881:
556:
724:
696:
1219:
286:
1283:
1022:
1109:
437:
395:
172:
876:{\displaystyle \int _{\Omega }\nabla u_{f}(x)\cdot \nabla v(x)\,\mathrm {d} x=\int _{\Omega }f(x)v(x)\,\mathrm {d} x{\mbox{ for all }}v\in H_{0}^{1}(\Omega ).}
291:
The achievement of Lax and
Milgram in their 1954 result was to specify sufficient conditions for this weak formulation to have a unique solution that depends
608:
891:
In 1971, Babuška provided the following generalization of Lax and
Milgram's earlier result, which begins by dispensing with the requirement that
1145:
219:
1469:
1387:
1230:
945:
1294:
1041:
551:{\displaystyle {\begin{cases}-\Delta u(x)=f(x),&x\in \Omega ;\\u(x)=0,&x\in \partial \Omega ;\end{cases}}}
71:, one does not attempt to solve a given partial differential equation directly, but by using the structure of the
1438:
68:
1464:
1448:
333:
112:
1443:
1403:
446:
145:
25:
37:
181:, this equation is only required to hold when "tested" against all other possible elements of
1316:
97:
1411:
1355:
1419:
1371:
8:
292:
64:
1392:
1359:
1363:
1343:
324:
1415:
1399:
1367:
1333:
1325:
416:
178:
691:{\displaystyle B(u,v)=\int _{\Omega }\nabla u(x)\cdot \nabla v(x)\,\mathrm {d} x.}
1407:
1351:
1383:
52:
1338:
1307:
44:
1458:
1347:
599:
312:
76:
41:
33:
29:
86:
72:
420:
83:
17:
1423:
1329:
1214:{\displaystyle B(u_{f},v)=\langle f,v\rangle {\mbox{ for all }}v\in V.}
1379:
48:
577:
424:
281:{\displaystyle B(u,v)=\langle f,v\rangle {\mbox{ for all }}v\in V.}
185:. This "testing" is accomplished by means of a bilinear function
1311:
1224:
Moreover, the solution depends continuously on the given data:
1394:
Contributions to the theory of partial differential equations
32:
can be "inverted" to show the existence and uniqueness of a
701:
Hence, the weak formulation of the
Poisson equation, given
544:
923:
be a continuous bilinear functional. Suppose also that
1193:
834:
260:
1233:
1148:
1044:
948:
727:
611:
440:
336:
222:
148:
1391:
1278:{\displaystyle \|u_{f}\|\leq {\frac {1}{c}}\|f\|.}
1277:
1213:
1103:
1017:{\displaystyle \sup _{\|v\|=1}|B(u,v)|\geq c\|u\|}
1016:
875:
690:
550:
389:
280:
166:
1456:
1046:
950:
1398:, Annals of Mathematics Studies, vol. 33,
111:is the space of possible solutions; given some
201:which encodes the differential operator Λ; a
1306:
1269:
1263:
1247:
1234:
1189:
1177:
1104:{\displaystyle \sup _{\|u\|=1}|B(u,v)|>0}
1056:
1050:
1011:
1005:
960:
954:
378:
371:
256:
244:
886:
1378:
1337:
824:
778:
676:
1312:"Error-bounds for finite element method"
565:could be taken to be the Sobolev space
390:{\displaystyle |B(u,u)|\geq c\|u\|^{2}}
28:, which gives conditions under which a
1457:
927:is weakly coercive: for some constant
1436:
576:(Ω); the former is a subspace of the
107:respectively. In many applications,
415:For example, in the solution of the
907:be two real Hilbert spaces and let
13:
864:
826:
795:
780:
763:
738:
733:
678:
661:
643:
638:
535:
532:
493:
452:
149:
24:is a generalization of the famous
14:
1481:
1430:
1122:, there exists a unique solution
40:. The result is named after the
1470:Partial differential equations
1171:
1152:
1091:
1087:
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995:
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821:
815:
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627:
615:
594:associated to −Δ is the
512:
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479:
473:
464:
458:
361:
357:
345:
338:
238:
226:
75:of possible solutions, e.g. a
69:partial differential equations
1:
1439:"Babuška–Lax–Milgram theorem"
1300:
131:, the objective is to find a
113:partial differential operator
58:
205:to the problem is to find a
167:{\displaystyle \Lambda u=f.}
82:. Abstractly, consider two
7:
1444:Encyclopedia of Mathematics
1288:
1027:and, for all 0 ≠
22:Babuška–Lax–Milgram theorem
10:
1486:
1404:Princeton University Press
931: > 0 and all
404: > 0 and all
1295:Lions–Lax–Milgram theorem
295:upon the specified datum
67:approach to the study of
899:be the same space. Let
887:Statement of the theorem
319:is continuous, and that
123:and a specified element
590:(Ω); the bilinear form
1279:
1215:
1105:
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877:
692:
552:
391:
282:
168:
98:continuous dual spaces
38:boundary value problem
1437:Roşca, Ioan (2001) ,
1388:"Parabolic equations"
1317:Numerische Mathematik
1280:
1216:
1106:
1019:
878:
693:
553:
427:domain Ω ⊂
392:
283:
169:
1465:Theorems in analysis
1406:, pp. 167–190,
1231:
1146:
1139:to the weak problem
1042:
946:
725:
609:
602:of the derivatives:
438:
334:
220:
146:
1195: for all
860:
836: for all
303:: it suffices that
262: for all
65:functional-analytic
26:Lax–Milgram theorem
1384:Milgram, Arthur N.
1339:10338.dmlcz/103498
1330:10.1007/BF02165003
1275:
1211:
1197:
1101:
1066:
1014:
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915: ×
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400:for some constant
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193: ×
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1400:Princeton, N. J.
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998:
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859:
854:
839:
835:
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709:(Ω), is to find
697:
695:
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642:
641:
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555:
554:
549:
547:
546:
417:Poisson equation
396:
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279:
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179:weak formulation
177:However, in the
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1134:
1090:
1067:
1049:
1043:
1040:
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994:
971:
953:
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944:
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889:
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850:
833:
825:
794:
790:
779:
745:
741:
732:
728:
726:
723:
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717:
677:
637:
633:
610:
607:
606:
571:
542:
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500:
499:
485:
442:
441:
439:
436:
435:
381:
377:
360:
337:
335:
332:
331:
259:
221:
218:
217:
147:
144:
143:
63:In the modern,
61:
12:
11:
5:
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1472:
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1453:
1452:
1432:
1431:External links
1429:
1428:
1427:
1376:
1324:(4): 322–333.
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1182:
1179:
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1130:
1114:Then, for all
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1080:
1077:
1074:
1070:
1064:
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572:(Ω) with dual
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115:Λ :
60:
57:
53:Arthur Milgram
42:mathematicians
9:
6:
4:
3:
2:
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1471:
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1460:
1450:
1446:
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1440:
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1421:
1417:
1413:
1409:
1405:
1401:
1396:
1395:
1389:
1385:
1381:
1380:Lax, Peter D.
1377:
1373:
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1357:
1353:
1349:
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1335:
1331:
1327:
1323:
1319:
1318:
1313:
1310:(1970–1971).
1309:
1305:
1304:
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1293:
1292:
1272:
1266:
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1255:
1250:
1242:
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1202:
1199:
1186:
1183:
1180:
1174:
1168:
1165:
1160:
1156:
1149:
1142:
1141:
1140:
1138:
1135: ∈
1133:
1129:
1126: =
1125:
1121:
1118: ∈
1117:
1098:
1095:
1084:
1081:
1078:
1072:
1062:
1059:
1053:
1038:
1037:
1036:
1034:
1031: ∈
1030:
1008:
1002:
999:
988:
985:
982:
976:
966:
963:
957:
942:
941:
940:
938:
935: ∈
934:
930:
926:
922:
919: →
918:
914:
911: :
910:
906:
902:
898:
894:
870:
856:
851:
847:
843:
840:
830:
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806:
800:
791:
787:
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772:
766:
760:
754:
746:
742:
729:
721:
720:
719:
716:
712:
708:
705: ∈
704:
685:
682:
670:
664:
658:
652:
646:
634:
630:
624:
621:
618:
612:
605:
604:
603:
601:
600:inner product
597:
593:
589:
586: =
585:
582:
580:
575:
568:
564:
538:
529:
526:
521:
518:
515:
509:
503:
496:
490:
487:
482:
476:
470:
467:
461:
455:
449:
443:
434:
433:
432:
430:
426:
422:
418:
413:
411:
408: ∈
407:
403:
382:
374:
368:
365:
354:
351:
348:
342:
330:
329:
328:
326:
322:
318:
314:
313:Hilbert space
310:
307: =
306:
302:
299: ∈
298:
294:
275:
272:
269:
266:
253:
250:
247:
241:
235:
232:
229:
223:
216:
215:
214:
212:
209: ∈
208:
204:
203:weak solution
200:
197: →
196:
192:
189: :
188:
184:
180:
161:
158:
155:
152:
142:
141:
140:
138:
135: ∈
134:
130:
127: ∈
126:
122:
119: →
118:
114:
110:
106:
102:
99:
95:
91:
88:
87:normed spaces
85:
81:
78:
77:Sobolev space
74:
70:
66:
56:
54:
50:
46:
43:
39:
35:
34:weak solution
31:
30:bilinear form
27:
23:
19:
1442:
1422:– via
1393:
1321:
1315:
1308:Babuška, Ivo
1223:
1136:
1131:
1127:
1123:
1119:
1115:
1113:
1032:
1028:
1026:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
890:
714:
710:
706:
702:
700:
595:
591:
587:
583:
578:
573:
566:
562:
560:
428:
414:
409:
405:
401:
399:
323:is strongly
320:
316:
308:
304:
300:
296:
293:continuously
290:
210:
206:
202:
198:
194:
190:
186:
182:
176:
136:
132:
128:
124:
120:
116:
108:
104:
100:
93:
89:
79:
73:vector space
62:
21:
15:
96:with their
45:Ivo Babuška
36:to a given
18:mathematics
1459:Categories
1424:De Gruyter
1420:0058.08703
1372:0214.42001
1301:References
718:such that
561:the space
213:such that
139:such that
59:Background
1449:EMS Press
1364:122191183
1348:0029-599X
1270:‖
1264:‖
1251:≤
1248:‖
1235:‖
1203:∈
1190:⟩
1178:⟨
1057:‖
1051:‖
1012:‖
1006:‖
1000:≥
961:‖
955:‖
865:Ω
844:∈
796:Ω
792:∫
764:∇
761:⋅
739:∇
734:Ω
730:∫
662:∇
659:⋅
644:∇
639:Ω
635:∫
536:Ω
533:∂
530:∈
494:Ω
491:∈
453:Δ
450:−
379:‖
372:‖
366:≥
270:∈
257:⟩
245:⟨
150:Λ
49:Peter Lax
1386:(1954),
1289:See also
325:coercive
1412:0067317
1356:0288971
421:bounded
327:, i.e.
315:, that
1418:
1410:
1370:
1362:
1354:
1346:
20:, the
1360:S2CID
581:space
419:on a
311:is a
1344:ISSN
1096:>
903:and
895:and
598:(Ω)
425:open
103:and
92:and
84:real
51:and
1416:Zbl
1368:Zbl
1334:hdl
1326:doi
1047:sup
1035:,
951:sup
16:In
1461::
1447:,
1441:,
1414:,
1408:MR
1402::
1390:,
1382:;
1366:.
1358:.
1352:MR
1350:.
1342:.
1332:.
1322:16
1320:.
1314:.
939:,
431:,
423:,
412:.
55:.
47:,
1374:.
1336::
1328::
1273:.
1267:f
1259:c
1256:1
1243:f
1239:u
1209:.
1206:V
1200:v
1187:v
1184:,
1181:f
1175:=
1172:)
1169:v
1166:,
1161:f
1157:u
1153:(
1150:B
1137:U
1132:f
1128:u
1124:u
1120:V
1116:f
1099:0
1092:|
1088:)
1085:v
1082:,
1079:u
1076:(
1073:B
1069:|
1063:1
1060:=
1054:u
1033:V
1029:v
1009:u
1003:c
996:|
992:)
989:v
986:,
983:u
980:(
977:B
973:|
967:1
964:=
958:v
937:U
933:u
929:c
925:B
921:R
917:V
913:U
909:B
905:V
901:U
897:V
893:U
871:.
868:)
862:(
857:1
852:0
848:H
841:v
831:x
827:d
822:)
819:x
816:(
813:v
810:)
807:x
804:(
801:f
788:=
785:x
781:d
776:)
773:x
770:(
767:v
758:)
755:x
752:(
747:f
743:u
715:f
711:u
707:L
703:f
686:.
683:x
679:d
674:)
671:x
668:(
665:v
656:)
653:x
650:(
647:u
631:=
628:)
625:v
622:,
619:u
616:(
613:B
596:L
592:B
588:L
584:V
579:L
574:H
570:0
567:H
563:U
539:;
527:x
522:,
519:0
516:=
513:)
510:x
507:(
504:u
497:;
488:x
483:,
480:)
477:x
474:(
471:f
468:=
465:)
462:x
459:(
456:u
444:{
429:R
410:U
406:u
402:c
383:2
375:u
369:c
362:|
358:)
355:u
352:,
349:u
346:(
343:B
339:|
321:B
317:B
309:V
305:U
301:V
297:f
276:.
273:V
267:v
254:v
251:,
248:f
242:=
239:)
236:v
233:,
230:u
227:(
224:B
211:U
207:u
199:R
195:V
191:U
187:B
183:V
162:.
159:f
156:=
153:u
137:U
133:u
129:V
125:f
121:V
117:U
109:U
105:V
101:U
94:V
90:U
80:W
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