267:
99:
48:
in the domain then the function is constant. The simple idea behind Hopf's proof, the comparison technique he introduced for this purpose, has led to an enormous range of important applications and generalizations.
619:
515:
553:
645:
765:
466:
442:
262:{\displaystyle Lu=\sum _{ij}a_{ij}(x){\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+\sum _{i}b_{i}(x){\frac {\partial u}{\partial x_{i}}}\geq 0}
25:
400:. In the later sections of his original paper, however, Hopf considered a more general situation which permits certain nonlinear operators
396:
565:
699:
675:
770:
40:
proved in 1927 that if a function satisfies a second order partial differential inequality of a certain kind in a domain of
368:
are just functions. If they are known to be continuous then it is sufficient to demand pointwise positive definiteness of
413:
381:
471:
28:
and has been described as the "classic and bedrock result" of that theory. Generalizing the maximum principle for
306:
445:
535:
726:
33:
624:
695:
671:
405:
29:
21:
451:
427:
734:
468:
has a smooth boundary), slightly more can be said. If in addition to the assumptions above,
281:
748:
709:
744:
705:
388:
730:
690:
Hopf, Eberhard (2002), Morawetz, Cathleen S.; Serrin, James B.; Sinai, Yakov G. (eds.),
409:
759:
392:
37:
717:
Pucci, Patrizia; Serrin, James (2004), "The strong maximum principle revisited",
665:
328:
739:
273:
45:
380:
It is usually thought that the Hopf maximum principle applies only to
614:{\displaystyle {\frac {\partial u}{\partial \nu }}(x_{0})>0}
404:
and, in some cases, leads to uniqueness statements in the
627:
568:
538:
474:
454:
430:
102:
93:
function which satisfies the differential inequality
387:. In particular, this is the point of view taken by
639:
613:
547:
509:
460:
436:
261:
694:, Providence, RI: American Mathematical Society,
692:Selected works of Eberhard Hopf with commentaries
510:{\displaystyle u\in C^{1}({\overline {\Omega }})}
8:
738:
670:. American Mathematical Soc. p. 28.
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569:
567:
537:
494:
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129:
116:
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766:Elliptic partial differential equations
667:Elliptic Partial Differential Equations
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26:elliptic partial differential equations
555:, then for any outward direction ν at
7:
580:
572:
542:
539:
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455:
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397:Methoden der mathematischen Physik
237:
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178:
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151:
14:
719:Journal of Differential Equations
664:Han, Qing; Lin, Fanghua (2011).
548:{\displaystyle \partial \Omega }
309:in Ω and the coefficients
602:
589:
504:
491:
223:
217:
144:
138:
24:in the theory of second order
1:
382:linear differential operators
499:
32:which was already known to
787:
276:(connected open subset of
740:10.1016/j.jde.2003.05.001
640:{\displaystyle u\equiv M}
446:interior sphere property
53:Mathematical formulation
771:Mathematical principles
461:{\displaystyle \Omega }
437:{\displaystyle \Omega }
305:) is locally uniformly
641:
615:
549:
521:takes a maximum value
511:
462:
438:
335:takes a maximum value
263:
18:Hopf maximum principle
642:
616:
550:
512:
463:
439:
414:Monge–Ampère equation
264:
625:
566:
536:
472:
452:
428:
280:) Ω, where the
100:
731:2004JDE...196....1P
637:
611:
545:
507:
458:
434:
420:Boundary behaviour
259:
206:
124:
30:harmonic functions
587:
502:
448:(for example, if
412:operator and the
406:Dirichlet problem
350:The coefficients
307:positive definite
251:
197:
192:
112:
22:maximum principle
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467:
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443:
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282:symmetric matrix
268:
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137:
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123:
786:
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689:
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663:
662:
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623:
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592:
579:
571:
564:
563:
561:
534:
533:
531:
481:
470:
469:
450:
449:
426:
425:
422:
377:on the domain.
376:
367:
358:
339:in Ω then
326:
317:
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291:
240:
236:
228:
207:
181:
168:
164:
150:
149:
125:
98:
97:
88:
79:
55:
12:
11:
5:
784:
782:
774:
773:
768:
758:
757:
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753:
714:
700:
684:
683:
676:
655:
654:
652:
649:
636:
633:
630:
610:
607:
604:
599:
595:
591:
585:
582:
577:
574:
562:, there holds
559:
544:
541:
529:
506:
501:
498:
493:
488:
484:
480:
477:
457:
433:
424:If the domain
421:
418:
410:mean curvature
372:
363:
354:
322:
313:
296:
287:
270:
269:
258:
255:
247:
243:
239:
234:
231:
225:
222:
219:
214:
210:
204:
200:
196:
188:
184:
180:
175:
171:
167:
162:
157:
153:
146:
143:
140:
135:
132:
128:
122:
119:
115:
111:
108:
105:
84:
77:
54:
51:
44:and attains a
13:
10:
9:
6:
4:
3:
2:
783:
772:
769:
767:
764:
763:
761:
750:
746:
741:
736:
732:
728:
724:
720:
715:
711:
707:
703:
701:0-8218-2077-X
697:
693:
688:
687:
679:
677:9780821853139
673:
669:
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650:
648:
634:
631:
628:
608:
605:
597:
593:
583:
575:
558:
528:
524:
520:
486:
482:
478:
475:
447:
419:
417:
415:
411:
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398:
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371:
366:
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241:
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202:
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186:
182:
173:
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155:
141:
133:
130:
126:
120:
117:
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109:
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103:
96:
95:
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83:
76:
72:
68:
64:
60:
52:
50:
47:
43:
39:
38:Eberhard Hopf
35:
31:
27:
23:
19:
722:
718:
691:
666:
659:
556:
526:
522:
518:
423:
401:
395:
384:
379:
373:
369:
364:
360:
355:
351:
349:
344:
340:
336:
332:
327:are locally
323:
319:
314:
310:
302:
297:
293:
288:
284:
277:
271:
90:
85:
81:
74:
70:
66:
62:
58:
56:
41:
17:
15:
725:(1): 1–66,
525:at a point
274:open domain
760:Categories
651:References
632:≡
584:ν
581:∂
573:∂
543:Ω
540:∂
500:¯
497:Ω
479:∈
456:Ω
432:Ω
393:Hilbert's
254:≥
238:∂
230:∂
199:∑
179:∂
166:∂
152:∂
114:∑
36:in 1839,
444:has the
408:for the
343:≡
749:2025185
727:Bibcode
710:1985954
621:unless
389:Courant
329:bounded
89:) be a
80:, ...,
46:maximum
747:
708:
698:
674:
272:in an
331:. If
34:Gauss
20:is a
696:ISBN
672:ISBN
606:>
517:and
391:and
57:Let
16:The
735:doi
723:196
532:in
73:= (
69:),
762::
745:MR
743:,
733:,
721:,
706:MR
704:,
647:.
416:.
374:ij
359:,
356:ij
347:.
318:,
315:ij
298:ji
292:=
289:ij
61:=
752:.
737::
729::
713:.
680:.
635:M
629:u
609:0
603:)
598:0
594:x
590:(
576:u
560:0
557:x
530:0
527:x
523:M
519:u
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492:(
487:1
483:C
476:u
402:L
385:L
370:a
365:i
361:b
352:a
345:M
341:u
337:M
333:u
324:i
320:b
311:a
303:x
301:(
294:a
285:a
278:R
257:0
246:i
242:x
233:u
224:)
221:x
218:(
213:i
209:b
203:i
195:+
187:j
183:x
174:i
170:x
161:u
156:2
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142:x
139:(
134:j
131:i
127:a
121:j
118:i
110:=
107:u
104:L
91:C
86:n
82:x
78:1
75:x
71:x
67:x
65:(
63:u
59:u
42:R
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