σ
2017:∈
2009:∀
2002:≥
1991:−
1977:−
1966:−
1914:Σ
1839:∂
1560:→
1545:×
1530::
1527:⟩
1524:⋅
1518:⋅
1515:⟨
1489:∈
1483:∀
1476:≥
1473:⟩
1467:−
1446:⟨
1318:→
1312::
1176:∈
1170:∀
1145:≥
1137:∗
1129:−
1115:∗
1102:′
1071:∈
1066:∗
1035:≤
1027:∗
1014:′
978:∗
945:≥
937:∗
924:′
888:∗
847:∗
834:′
798:∗
741:∗
584:→
576:×
571:∗
561::
558:⟩
555:⋅
549:⋅
546:⟨
518:∈
512:∀
505:≥
502:⟩
496:−
475:⟨
375:∗
321:∗
311:→
303::
84:economics
3914:(1964),
3824:(1967),
3784:(1965),
3732:New York
3716:(1980),
3669:New York
3569:(1963),
3538:New York
3496:New York
3403:(1971),
3282:See also
1829:boundary
647:Examples
614:infinite
417:variable
415:for the
220:Given a
49:variable
3937:Gallica
3896:3014456
3807:Gallica
3748:Toronto
3695:0635927
3587:0176661
3369:0714990
3273:is the
3193:is the
2990:is the
2938:is the
2719:is the
2596:is the
2473:surface
2470:contact
2468:is the
1862:surface
1581:on the
1342:, the
1300:mapping
687:over a
639:origin.
601:is the
413:solving
357:to the
130:) and (
102:History
88:finance
3930:
3902:
3894:
3884:
3850:
3800:
3768:
3758:
3740:Sydney
3728:London
3724:Boston
3701:
3693:
3683:
3673:Oxford
3617:
3593:
3585:
3558:
3548:
3530:Berlin
3516:
3506:
3488:Berlin
3375:
3367:
3103:where
2046:where
1827:whose
1799:subset
1787:of an
1507:where
1394:vector
1344:finite
1298:and a
1248:subset
728:. Let
664:of a
610:finite
538:where
247:subset
184:France
176:Brixen
160:coined
60:theory
55:. The
45:solved
35:is an
3744:Tokyo
3430:(PDF)
3419:Paris
1856:rigid
1648:plane
1642:rigid
1001:then
911:then
821:then
612:– or
335:from
3882:ISBN
3756:ISBN
3681:ISBN
3640:Rome
3626:and
3546:ISBN
3504:ISBN
3459:Roma
3384:and
3197:and
2085:and
1269:of
1161:for
803:<
790:<
245:, a
158:and
154:for
146:in (
134:), (
94:and
31:, a
3928:Zbl
3924:258
3900:Zbl
3874:doi
3848:Zbl
3840:doi
3798:Zbl
3794:261
3766:Zbl
3699:Zbl
3615:Zbl
3591:Zbl
3556:Zbl
3514:Zbl
3392:by
3373:Zbl
3355:doi
2477:set
2445:,
1894:of
1892:set
1831:is
1658:),
965:if
875:if
779:if
270:of
202:).
27:In
3984::
3964:,
3958:,
3922:,
3918:,
3898:,
3892:MR
3890:,
3880:,
3846:,
3836:20
3834:,
3828:,
3820:;
3813:).
3792:,
3788:,
3780:;
3764:,
3754:,
3750::
3712:;
3697:,
3691:MR
3689:,
3675::
3655:;
3648:).
3638:,
3589:,
3583:MR
3579:34
3554:,
3544:,
3540::
3512:,
3502:,
3498::
3467:.
3461::
3421::
3413:,
3407:,
3371:,
3365:MR
3363:,
3349:,
2479:),
1613:.
1038:0.
605:.
464::
98:.
90:,
86:,
3974:,
3943:.
3907:.
3876::
3842::
3773:.
3746:–
3742:–
3738:–
3734:–
3730:–
3726:–
3671:–
3667:–
3611:7
3536:–
3532:–
3494:–
3490:–
3357::
3351:9
3277:.
3260:)
3256:)
3252:x
3248:(
3243:h
3240:j
3237:k
3234:i
3230:a
3225:(
3221:=
3218:)
3214:x
3210:(
3206:a
3179:h
3176:j
3166:k
3163:i
3155:)
3151:x
3147:(
3142:h
3139:j
3136:k
3133:i
3129:a
3125:=
3122:)
3114:(
3111:W
3085:3
3082:,
3079:2
3076:,
3073:1
3070:=
3067:k
3064:,
3061:i
3049:k
3046:i
3033:W
3021:=
3016:k
3013:i
2977:)
2972:k
2969:i
2961:(
2957:=
2942:,
2925:)
2920:)
2911:i
2907:x
2896:k
2892:u
2882:+
2874:k
2870:x
2859:i
2855:u
2844:(
2838:2
2835:1
2829:(
2825:=
2821:)
2817:)
2813:u
2809:(
2804:k
2801:i
2792:(
2788:=
2785:)
2781:u
2777:(
2769:=
2754:,
2734:A
2706:)
2702:)
2698:x
2694:(
2689:3
2685:g
2681:,
2678:)
2674:x
2670:(
2665:2
2661:g
2657:,
2654:)
2650:x
2646:(
2641:1
2637:g
2632:(
2628:=
2625:)
2621:x
2617:(
2613:g
2583:)
2579:)
2575:x
2571:(
2566:3
2562:f
2558:,
2555:)
2551:x
2547:(
2542:2
2538:f
2534:,
2531:)
2527:x
2523:(
2518:1
2514:f
2509:(
2505:=
2502:)
2498:x
2494:(
2490:f
2433:A
2426:x
2394:U
2384:v
2380:,
2376:u
2351:d
2344:i
2340:g
2334:i
2330:v
2313:A
2302:+
2299:x
2295:d
2288:i
2284:f
2278:i
2274:v
2268:A
2260:=
2257:)
2253:v
2249:(
2246:F
2226:x
2222:d
2217:)
2213:v
2209:(
2204:k
2201:i
2193:)
2189:u
2185:(
2180:k
2177:i
2167:A
2156:=
2153:)
2149:v
2145:,
2141:u
2137:(
2134:B
2104:)
2100:v
2096:(
2093:F
2073:)
2069:v
2065:,
2061:u
2057:(
2054:B
2023:U
2013:v
2005:0
1999:)
1995:u
1987:v
1983:(
1980:F
1974:)
1970:u
1962:v
1958:,
1954:u
1950:(
1947:B
1908:U
1877:u
1842:A
1808:A
1774:)
1770:)
1766:x
1762:(
1757:3
1753:u
1749:,
1746:)
1742:x
1738:(
1733:2
1729:u
1725:,
1722:)
1718:x
1714:(
1709:1
1705:u
1700:(
1696:=
1693:)
1689:x
1685:(
1681:u
1650:?
1599:n
1594:R
1564:R
1555:n
1550:R
1540:n
1535:R
1521:,
1492:K
1486:y
1479:0
1470:x
1464:y
1461:,
1458:)
1455:x
1452:(
1449:F
1423:K
1403:x
1379:n
1358:K
1346:-
1328:n
1323:R
1315:K
1309:F
1284:n
1279:R
1257:K
1232:n
1227:R
1182:.
1179:I
1173:y
1148:0
1142:)
1133:x
1126:y
1123:(
1120:)
1111:x
1107:(
1098:f
1074:I
1062:x
1032:)
1023:x
1019:(
1010:f
989:,
986:b
983:=
974:x
951:;
948:0
942:)
933:x
929:(
920:f
899:,
896:a
893:=
884:x
861:;
858:0
855:=
852:)
843:x
839:(
830:f
809:,
806:b
794:x
787:a
763:I
737:x
716:]
713:b
710:,
707:a
704:[
701:=
698:I
675:f
616:-
588:R
580:E
566:E
552:,
522:K
515:y
508:0
499:x
493:y
490:,
487:)
484:x
481:(
478:F
447:K
426:x
398:E
370:E
344:K
316:E
307:K
300:F
279:E
257:K
232:E
20:)
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