4184:
3667:
4179:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{\frac {\partial ^{2}h}{\partial x^{i}\,\partial x^{j}}}+\sum _{i=1}^{n}b_{i}{\frac {\partial h}{\partial x^{i}}}=\varepsilon \alpha e^{-\alpha |x-x_{\text{c}}|^{2}}\left(4\alpha \sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}(x){\big (}x^{i}-x_{\text{c}}^{i}{\big )}{\big (}x^{j}-x_{\text{c}}^{j}{\big )}-2\sum _{i=1}^{n}a_{ii}-2\sum _{i=1}^{n}b_{i}{\big (}x^{i}-x_{\text{c}}^{i}{\big )}\right).}
1283:
1071:
4960:
4523:
875:
1464:
656:
The essence of the maximum principle is the simple observation that if each eigenvalue is positive (which amounts to a certain formulation of "ellipticity" of the differential equation) then the above equation imposes a certain balancing of the directional second derivatives of the solution. In
215:
Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about
57:
The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential
3047:
2806:
651:
400:
1292:, only says that a weighted average of manifestly nonpositive quantities is nonpositive. This is trivially true, and so one cannot draw any nontrivial conclusion from it. This is reflected by any number of concrete examples, such as the fact that
3535:
171:
1098:
886:
4778:
4341:
690:
4544:
The point of the continuity assumption is that continuous functions are bounded on compact sets, the relevant compact set here being the spherical annulus appearing in the proof. Furthermore, by the same principle, there is a number
3309:
1298:
1649:
form a contradiction to this algebraic relation. This is the essence of the maximum principle. Clearly, the applicability of this idea depends strongly on the particular partial differential equation in question.
4576:
These continuity assumptions are clearly not the most general possible in order for the proof to work. For instance, the following is
Gilbarg and Trudinger's statement of the theorem, following the same proof:
661:
is maximized, all directional second derivatives are automatically nonpositive, and the "balancing" represented by the above equation then requires all directional second derivatives to be identically zero.
5199:
Linear and quasilinear equations of parabolic type. Translated from the
Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968 xi+648
2031:
4744:
2925:
2687:
4696:
4981:
One cannot naively extend these statements to the general second-order linear elliptic equation, as already seen in the one-dimensional case. For instance, the ordinary differential equation
2543:
above was very inessential. All that mattered was to have a function which extends continuously to the boundary and whose
Laplacian is strictly positive. So we could have used, for instance,
5182:
Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in
Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp.
665:
This elementary reasoning could be argued to represent an infinitesimal formulation of the strong maximum principle, which states, under some extra assumptions (such as the continuity of
511:
5210:
Linear and quasilinear elliptic equations. Translated from the
Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York-London 1968 xviii+495 pp.
2910:
2211:
2273:
1635:
2621:
279:
5118:
E. Hopf. Elementare
Bemerkungen Über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitber. Preuss. Akad. Wiss. Berlin 19 (1927), 147-152.
1896:
1716:
2675:
1937:
1529:
5255:
Smoller, Joel. Shock waves and reaction-diffusion equations. Second edition. Grundlehren der
Mathematischen Wissenschaften , 258. Springer-Verlag, New York, 1994. xxiv+632 pp.
2380:
2087:
1834:
1748:
1278:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{\frac {\partial ^{2}u}{\partial x^{i}\,\partial x^{j}}}+\sum _{i=1}^{n}b_{i}{\frac {\partial u}{\partial x^{i}}}\leq 0,}
1066:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{\frac {\partial ^{2}u}{\partial x^{i}\,\partial x^{j}}}+\sum _{i=1}^{n}b_{i}{\frac {\partial u}{\partial x^{i}}}\geq 0,}
4955:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{\frac {\partial ^{2}u}{\partial x^{i}\,\partial x^{j}}}+\sum _{i=1}^{n}b_{i}{\frac {\partial u}{\partial x^{i}}}\geq 0}
4518:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{\frac {\partial ^{2}u}{\partial x^{i}\,\partial x^{j}}}+\sum _{i=1}^{n}b_{i}{\frac {\partial u}{\partial x^{i}}}\geq 0}
5169:
Evans, Lawrence C. Partial differential equations. Second edition. Graduate
Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp.
3416:
2492:
2462:
2436:
82:
5233:
Protter, Murray H.; Weinberger, Hans F. Maximum principles in differential equations. Corrected reprint of the 1967 original. Springer-Verlag, New York, 1984. x+261 pp.
5223:
Morrey, Charles B., Jr. Multiple integrals in the calculus of variations. Reprint of the 1966 edition. Classics in
Mathematics. Springer-Verlag, Berlin, 2008. x+506 pp.
2116:
4992:
has sinusoidal solutions, which certainly have interior maxima. This extends to the higher-dimensional case, where one often has solutions to "eigenfunction" equations
2541:
1580:
870:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{\frac {\partial ^{2}u}{\partial x^{i}\,\partial x^{j}}}+\sum _{i=1}^{n}b_{i}{\frac {\partial u}{\partial x^{i}}}=0,}
4561:, as appearing in the proof, to be large relative to these bounds. Evans's book has a slightly weaker formulation, in which there is assumed to be a positive number
2302:
1777:
1459:{\displaystyle {\frac {\partial ^{2}}{\partial x^{2}}}{\big (}{-x}^{2}-y^{2}{\big )}+{\frac {\partial ^{2}}{\partial y^{2}}}{\big (}{-x}^{2}-y^{2}{\big )}\leq 0,}
4205:
along the inward-pointing radial line of the annulus is strictly positive. As described in the above summary, this will ensure that a directional derivative of
3155:
2323:
also cannot attain a maximum value. However, it is possible for the pointwise limit of a sequence of functions without maxima to have a maxima. Nonetheless, if
2036:
which is a sort of "non-local" differential equation, then the automatic strict positivity of the right-hand side shows, by the same analysis as above, that
880:
since the added term is automatically zero at any hypothetical maximum point. The reasoning is also unaffected if one considers the more general condition
5102:
Cheng, S.Y.; Yau, S.T. Differential equations on
Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354.
19:
This article describes the maximum principle in the theory of partial differential equations. For the maximum principle in optimal control theory, see
5280:
5134:
1084:) in this condition at the hypothetical maximum point. This phenomenon is important in the formal proof of the classical weak maximum principle.
5105:
Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), no. 3, 209–243.
1641:
One can view a partial differential equation as the imposition of an algebraic relation between the various derivatives of a function. So, if
5228:
5174:
5062:
5213:
Lieberman, Gary M. Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. xii+439 pp.
5112:. Mathematical analysis and applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.
4189:
There are various conditions under which the right-hand side can be guaranteed to be nonnegative; see the statement of the theorem below.
1645:
is the solution of a partial differential equation, then it is possible that the above conditions on the first and second derivatives of
5133:
Kreyberg, H. J. A. On the maximum principle of optimal control in economic processes, 1969 (Trondheim, NTH, Sosialøkonomisk institutt
5260:
5238:
5218:
5187:
5160:
4243:
The following is the statement of the theorem in the books of Morrey and Smoller, following the original statement of Hopf (1927):
657:
particular, if one of the directional second derivatives is negative, then another must be positive. At a hypothetical point where
20:
5099:
Calabi, E. An extension of E. Hopf's maximum principle with an application to Riemannian geometry. Duke Math. J. 25 (1958), 45–56.
5285:
5179:
Friedman, Avner. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp.
3042:{\displaystyle a_{ij}{\frac {\partial ^{2}h}{\partial x^{i}\,\partial x^{j}}}+b_{i}{\frac {\partial h}{\partial x^{i}}}\geq 0,}
2801:{\displaystyle a_{ij}{\frac {\partial ^{2}u}{\partial x^{i}\,\partial x^{j}}}+b_{i}{\frac {\partial u}{\partial x^{i}}}\geq 0.}
1945:
5121:
Hopf, Eberhard. A remark on linear elliptic differential equations of second order. Proc. Amer. Math. Soc. 3 (1952), 791–793.
4701:
4662:
5019:
are exponentials, and the character of the maxima of such functions is quite different from that of sinusoidal functions.
5275:
5115:
Hamilton, Richard S. Four-manifolds with positive curvature operator. J. Differential Geom. 24 (1986), no. 2, 153–179.
3326:
so that the right-hand side has a manifestly positive nature, then this will provide a contradiction to the fact that
2047:
is a harmonic function, then the above sort of contradiction does not directly occur, since the existence of a point
1836:
then one would not have such a contradiction, and the analysis given so far does not imply anything interesting. If
1076:
in which one can even note the extra phenomena of having an outright contradiction if there is a strict inequality (
646:{\displaystyle \sum _{i=1}^{n}\lambda _{i}\left.{\frac {d^{2}}{dt^{2}}}\right|_{t=0}{\big (}u(x+tv_{i}){\big )}=0.}
5028:
24:
5130:
Yau, Shing Tung. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28 (1975), 201–228.
5124:
Nirenberg, Louis. A strong maximum principle for parabolic equations. Comm. Pure Appl. Math. 6 (1953), 167–177.
2881:
2139:
5135:
https://www.worldcat.org/title/on-the-maximum-principle-of-optimal-control-in-economic-processes/oclc/23714026
2222:
2043:
There are many methods to extend the applicability of this kind of analysis in various ways. For instance, if
1586:
395:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{\frac {\partial ^{2}u}{\partial x^{i}\,\partial x^{j}}}=0}
255:
Here we consider the simplest case, although the same thinking can be extended to more general scenarios. Let
2549:
684:
Note that the above reasoning is unaffected if one considers the more general partial differential equation
42:
is one of the most useful and best known tools of study. Solutions of a differential inequality in a domain
5244:
5078:
5033:
1843:
1663:
31:
5108:
Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry of positive solutions of nonlinear elliptic equations in
2646:
1908:
1905:
The possibility of such analysis is not even limited to partial differential equations. For instance, if
1500:
2353:
2054:
1790:
3530:{\displaystyle h(x)=\varepsilon {\Big (}e^{-\alpha |x-x_{\text{c}}|^{2}}-e^{-\alpha R^{2}}{\Big )}.}
1724:
166:{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.}
5127:
Omori, Hideki. Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19 (1967), 205–214.
239:
224:
5203:
5192:
2471:
2441:
2415:
467:
of linear algebra, all eigenvalues of the matrix are real, and there is an orthonormal basis of
35:
2092:
220:. There is no single or most general maximum principle which applies to all situations at once.
2512:
2498:
for harmonic functions. This does not, by itself, rule out the possibility that the maximum of
1541:
5256:
5234:
5224:
5214:
5183:
5170:
5156:
5058:
2281:
464:
5148:
3304:{\displaystyle -{\frac {u(x)-u(x_{0})}{|x-x_{0}|}}\geq {\frac {h(x)-h(x_{0})}{|x-x_{0}|}}}
2506:. That is the content of the "strong maximum principle," which requires further analysis.
1753:
228:
5207:
5196:
1779:
at any point of the domain. So, following the above observation, it is impossible for
5269:
5008:
is relevant, as also seen in the one-dimensional case; for instance the solutions to
1288:
since now the "balancing" condition, as evaluated at a hypothetical maximum point of
58:
equations and in the determination of bounds for the errors in such approximations.
232:
2335:
can be continuously extended to the boundary, it follows immediately that both
1092:
However, the above reasoning no longer applies if one considers the condition
235:
3406:
be a point on this latter set which realizes the distance. The inner radius
216:
solutions of differential equations, such as control over the size of their
5155:. Providence, Rhode Island: American Mathematical Society. pp. 31–41.
4553:
in the annulus, the matrix has all eigenvalues greater than or equal to
217:
208:
is a constant function, the maximum cannot also be achieved anywhere on
2315:
cannot attain a maximum value. One might wish to consider the limit as
61:
In a simple two-dimensional case, consider a function of two variables
2118:
everywhere. However, one could consider, for an arbitrary real number
227:, there is an analogous statement which asserts that the maximum of a
2677:
be a twice-differentiable function which attains its maximum value
3580:
holds on this part of the boundary, together with the requirement
2397:, does not have a maximum, it follows that the maximum point of
1497:
denote an open subset of Euclidean space. If a smooth function
2630:
The classical strong maximum principle for linear elliptic PDE
1484:
The classical weak maximum principle for linear elliptic PDE
180:, in this setting, says that for any open precompact subset
2331:
together with its boundary is compact, then supposing that
1469:
and on any open region containing the origin, the function
548:
3654:
on the inner sphere, and hence on the entire boundary of
3608:
and the compactness of the inner sphere, one can select
2026:{\displaystyle \Delta u-|du|^{4}=\int _{M}e^{u(x)}\,dx,}
2811:
Suppose that one can find (or prove the existence of):
4739:{\displaystyle \textstyle {\frac {|b_{i}|}{\lambda }}}
4705:
4666:
4655:, the symmetric matrix is positive-definite, and let
4565:
which is a lower bound of the eigenvalues of for all
3544:
consists of two spheres; on the outer sphere, one has
471:
consisting of eigenvectors. Denote the eigenvalues by
4781:
4704:
4691:{\displaystyle \textstyle {\frac {a_{ii}}{\lambda }}}
4665:
4344:
3670:
3419:
3158:
2928:
2884:
2690:
2649:
2552:
2515:
2474:
2444:
2418:
2356:
2284:
2225:
2142:
2095:
2057:
1948:
1911:
1846:
1793:
1756:
1727:
1666:
1589:
1544:
1503:
1301:
1101:
889:
693:
514:
282:
251:
A partial formulation of the strong maximum principle
85:
5053:Protter, Murray H.; Weinberger, Hans Felix (1984).
3386:is selected to be the distance from this center to
3130:; according to the weak maximum principle, one has
4954:
4738:
4690:
4517:
4178:
3529:
3355:to be a spherical annulus; one selects its center
3303:
3041:
2904:
2800:
2669:
2615:
2535:
2486:
2456:
2430:
2374:
2296:
2267:
2205:
2110:
2081:
2025:
1931:
1890:
1828:
1771:
1742:
1710:
1629:
1574:
1523:
1458:
1277:
1065:
869:
645:
394:
165:
4321:, the symmetric matrix is positive-definite. If
3633:is constant on this inner sphere, one can select
3519:
3440:
2912:which is twice-differentiable on the interior of
1088:Non-applicability of the strong maximum principle
4192:Lastly, note that the directional derivative of
50:if they achieve their maxima at the boundary of
4579:
4245:
3351:The above "program" can be carried out. Choose
501:. Then the differential equation, at the point
4659:denote its smallest eigenvalue. Suppose that
4163:
4128:
4044:
4009:
4002:
3967:
1442:
1407:
1368:
1333:
632:
597:
259:be an open subset of Euclidean space and let
8:
5055:Maximum principles in differential equations
2089:is not in contradiction to the requirement
23:. For the theorem in complex analysis, see
2643:be an open subset of Euclidean space. Let
5057:. New York Berlin Heidelberg : Springer.
4937:
4919:
4913:
4903:
4892:
4876:
4868:
4862:
4844:
4837:
4828:
4818:
4807:
4797:
4786:
4780:
4724:
4718:
4709:
4706:
4703:
4673:
4667:
4664:
4500:
4482:
4476:
4466:
4455:
4439:
4431:
4425:
4407:
4400:
4391:
4381:
4370:
4360:
4349:
4343:
4162:
4161:
4155:
4150:
4137:
4127:
4126:
4120:
4110:
4099:
4080:
4070:
4059:
4043:
4042:
4036:
4031:
4018:
4008:
4007:
4001:
4000:
3994:
3989:
3976:
3966:
3965:
3947:
3937:
3926:
3916:
3905:
3882:
3877:
3870:
3855:
3848:
3826:
3808:
3802:
3792:
3781:
3765:
3757:
3751:
3733:
3726:
3717:
3707:
3696:
3686:
3675:
3669:
3518:
3517:
3509:
3498:
3483:
3478:
3471:
3456:
3449:
3439:
3438:
3418:
3293:
3287:
3272:
3261:
3233:
3222:
3216:
3201:
3190:
3162:
3157:
3021:
3003:
2997:
2981:
2973:
2967:
2949:
2942:
2933:
2927:
2905:{\displaystyle h:\Omega \to \mathbb {R} }
2898:
2897:
2883:
2783:
2765:
2759:
2743:
2735:
2729:
2711:
2704:
2695:
2689:
2663:
2662:
2648:
2607:
2602:
2593:
2557:
2551:
2525:
2520:
2514:
2473:
2443:
2417:
2355:
2283:
2254:
2249:
2233:
2224:
2206:{\displaystyle u_{s}(x)=u(x)+se^{x_{1}}.}
2192:
2187:
2147:
2141:
2094:
2056:
2013:
1998:
1988:
1975:
1970:
1958:
1947:
1925:
1924:
1910:
1873:
1868:
1856:
1845:
1820:
1815:
1803:
1792:
1755:
1726:
1693:
1688:
1676:
1665:
1597:
1588:
1543:
1517:
1516:
1502:
1441:
1440:
1434:
1421:
1413:
1406:
1405:
1396:
1382:
1376:
1367:
1366:
1360:
1347:
1339:
1332:
1331:
1322:
1308:
1302:
1300:
1257:
1239:
1233:
1223:
1212:
1196:
1188:
1182:
1164:
1157:
1148:
1138:
1127:
1117:
1106:
1100:
1045:
1027:
1021:
1011:
1000:
984:
976:
970:
952:
945:
936:
926:
915:
905:
894:
888:
849:
831:
825:
815:
804:
788:
780:
774:
756:
749:
740:
730:
719:
709:
698:
692:
631:
630:
621:
596:
595:
583:
570:
556:
550:
540:
530:
519:
513:
377:
369:
363:
345:
338:
329:
319:
308:
298:
287:
281:
148:
130:
123:
111:
93:
86:
84:
5251:. Princeton: Princeton University Press.
5004:which have interior maxima. The sign of
2268:{\displaystyle \Delta u_{s}=se^{x_{1}}.}
1783:to take on a maximum value. If, instead
1630:{\displaystyle (\nabla ^{2}u)(p)\leq 0,}
673:must be constant if there is a point of
5045:
3364:to be a point closer to the closed set
1898:then the same analysis would show that
2616:{\displaystyle u_{s}(x)=u(x)+s|x|^{2}}
1721:then it is clearly impossible to have
482:and the corresponding eigenvectors by
4585:be an open subset of Euclidean space
4251:be an open subset of Euclidean space
7:
3343:, so that its gradient must vanish.
2823:, with nonempty interior, such that
1891:{\displaystyle \Delta u=|du|^{2}-2,}
1711:{\displaystyle \Delta u=|du|^{2}+2,}
4973:does not attain a maximum value on
4536:does not attain a maximum value on
2670:{\displaystyle u:M\to \mathbb {R} }
1932:{\displaystyle u:M\to \mathbb {R} }
1524:{\displaystyle u:M\to \mathbb {R} }
5153:Fully Nonlinear Elliptic Equations
4930:
4922:
4869:
4855:
4841:
4493:
4485:
4432:
4418:
4404:
3819:
3811:
3758:
3744:
3730:
3014:
3006:
2974:
2960:
2946:
2891:
2776:
2768:
2736:
2722:
2708:
2475:
2445:
2419:
2363:
2226:
2216:It is straightforward to see that
2096:
2058:
1949:
1847:
1794:
1728:
1667:
1594:
1389:
1379:
1315:
1305:
1250:
1242:
1189:
1175:
1161:
1038:
1030:
977:
963:
949:
842:
834:
781:
767:
753:
370:
356:
342:
141:
127:
104:
90:
14:
3149:. This can be reorganized to say
2509:The use of the specific function
2438:By the sequential compactness of
2375:{\displaystyle M\cup \partial M.}
2082:{\displaystyle \Delta u(p)\leq 0}
1840:solved the differential equation
1829:{\displaystyle \Delta u=|du|^{2}}
1787:solved the differential equation
1657:solves the differential equation
4218:is nonzero, in contradiction to
3322:. If one can make the choice of
1902:cannot take on a minimum value.
3594:. On the inner sphere, one has
2464:it follows that the maximum of
2319:to 0 in order to conclude that
2040:cannot attain a maximum value.
196:is achieved on the boundary of
5281:Partial differential equations
4725:
4710:
3962:
3956:
3878:
3856:
3479:
3457:
3429:
3423:
3294:
3273:
3267:
3254:
3245:
3239:
3223:
3202:
3196:
3183:
3174:
3168:
2894:
2659:
2603:
2594:
2584:
2578:
2569:
2563:
2502:is also attained somewhere on
2174:
2168:
2159:
2153:
2070:
2064:
2008:
2002:
1971:
1959:
1921:
1869:
1857:
1816:
1804:
1743:{\displaystyle \Delta u\leq 0}
1689:
1677:
1615:
1609:
1606:
1590:
1563:
1557:
1554:
1545:
1535:, then one automatically has:
1513:
627:
605:
30:In the mathematical fields of
21:Pontryagin's maximum principle
1:
2845:, and such that there exists
4289:be continuous functions on
3604:. Due to the continuity of
2487:{\displaystyle \partial M.}
2457:{\displaystyle \partial M,}
2431:{\displaystyle \partial M.}
16:Theorem in complex analysis
5302:
3551:; due to the selection of
2350:attain a maximum value on
2278:By the above analysis, if
2111:{\displaystyle \Delta u=0}
18:
5029:Maximum modulus principle
4746:are bounded functions on
4227:being a maximum point of
3661:Direct calculation shows
2536:{\displaystyle e^{x_{1}}}
2382:Since we have shown that
2327:has a boundary such that
1575:{\displaystyle (du)(p)=0}
1480:certainly has a maximum.
25:Maximum modulus principle
4647:. Suppose that for all
4313:. Suppose that for all
4239:Statement of the theorem
1939:is a function such that
1531:is maximized at a point
202:strong maximum principle
5286:Mathematical principles
5151:; Xavier Cabre (1995).
3565:on this sphere, and so
3382:, and the outer radius
3375:than to the closed set
1637:as a matrix inequality.
5204:Ladyzhenskaya, Olga A.
5034:Hopf maximum principle
4979:
4956:
4908:
4823:
4802:
4740:
4692:
4542:
4519:
4471:
4386:
4365:
4180:
4115:
4075:
3942:
3921:
3797:
3712:
3691:
3531:
3335:is a maximum point of
3305:
3053:and such that one has
3043:
2906:
2878:a continuous function
2802:
2671:
2626:with the same effect.
2617:
2537:
2496:weak maximum principle
2488:
2458:
2432:
2376:
2298:
2297:{\displaystyle s>0}
2269:
2207:
2112:
2083:
2027:
1933:
1892:
1830:
1773:
1744:
1712:
1631:
1576:
1525:
1460:
1279:
1228:
1143:
1122:
1067:
1016:
931:
910:
871:
820:
735:
714:
647:
535:
505:, can be rephrased as
396:
324:
303:
178:weak maximum principle
167:
32:differential equations
5195:; Solonnikov, V. A.;
4957:
4888:
4803:
4782:
4741:
4693:
4520:
4451:
4366:
4345:
4181:
4095:
4055:
3922:
3901:
3777:
3692:
3671:
3532:
3410:is arbitrary. Define
3306:
3044:
2907:
2803:
2672:
2618:
2538:
2489:
2459:
2433:
2377:
2299:
2270:
2208:
2113:
2084:
2028:
1934:
1893:
1831:
1774:
1745:
1713:
1632:
1577:
1526:
1461:
1280:
1208:
1123:
1102:
1068:
996:
911:
890:
872:
800:
715:
694:
648:
515:
397:
304:
283:
168:
4779:
4702:
4663:
4342:
3668:
3540:Now the boundary of
3417:
3156:
2926:
2882:
2688:
2647:
2550:
2513:
2472:
2442:
2416:
2354:
2282:
2223:
2140:
2093:
2055:
1946:
1909:
1844:
1791:
1772:{\displaystyle du=0}
1754:
1725:
1664:
1587:
1542:
1501:
1299:
1099:
887:
691:
512:
280:
83:
5208:Ural'tseva, Nina N.
5193:Ladyženskaja, O. A.
5149:Caffarelli, Luis A.
4160:
4041:
3999:
3126:on the boundary of
3067:on the boundary of
2854:on the boundary of
2841:in the interior of
2393:, as a function on
463:. According to the
455:Fix some choice of
238:is attained on the
225:convex optimization
5276:Harmonic functions
5245:Rockafellar, R. T.
4952:
4736:
4735:
4688:
4687:
4549:such that for all
4515:
4176:
4146:
4027:
3985:
3527:
3301:
3039:
2902:
2798:
2667:
2613:
2533:
2484:
2454:
2428:
2372:
2294:
2265:
2203:
2108:
2079:
2023:
1929:
1888:
1826:
1769:
1740:
1708:
1627:
1572:
1521:
1489:The essential idea
1456:
1275:
1063:
867:
643:
392:
204:says that, unless
192:on the closure of
163:
36:geometric analysis
5229:978-3-540-69915-6
5175:978-0-8218-4974-3
5094:Research articles
5064:978-3-540-96068-3
4944:
4883:
4762:is a nonconstant
4733:
4685:
4557:. One then takes
4507:
4446:
4325:is a nonconstant
4153:
4034:
3992:
3873:
3833:
3772:
3474:
3299:
3228:
3028:
2988:
2815:a compact subset
2790:
2750:
1653:For instance, if
1403:
1329:
1264:
1203:
1052:
991:
856:
795:
577:
428:is a function on
384:
188:, the maximum of
184:of the domain of
155:
118:
48:maximum principle
40:maximum principle
5293:
5252:
5197:Uralʹceva, N. N.
5166:
5111:
5082:
5075:
5069:
5068:
5050:
5018:
5003:
4991:
4976:
4972:
4968:
4961:
4959:
4958:
4953:
4945:
4943:
4942:
4941:
4928:
4920:
4918:
4917:
4907:
4902:
4884:
4882:
4881:
4880:
4867:
4866:
4853:
4849:
4848:
4838:
4836:
4835:
4822:
4817:
4801:
4796:
4771:
4767:
4761:
4757:
4753:
4749:
4745:
4743:
4742:
4737:
4734:
4729:
4728:
4723:
4722:
4713:
4707:
4697:
4695:
4694:
4689:
4686:
4681:
4680:
4668:
4658:
4654:
4650:
4646:
4626:
4623:be functions on
4622:
4611:
4600:
4596:
4592:
4588:
4584:
4572:
4568:
4564:
4560:
4556:
4552:
4548:
4539:
4535:
4531:
4524:
4522:
4521:
4516:
4508:
4506:
4505:
4504:
4491:
4483:
4481:
4480:
4470:
4465:
4447:
4445:
4444:
4443:
4430:
4429:
4416:
4412:
4411:
4401:
4399:
4398:
4385:
4380:
4364:
4359:
4334:
4330:
4324:
4320:
4316:
4312:
4292:
4288:
4277:
4266:
4262:
4258:
4254:
4250:
4234:
4231:on the open set
4230:
4226:
4217:
4208:
4204:
4195:
4185:
4183:
4182:
4177:
4172:
4168:
4167:
4166:
4159:
4154:
4151:
4142:
4141:
4132:
4131:
4125:
4124:
4114:
4109:
4088:
4087:
4074:
4069:
4048:
4047:
4040:
4035:
4032:
4023:
4022:
4013:
4012:
4006:
4005:
3998:
3993:
3990:
3981:
3980:
3971:
3970:
3955:
3954:
3941:
3936:
3920:
3915:
3889:
3888:
3887:
3886:
3881:
3875:
3874:
3871:
3859:
3834:
3832:
3831:
3830:
3817:
3809:
3807:
3806:
3796:
3791:
3773:
3771:
3770:
3769:
3756:
3755:
3742:
3738:
3737:
3727:
3725:
3724:
3711:
3706:
3690:
3685:
3657:
3653:
3639:
3632:
3628:
3614:
3607:
3603:
3593:
3579:
3564:
3554:
3550:
3543:
3536:
3534:
3533:
3528:
3523:
3522:
3516:
3515:
3514:
3513:
3490:
3489:
3488:
3487:
3482:
3476:
3475:
3472:
3460:
3444:
3443:
3409:
3405:
3396:
3385:
3381:
3374:
3363:
3354:
3342:
3338:
3334:
3325:
3321:
3317:
3310:
3308:
3307:
3302:
3300:
3298:
3297:
3292:
3291:
3276:
3270:
3266:
3265:
3234:
3229:
3227:
3226:
3221:
3220:
3205:
3199:
3195:
3194:
3163:
3148:
3144:
3129:
3125:
3110:
3106:
3084:
3070:
3066:
3048:
3046:
3045:
3040:
3029:
3027:
3026:
3025:
3012:
3004:
3002:
3001:
2989:
2987:
2986:
2985:
2972:
2971:
2958:
2954:
2953:
2943:
2941:
2940:
2915:
2911:
2909:
2908:
2903:
2901:
2874:
2857:
2853:
2844:
2840:
2836:
2822:
2818:
2807:
2805:
2804:
2799:
2791:
2789:
2788:
2787:
2774:
2766:
2764:
2763:
2751:
2749:
2748:
2747:
2734:
2733:
2720:
2716:
2715:
2705:
2703:
2702:
2680:
2676:
2674:
2673:
2668:
2666:
2642:
2635:Summary of proof
2622:
2620:
2619:
2614:
2612:
2611:
2606:
2597:
2562:
2561:
2542:
2540:
2539:
2534:
2532:
2531:
2530:
2529:
2505:
2501:
2493:
2491:
2490:
2485:
2467:
2463:
2461:
2460:
2455:
2437:
2435:
2434:
2429:
2411:
2407:
2396:
2392:
2381:
2379:
2378:
2373:
2349:
2338:
2334:
2330:
2326:
2322:
2318:
2314:
2303:
2301:
2300:
2295:
2274:
2272:
2271:
2266:
2261:
2260:
2259:
2258:
2238:
2237:
2212:
2210:
2209:
2204:
2199:
2198:
2197:
2196:
2152:
2151:
2132:
2121:
2117:
2115:
2114:
2109:
2088:
2086:
2085:
2080:
2050:
2046:
2039:
2032:
2030:
2029:
2024:
2012:
2011:
1993:
1992:
1980:
1979:
1974:
1962:
1938:
1936:
1935:
1930:
1928:
1901:
1897:
1895:
1894:
1889:
1878:
1877:
1872:
1860:
1839:
1835:
1833:
1832:
1827:
1825:
1824:
1819:
1807:
1786:
1782:
1778:
1776:
1775:
1770:
1749:
1747:
1746:
1741:
1717:
1715:
1714:
1709:
1698:
1697:
1692:
1680:
1656:
1648:
1644:
1636:
1634:
1633:
1628:
1602:
1601:
1581:
1579:
1578:
1573:
1534:
1530:
1528:
1527:
1522:
1520:
1496:
1479:
1465:
1463:
1462:
1457:
1446:
1445:
1439:
1438:
1426:
1425:
1420:
1411:
1410:
1404:
1402:
1401:
1400:
1387:
1386:
1377:
1372:
1371:
1365:
1364:
1352:
1351:
1346:
1337:
1336:
1330:
1328:
1327:
1326:
1313:
1312:
1303:
1291:
1284:
1282:
1281:
1276:
1265:
1263:
1262:
1261:
1248:
1240:
1238:
1237:
1227:
1222:
1204:
1202:
1201:
1200:
1187:
1186:
1173:
1169:
1168:
1158:
1156:
1155:
1142:
1137:
1121:
1116:
1083:
1079:
1072:
1070:
1069:
1064:
1053:
1051:
1050:
1049:
1036:
1028:
1026:
1025:
1015:
1010:
992:
990:
989:
988:
975:
974:
961:
957:
956:
946:
944:
943:
930:
925:
909:
904:
876:
874:
873:
868:
857:
855:
854:
853:
840:
832:
830:
829:
819:
814:
796:
794:
793:
792:
779:
778:
765:
761:
760:
750:
748:
747:
734:
729:
713:
708:
680:
676:
672:
668:
660:
652:
650:
649:
644:
636:
635:
626:
625:
601:
600:
594:
593:
582:
578:
576:
575:
574:
561:
560:
551:
545:
544:
534:
529:
504:
500:
496:
492:
481:
470:
465:spectral theorem
462:
458:
451:
431:
427:
416:
412:
408:
401:
399:
398:
393:
385:
383:
382:
381:
368:
367:
354:
350:
349:
339:
337:
336:
323:
318:
302:
297:
272:
268:
262:
258:
223:In the field of
211:
207:
199:
195:
191:
187:
183:
172:
170:
169:
164:
156:
154:
153:
152:
139:
135:
134:
124:
119:
117:
116:
115:
102:
98:
97:
87:
75:
5301:
5300:
5296:
5295:
5294:
5292:
5291:
5290:
5266:
5265:
5249:Convex analysis
5243:
5163:
5147:
5144:
5109:
5096:
5091:
5086:
5085:
5076:
5072:
5065:
5052:
5051:
5047:
5042:
5025:
5009:
4993:
4982:
4974:
4970:
4966:
4933:
4929:
4921:
4909:
4872:
4858:
4854:
4840:
4839:
4824:
4777:
4776:
4769:
4763:
4759:
4755:
4751:
4747:
4714:
4708:
4700:
4699:
4669:
4661:
4660:
4656:
4652:
4648:
4645:
4636:
4628:
4624:
4621:
4613:
4610:
4602:
4598:
4594:
4590:
4586:
4582:
4570:
4566:
4562:
4558:
4554:
4550:
4546:
4537:
4533:
4529:
4496:
4492:
4484:
4472:
4435:
4421:
4417:
4403:
4402:
4387:
4340:
4339:
4332:
4326:
4322:
4318:
4314:
4311:
4302:
4294:
4290:
4287:
4279:
4276:
4268:
4264:
4260:
4256:
4252:
4248:
4241:
4232:
4228:
4225:
4219:
4216:
4210:
4206:
4203:
4197:
4193:
4133:
4116:
4076:
4014:
3972:
3943:
3894:
3890:
3876:
3866:
3844:
3822:
3818:
3810:
3798:
3761:
3747:
3743:
3729:
3728:
3713:
3666:
3665:
3655:
3641:
3634:
3630:
3616:
3609:
3605:
3595:
3591:
3581:
3566:
3556:
3552:
3545:
3541:
3505:
3494:
3477:
3467:
3445:
3415:
3414:
3407:
3404:
3398:
3387:
3383:
3376:
3365:
3362:
3356:
3352:
3349:
3340:
3336:
3333:
3327:
3323:
3319:
3315:
3283:
3271:
3257:
3235:
3212:
3200:
3186:
3164:
3154:
3153:
3146:
3131:
3127:
3112:
3108:
3089:
3082:
3072:
3068:
3054:
3017:
3013:
3005:
2993:
2977:
2963:
2959:
2945:
2944:
2929:
2924:
2923:
2913:
2880:
2879:
2869:
2859:
2855:
2852:
2846:
2842:
2838:
2824:
2820:
2816:
2779:
2775:
2767:
2755:
2739:
2725:
2721:
2707:
2706:
2691:
2686:
2685:
2681:. Suppose that
2678:
2645:
2644:
2640:
2637:
2632:
2601:
2553:
2548:
2547:
2521:
2516:
2511:
2510:
2503:
2499:
2470:
2469:
2468:is attained on
2465:
2440:
2439:
2414:
2413:
2409:
2406:
2398:
2394:
2391:
2383:
2352:
2351:
2348:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2313:
2305:
2280:
2279:
2250:
2245:
2229:
2221:
2220:
2188:
2183:
2143:
2138:
2137:
2131:
2123:
2122:, the function
2119:
2091:
2090:
2053:
2052:
2048:
2044:
2037:
1994:
1984:
1969:
1944:
1943:
1907:
1906:
1899:
1867:
1842:
1841:
1837:
1814:
1789:
1788:
1784:
1780:
1752:
1751:
1723:
1722:
1687:
1662:
1661:
1654:
1646:
1642:
1593:
1585:
1584:
1540:
1539:
1532:
1499:
1498:
1494:
1491:
1486:
1470:
1430:
1412:
1392:
1388:
1378:
1356:
1338:
1318:
1314:
1304:
1297:
1296:
1289:
1253:
1249:
1241:
1229:
1192:
1178:
1174:
1160:
1159:
1144:
1097:
1096:
1090:
1081:
1077:
1041:
1037:
1029:
1017:
980:
966:
962:
948:
947:
932:
885:
884:
845:
841:
833:
821:
784:
770:
766:
752:
751:
736:
689:
688:
678:
674:
670:
666:
658:
617:
566:
562:
552:
547:
546:
536:
510:
509:
502:
498:
494:
491:
483:
480:
472:
468:
460:
456:
450:
441:
433:
429:
426:
418:
414:
410:
406:
405:where for each
373:
359:
355:
341:
340:
325:
278:
277:
270:
264:
260:
256:
253:
248:
229:convex function
209:
205:
197:
193:
189:
185:
181:
144:
140:
126:
125:
107:
103:
89:
88:
81:
80:
62:
28:
17:
12:
11:
5:
5299:
5297:
5289:
5288:
5283:
5278:
5268:
5267:
5264:
5263:
5253:
5241:
5231:
5221:
5211:
5201:
5190:
5180:
5177:
5167:
5161:
5143:
5140:
5139:
5138:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5106:
5103:
5100:
5095:
5092:
5090:
5087:
5084:
5083:
5077:Chapter 32 of
5070:
5063:
5044:
5043:
5041:
5038:
5037:
5036:
5031:
5024:
5021:
4963:
4962:
4951:
4948:
4940:
4936:
4932:
4927:
4924:
4916:
4912:
4906:
4901:
4898:
4895:
4891:
4887:
4879:
4875:
4871:
4865:
4861:
4857:
4852:
4847:
4843:
4834:
4831:
4827:
4821:
4816:
4813:
4810:
4806:
4800:
4795:
4792:
4789:
4785:
4754:between 1 and
4732:
4727:
4721:
4717:
4712:
4684:
4679:
4676:
4672:
4641:
4632:
4617:
4606:
4597:between 1 and
4526:
4525:
4514:
4511:
4503:
4499:
4495:
4490:
4487:
4479:
4475:
4469:
4464:
4461:
4458:
4454:
4450:
4442:
4438:
4434:
4428:
4424:
4420:
4415:
4410:
4406:
4397:
4394:
4390:
4384:
4379:
4376:
4373:
4369:
4363:
4358:
4355:
4352:
4348:
4307:
4298:
4283:
4272:
4263:between 1 and
4240:
4237:
4223:
4214:
4201:
4187:
4186:
4175:
4171:
4165:
4158:
4149:
4145:
4140:
4136:
4130:
4123:
4119:
4113:
4108:
4105:
4102:
4098:
4094:
4091:
4086:
4083:
4079:
4073:
4068:
4065:
4062:
4058:
4054:
4051:
4046:
4039:
4030:
4026:
4021:
4017:
4011:
4004:
3997:
3988:
3984:
3979:
3975:
3969:
3964:
3961:
3958:
3953:
3950:
3946:
3940:
3935:
3932:
3929:
3925:
3919:
3914:
3911:
3908:
3904:
3900:
3897:
3893:
3885:
3880:
3869:
3865:
3862:
3858:
3854:
3851:
3847:
3843:
3840:
3837:
3829:
3825:
3821:
3816:
3813:
3805:
3801:
3795:
3790:
3787:
3784:
3780:
3776:
3768:
3764:
3760:
3754:
3750:
3746:
3741:
3736:
3732:
3723:
3720:
3716:
3710:
3705:
3702:
3699:
3695:
3689:
3684:
3681:
3678:
3674:
3589:
3538:
3537:
3526:
3521:
3512:
3508:
3504:
3501:
3497:
3493:
3486:
3481:
3470:
3466:
3463:
3459:
3455:
3452:
3448:
3442:
3437:
3434:
3431:
3428:
3425:
3422:
3402:
3360:
3348:
3345:
3331:
3312:
3311:
3296:
3290:
3286:
3282:
3279:
3275:
3269:
3264:
3260:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3232:
3225:
3219:
3215:
3211:
3208:
3204:
3198:
3193:
3189:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3161:
3086:
3085:
3080:
3051:
3050:
3049:
3038:
3035:
3032:
3024:
3020:
3016:
3011:
3008:
3000:
2996:
2992:
2984:
2980:
2976:
2970:
2966:
2962:
2957:
2952:
2948:
2939:
2936:
2932:
2918:
2917:
2900:
2896:
2893:
2890:
2887:
2876:
2867:
2850:
2809:
2808:
2797:
2794:
2786:
2782:
2778:
2773:
2770:
2762:
2758:
2754:
2746:
2742:
2738:
2732:
2728:
2724:
2719:
2714:
2710:
2701:
2698:
2694:
2665:
2661:
2658:
2655:
2652:
2636:
2633:
2631:
2628:
2624:
2623:
2610:
2605:
2600:
2596:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2560:
2556:
2528:
2524:
2519:
2483:
2480:
2477:
2453:
2450:
2447:
2427:
2424:
2421:
2402:
2387:
2371:
2368:
2365:
2362:
2359:
2344:
2309:
2293:
2290:
2287:
2276:
2275:
2264:
2257:
2253:
2248:
2244:
2241:
2236:
2232:
2228:
2214:
2213:
2202:
2195:
2191:
2186:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2150:
2146:
2127:
2107:
2104:
2101:
2098:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2034:
2033:
2022:
2019:
2016:
2010:
2007:
2004:
2001:
1997:
1991:
1987:
1983:
1978:
1973:
1968:
1965:
1961:
1957:
1954:
1951:
1927:
1923:
1920:
1917:
1914:
1887:
1884:
1881:
1876:
1871:
1866:
1863:
1859:
1855:
1852:
1849:
1823:
1818:
1813:
1810:
1806:
1802:
1799:
1796:
1768:
1765:
1762:
1759:
1739:
1736:
1733:
1730:
1719:
1718:
1707:
1704:
1701:
1696:
1691:
1686:
1683:
1679:
1675:
1672:
1669:
1639:
1638:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1600:
1596:
1592:
1582:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1519:
1515:
1512:
1509:
1506:
1490:
1487:
1485:
1482:
1467:
1466:
1455:
1452:
1449:
1444:
1437:
1433:
1429:
1424:
1419:
1416:
1409:
1399:
1395:
1391:
1385:
1381:
1375:
1370:
1363:
1359:
1355:
1350:
1345:
1342:
1335:
1325:
1321:
1317:
1311:
1307:
1286:
1285:
1274:
1271:
1268:
1260:
1256:
1252:
1247:
1244:
1236:
1232:
1226:
1221:
1218:
1215:
1211:
1207:
1199:
1195:
1191:
1185:
1181:
1177:
1172:
1167:
1163:
1154:
1151:
1147:
1141:
1136:
1133:
1130:
1126:
1120:
1115:
1112:
1109:
1105:
1089:
1086:
1074:
1073:
1062:
1059:
1056:
1048:
1044:
1040:
1035:
1032:
1024:
1020:
1014:
1009:
1006:
1003:
999:
995:
987:
983:
979:
973:
969:
965:
960:
955:
951:
942:
939:
935:
929:
924:
921:
918:
914:
908:
903:
900:
897:
893:
878:
877:
866:
863:
860:
852:
848:
844:
839:
836:
828:
824:
818:
813:
810:
807:
803:
799:
791:
787:
783:
777:
773:
769:
764:
759:
755:
746:
743:
739:
733:
728:
725:
722:
718:
712:
707:
704:
701:
697:
681:is maximized.
654:
653:
642:
639:
634:
629:
624:
620:
616:
613:
610:
607:
604:
599:
592:
589:
586:
581:
573:
569:
565:
559:
555:
549:
543:
539:
533:
528:
525:
522:
518:
487:
476:
446:
437:
422:
413:between 1 and
403:
402:
391:
388:
380:
376:
372:
366:
362:
358:
353:
348:
344:
335:
332:
328:
322:
317:
314:
311:
307:
301:
296:
293:
290:
286:
252:
249:
247:
244:
174:
173:
162:
159:
151:
147:
143:
138:
133:
129:
122:
114:
110:
106:
101:
96:
92:
15:
13:
10:
9:
6:
4:
3:
2:
5298:
5287:
5284:
5282:
5279:
5277:
5274:
5273:
5271:
5262:
5261:0-387-94259-9
5258:
5254:
5250:
5246:
5242:
5240:
5239:0-387-96068-6
5236:
5232:
5230:
5226:
5222:
5220:
5219:981-02-2883-X
5216:
5212:
5209:
5205:
5202:
5198:
5194:
5191:
5189:
5188:3-540-41160-7
5185:
5181:
5178:
5176:
5172:
5168:
5164:
5162:0-8218-0437-5
5158:
5154:
5150:
5146:
5145:
5141:
5136:
5132:
5129:
5126:
5123:
5120:
5117:
5114:
5107:
5104:
5101:
5098:
5097:
5093:
5088:
5080:
5074:
5071:
5066:
5060:
5056:
5049:
5046:
5039:
5035:
5032:
5030:
5027:
5026:
5022:
5020:
5016:
5012:
5007:
5001:
4997:
4989:
4985:
4978:
4949:
4946:
4938:
4934:
4925:
4914:
4910:
4904:
4899:
4896:
4893:
4889:
4885:
4877:
4873:
4863:
4859:
4850:
4845:
4832:
4829:
4825:
4819:
4814:
4811:
4808:
4804:
4798:
4793:
4790:
4787:
4783:
4775:
4774:
4773:
4766:
4730:
4719:
4715:
4682:
4677:
4674:
4670:
4644:
4640:
4635:
4631:
4620:
4616:
4609:
4605:
4578:
4574:
4541:
4512:
4509:
4501:
4497:
4488:
4477:
4473:
4467:
4462:
4459:
4456:
4452:
4448:
4440:
4436:
4426:
4422:
4413:
4408:
4395:
4392:
4388:
4382:
4377:
4374:
4371:
4367:
4361:
4356:
4353:
4350:
4346:
4338:
4337:
4336:
4329:
4310:
4306:
4301:
4297:
4286:
4282:
4275:
4271:
4244:
4238:
4236:
4222:
4213:
4200:
4190:
4173:
4169:
4156:
4147:
4143:
4138:
4134:
4121:
4117:
4111:
4106:
4103:
4100:
4096:
4092:
4089:
4084:
4081:
4077:
4071:
4066:
4063:
4060:
4056:
4052:
4049:
4037:
4028:
4024:
4019:
4015:
3995:
3986:
3982:
3977:
3973:
3959:
3951:
3948:
3944:
3938:
3933:
3930:
3927:
3923:
3917:
3912:
3909:
3906:
3902:
3898:
3895:
3891:
3883:
3867:
3863:
3860:
3852:
3849:
3845:
3841:
3838:
3835:
3827:
3823:
3814:
3803:
3799:
3793:
3788:
3785:
3782:
3778:
3774:
3766:
3762:
3752:
3748:
3739:
3734:
3721:
3718:
3714:
3708:
3703:
3700:
3697:
3693:
3687:
3682:
3679:
3676:
3672:
3664:
3663:
3662:
3659:
3652:
3648:
3644:
3637:
3627:
3623:
3619:
3612:
3602:
3598:
3588:
3584:
3577:
3573:
3569:
3563:
3559:
3548:
3524:
3510:
3506:
3502:
3499:
3495:
3491:
3484:
3468:
3464:
3461:
3453:
3450:
3446:
3435:
3432:
3426:
3420:
3413:
3412:
3411:
3401:
3394:
3390:
3380:
3372:
3368:
3359:
3346:
3344:
3330:
3288:
3284:
3280:
3277:
3262:
3258:
3251:
3248:
3242:
3236:
3230:
3217:
3213:
3209:
3206:
3191:
3187:
3180:
3177:
3171:
3165:
3159:
3152:
3151:
3150:
3142:
3138:
3134:
3123:
3119:
3115:
3104:
3100:
3096:
3092:
3079:
3075:
3065:
3061:
3057:
3052:
3036:
3033:
3030:
3022:
3018:
3009:
2998:
2994:
2990:
2982:
2978:
2968:
2964:
2955:
2950:
2937:
2934:
2930:
2922:
2921:
2920:
2919:
2888:
2885:
2877:
2873:
2866:
2862:
2849:
2835:
2831:
2827:
2814:
2813:
2812:
2795:
2792:
2784:
2780:
2771:
2760:
2756:
2752:
2744:
2740:
2730:
2726:
2717:
2712:
2699:
2696:
2692:
2684:
2683:
2682:
2656:
2653:
2650:
2634:
2629:
2627:
2608:
2598:
2590:
2587:
2581:
2575:
2572:
2566:
2558:
2554:
2546:
2545:
2544:
2526:
2522:
2517:
2507:
2497:
2481:
2478:
2451:
2448:
2425:
2422:
2405:
2401:
2390:
2386:
2369:
2366:
2360:
2357:
2347:
2343:
2312:
2308:
2291:
2288:
2285:
2262:
2255:
2251:
2246:
2242:
2239:
2234:
2230:
2219:
2218:
2217:
2200:
2193:
2189:
2184:
2180:
2177:
2171:
2165:
2162:
2156:
2148:
2144:
2136:
2135:
2134:
2130:
2126:
2105:
2102:
2099:
2076:
2073:
2067:
2061:
2041:
2020:
2017:
2014:
2005:
1999:
1995:
1989:
1985:
1981:
1976:
1966:
1963:
1955:
1952:
1942:
1941:
1940:
1918:
1915:
1912:
1903:
1885:
1882:
1879:
1874:
1864:
1861:
1853:
1850:
1821:
1811:
1808:
1800:
1797:
1766:
1763:
1760:
1757:
1737:
1734:
1731:
1705:
1702:
1699:
1694:
1684:
1681:
1673:
1670:
1660:
1659:
1658:
1651:
1624:
1621:
1618:
1612:
1603:
1598:
1583:
1569:
1566:
1560:
1551:
1548:
1538:
1537:
1536:
1510:
1507:
1504:
1488:
1483:
1481:
1478:
1474:
1453:
1450:
1447:
1435:
1431:
1427:
1422:
1417:
1414:
1397:
1393:
1383:
1373:
1361:
1357:
1353:
1348:
1343:
1340:
1323:
1319:
1309:
1295:
1294:
1293:
1272:
1269:
1266:
1258:
1254:
1245:
1234:
1230:
1224:
1219:
1216:
1213:
1209:
1205:
1197:
1193:
1183:
1179:
1170:
1165:
1152:
1149:
1145:
1139:
1134:
1131:
1128:
1124:
1118:
1113:
1110:
1107:
1103:
1095:
1094:
1093:
1087:
1085:
1060:
1057:
1054:
1046:
1042:
1033:
1022:
1018:
1012:
1007:
1004:
1001:
997:
993:
985:
981:
971:
967:
958:
953:
940:
937:
933:
927:
922:
919:
916:
912:
906:
901:
898:
895:
891:
883:
882:
881:
864:
861:
858:
850:
846:
837:
826:
822:
816:
811:
808:
805:
801:
797:
789:
785:
775:
771:
762:
757:
744:
741:
737:
731:
726:
723:
720:
716:
710:
705:
702:
699:
695:
687:
686:
685:
682:
663:
640:
637:
622:
618:
614:
611:
608:
602:
590:
587:
584:
579:
571:
567:
563:
557:
553:
541:
537:
531:
526:
523:
520:
516:
508:
507:
506:
490:
486:
479:
475:
466:
453:
449:
445:
440:
436:
425:
421:
389:
386:
378:
374:
364:
360:
351:
346:
333:
330:
326:
320:
315:
312:
309:
305:
299:
294:
291:
288:
284:
276:
275:
274:
267:
250:
245:
243:
241:
237:
234:
230:
226:
221:
219:
213:
203:
179:
160:
157:
149:
145:
136:
131:
120:
112:
108:
99:
94:
79:
78:
77:
73:
69:
65:
59:
55:
53:
49:
45:
41:
37:
33:
26:
22:
5248:
5152:
5073:
5054:
5048:
5014:
5010:
5005:
4999:
4995:
4987:
4983:
4980:
4964:
4768:function on
4764:
4642:
4638:
4633:
4629:
4618:
4614:
4607:
4603:
4580:
4575:
4543:
4527:
4331:function on
4327:
4308:
4304:
4299:
4295:
4284:
4280:
4273:
4269:
4246:
4242:
4220:
4211:
4198:
4191:
4188:
3660:
3650:
3646:
3642:
3635:
3625:
3621:
3617:
3610:
3600:
3596:
3586:
3582:
3575:
3571:
3567:
3561:
3557:
3546:
3539:
3399:
3392:
3388:
3378:
3370:
3366:
3357:
3350:
3328:
3313:
3140:
3136:
3132:
3121:
3117:
3113:
3102:
3098:
3094:
3090:
3087:
3077:
3073:
3063:
3059:
3055:
2871:
2864:
2860:
2847:
2833:
2829:
2825:
2810:
2638:
2625:
2508:
2495:
2494:This is the
2403:
2399:
2388:
2384:
2345:
2341:
2310:
2306:
2277:
2215:
2128:
2124:
2042:
2035:
1904:
1720:
1652:
1640:
1492:
1476:
1472:
1468:
1287:
1091:
1080:rather than
1075:
879:
683:
664:
655:
488:
484:
477:
473:
454:
447:
443:
438:
434:
423:
419:
404:
269:function on
265:
254:
222:
214:
201:
177:
175:
71:
67:
63:
60:
56:
51:
47:
46:satisfy the
43:
39:
29:
5079:Rockafellar
5013:″ - 2
4986:″ + 2
4589:. For each
4255:. For each
2133:defined by
5270:Categories
5089:References
4772:such that
4335:such that
3640:such that
3615:such that
3555:, one has
2408:, for any
497:from 1 to
273:such that
236:convex set
76:such that
5142:Textbooks
4947:≥
4931:∂
4923:∂
4890:∑
4870:∂
4856:∂
4842:∂
4805:∑
4784:∑
4750:for each
4731:λ
4683:λ
4510:≥
4494:∂
4486:∂
4453:∑
4433:∂
4419:∂
4405:∂
4368:∑
4347:∑
4144:−
4097:∑
4090:−
4057:∑
4050:−
4025:−
3983:−
3924:∑
3903:∑
3899:α
3864:−
3853:α
3850:−
3842:α
3839:ε
3820:∂
3812:∂
3779:∑
3759:∂
3745:∂
3731:∂
3694:∑
3673:∑
3503:α
3500:−
3492:−
3465:−
3454:α
3451:−
3436:ε
3281:−
3249:−
3231:≥
3210:−
3178:−
3160:−
3031:≥
3015:∂
3007:∂
2975:∂
2961:∂
2947:∂
2895:→
2892:Ω
2793:≥
2777:∂
2769:∂
2737:∂
2723:∂
2709:∂
2660:→
2476:∂
2446:∂
2420:∂
2364:∂
2361:∪
2227:Δ
2097:Δ
2074:≤
2059:Δ
1986:∫
1956:−
1950:Δ
1922:→
1880:−
1848:Δ
1795:Δ
1735:≤
1729:Δ
1668:Δ
1619:≤
1595:∇
1514:→
1448:≤
1428:−
1415:−
1390:∂
1380:∂
1354:−
1341:−
1316:∂
1306:∂
1267:≤
1251:∂
1243:∂
1210:∑
1190:∂
1176:∂
1162:∂
1125:∑
1104:∑
1055:≥
1039:∂
1031:∂
998:∑
978:∂
964:∂
950:∂
913:∑
892:∑
843:∂
835:∂
802:∑
782:∂
768:∂
754:∂
717:∑
696:∑
538:λ
517:∑
371:∂
357:∂
343:∂
306:∑
285:∑
246:Intuition
142:∂
128:∂
105:∂
91:∂
5247:(1970).
5023:See also
3629:. Since
3314:for all
2916:and with
2837:for all
2412:, is on
669:), that
240:boundary
218:gradient
212:itself.
5081:(1970).
4969:, then
4532:, then
2832:) <
233:compact
5259:
5237:
5227:
5217:
5186:
5173:
5159:
5061:
4601:, let
4267:, let
3638:> 0
3613:> 0
3397:; let
2051:where
677:where
493:, for
200:. The
38:, the
5040:Notes
4758:. If
4627:with
4293:with
3624:<
3599:<
3592:) = 0
3347:Proof
3111:with
3105:) ≥ 0
3088:Then
3083:) = 0
3071:with
2858:with
2304:then
432:with
263:be a
231:on a
5257:ISBN
5235:ISBN
5225:ISBN
5215:ISBN
5184:ISBN
5171:ISBN
5157:ISBN
5059:ISBN
4698:and
4657:λ(x)
4612:and
4593:and
4581:Let
4278:and
4259:and
4247:Let
2870:) =
2639:Let
2339:and
2289:>
1750:and
1493:Let
1078:>
409:and
176:The
34:and
5200:pp.
5017:= 0
5002:= 0
4990:= 0
4965:on
4651:in
4569:in
4528:on
4317:in
4209:at
4196:at
3578:≤ 0
3549:= 0
3339:on
3318:in
3145:on
3143:≤ 0
3124:≤ 0
3107:on
2819:of
459:in
5272::
5206:;
5000:cu
4998:+
4643:ji
4637:=
4634:ij
4608:ij
4573:.
4309:ji
4303:=
4300:ij
4274:ij
4235:.
3658:.
3649:≤
3645:+
3620:+
3574:−
3570:+
3560:≤
3139:−
3135:+
3120:−
3116:+
3101:−
3097:+
3062:≤
3058:+
2796:0.
641:0.
452:.
448:ji
442:=
439:ij
424:ij
417:,
242:.
161:0.
54:.
5165:.
5137:)
5110:R
5067:.
5015:y
5011:y
5006:c
4996:u
4994:Δ
4988:y
4984:y
4977:.
4975:M
4971:u
4967:M
4950:0
4939:i
4935:x
4926:u
4915:i
4911:b
4905:n
4900:1
4897:=
4894:i
4886:+
4878:j
4874:x
4864:i
4860:x
4851:u
4846:2
4833:j
4830:i
4826:a
4820:n
4815:1
4812:=
4809:j
4799:n
4794:1
4791:=
4788:i
4770:M
4765:C
4760:u
4756:n
4752:i
4748:M
4726:|
4720:i
4716:b
4711:|
4678:i
4675:i
4671:a
4653:M
4649:x
4639:a
4630:a
4625:M
4619:i
4615:b
4604:a
4599:n
4595:j
4591:i
4587:ℝ
4583:M
4571:M
4567:x
4563:λ
4559:α
4555:λ
4551:x
4547:λ
4540:.
4538:M
4534:u
4530:M
4513:0
4502:i
4498:x
4489:u
4478:i
4474:b
4468:n
4463:1
4460:=
4457:i
4449:+
4441:j
4437:x
4427:i
4423:x
4414:u
4409:2
4396:j
4393:i
4389:a
4383:n
4378:1
4375:=
4372:j
4362:n
4357:1
4354:=
4351:i
4333:M
4328:C
4323:u
4319:M
4315:x
4305:a
4296:a
4291:M
4285:i
4281:b
4270:a
4265:n
4261:j
4257:i
4253:ℝ
4249:M
4233:M
4229:u
4224:0
4221:x
4215:0
4212:x
4207:u
4202:0
4199:x
4194:h
4174:.
4170:)
4164:)
4157:i
4152:c
4148:x
4139:i
4135:x
4129:(
4122:i
4118:b
4112:n
4107:1
4104:=
4101:i
4093:2
4085:i
4082:i
4078:a
4072:n
4067:1
4064:=
4061:i
4053:2
4045:)
4038:j
4033:c
4029:x
4020:j
4016:x
4010:(
4003:)
3996:i
3991:c
3987:x
3978:i
3974:x
3968:(
3963:)
3960:x
3957:(
3952:j
3949:i
3945:a
3939:n
3934:1
3931:=
3928:j
3918:n
3913:1
3910:=
3907:i
3896:4
3892:(
3884:2
3879:|
3872:c
3868:x
3861:x
3857:|
3846:e
3836:=
3828:i
3824:x
3815:h
3804:i
3800:b
3794:n
3789:1
3786:=
3783:i
3775:+
3767:j
3763:x
3753:i
3749:x
3740:h
3735:2
3722:j
3719:i
3715:a
3709:n
3704:1
3701:=
3698:j
3688:n
3683:1
3680:=
3677:i
3656:Ω
3651:C
3647:h
3643:u
3636:ε
3631:h
3626:C
3622:δ
3618:u
3611:δ
3606:u
3601:C
3597:u
3590:0
3587:x
3585:(
3583:h
3576:C
3572:h
3568:u
3562:C
3558:u
3553:R
3547:h
3542:Ω
3525:.
3520:)
3511:2
3507:R
3496:e
3485:2
3480:|
3473:c
3469:x
3462:x
3458:|
3447:e
3441:(
3433:=
3430:)
3427:x
3424:(
3421:h
3408:ρ
3403:0
3400:x
3395:)
3393:C
3391:(
3389:u
3384:R
3379:M
3377:∂
3373:)
3371:C
3369:(
3367:u
3361:c
3358:x
3353:Ω
3341:M
3337:u
3332:0
3329:x
3324:h
3320:Ω
3316:x
3295:|
3289:0
3285:x
3278:x
3274:|
3268:)
3263:0
3259:x
3255:(
3252:h
3246:)
3243:x
3240:(
3237:h
3224:|
3218:0
3214:x
3207:x
3203:|
3197:)
3192:0
3188:x
3184:(
3181:u
3175:)
3172:x
3169:(
3166:u
3147:Ω
3141:C
3137:h
3133:u
3128:Ω
3122:C
3118:h
3114:u
3109:Ω
3103:C
3099:h
3095:u
3093:(
3091:L
3081:0
3078:x
3076:(
3074:h
3069:Ω
3064:C
3060:h
3056:u
3037:,
3034:0
3023:i
3019:x
3010:h
2999:i
2995:b
2991:+
2983:j
2979:x
2969:i
2965:x
2956:h
2951:2
2938:j
2935:i
2931:a
2914:Ω
2899:R
2889::
2886:h
2875:.
2872:C
2868:0
2865:x
2863:(
2861:u
2856:Ω
2851:0
2848:x
2843:Ω
2839:x
2834:C
2830:x
2828:(
2826:u
2821:M
2817:Ω
2785:i
2781:x
2772:u
2761:i
2757:b
2753:+
2745:j
2741:x
2731:i
2727:x
2718:u
2713:2
2700:j
2697:i
2693:a
2679:C
2664:R
2657:M
2654::
2651:u
2641:M
2609:2
2604:|
2599:x
2595:|
2591:s
2588:+
2585:)
2582:x
2579:(
2576:u
2573:=
2570:)
2567:x
2564:(
2559:s
2555:u
2527:1
2523:x
2518:e
2504:M
2500:u
2482:.
2479:M
2466:u
2452:,
2449:M
2426:.
2423:M
2410:s
2404:s
2400:u
2395:M
2389:s
2385:u
2370:.
2367:M
2358:M
2346:s
2342:u
2337:u
2333:u
2329:M
2325:M
2321:u
2317:s
2311:s
2307:u
2292:0
2286:s
2263:.
2256:1
2252:x
2247:e
2243:s
2240:=
2235:s
2231:u
2201:.
2194:1
2190:x
2185:e
2181:s
2178:+
2175:)
2172:x
2169:(
2166:u
2163:=
2160:)
2157:x
2154:(
2149:s
2145:u
2129:s
2125:u
2120:s
2106:0
2103:=
2100:u
2077:0
2071:)
2068:p
2065:(
2062:u
2049:p
2045:u
2038:u
2021:,
2018:x
2015:d
2009:)
2006:x
2003:(
2000:u
1996:e
1990:M
1982:=
1977:4
1972:|
1967:u
1964:d
1960:|
1953:u
1926:R
1919:M
1916::
1913:u
1900:u
1886:,
1883:2
1875:2
1870:|
1865:u
1862:d
1858:|
1854:=
1851:u
1838:u
1822:2
1817:|
1812:u
1809:d
1805:|
1801:=
1798:u
1785:u
1781:u
1767:0
1764:=
1761:u
1758:d
1738:0
1732:u
1706:,
1703:2
1700:+
1695:2
1690:|
1685:u
1682:d
1678:|
1674:=
1671:u
1655:u
1647:u
1643:u
1625:,
1622:0
1616:)
1613:p
1610:(
1607:)
1604:u
1599:2
1591:(
1570:0
1567:=
1564:)
1561:p
1558:(
1555:)
1552:u
1549:d
1546:(
1533:p
1518:R
1511:M
1508::
1505:u
1495:M
1477:y
1475:−
1473:x
1471:−
1454:,
1451:0
1443:)
1436:2
1432:y
1423:2
1418:x
1408:(
1398:2
1394:y
1384:2
1374:+
1369:)
1362:2
1358:y
1349:2
1344:x
1334:(
1324:2
1320:x
1310:2
1290:u
1273:,
1270:0
1259:i
1255:x
1246:u
1235:i
1231:b
1225:n
1220:1
1217:=
1214:i
1206:+
1198:j
1194:x
1184:i
1180:x
1171:u
1166:2
1153:j
1150:i
1146:a
1140:n
1135:1
1132:=
1129:j
1119:n
1114:1
1111:=
1108:i
1082:≥
1061:,
1058:0
1047:i
1043:x
1034:u
1023:i
1019:b
1013:n
1008:1
1005:=
1002:i
994:+
986:j
982:x
972:i
968:x
959:u
954:2
941:j
938:i
934:a
928:n
923:1
920:=
917:j
907:n
902:1
899:=
896:i
865:,
862:0
859:=
851:i
847:x
838:u
827:i
823:b
817:n
812:1
809:=
806:i
798:+
790:j
786:x
776:i
772:x
763:u
758:2
745:j
742:i
738:a
732:n
727:1
724:=
721:j
711:n
706:1
703:=
700:i
679:u
675:M
671:u
667:a
659:u
638:=
633:)
628:)
623:i
619:v
615:t
612:+
609:x
606:(
603:u
598:(
591:0
588:=
585:t
580:|
572:2
568:t
564:d
558:2
554:d
542:i
532:n
527:1
524:=
521:i
503:x
499:n
495:i
489:i
485:v
478:i
474:λ
469:ℝ
461:M
457:x
444:a
435:a
430:M
420:a
415:n
411:j
407:i
390:0
387:=
379:j
375:x
365:i
361:x
352:u
347:2
334:j
331:i
327:a
321:n
316:1
313:=
310:j
300:n
295:1
292:=
289:i
271:M
266:C
261:u
257:M
210:M
206:u
198:M
194:M
190:u
186:u
182:M
158:=
150:2
146:y
137:u
132:2
121:+
113:2
109:x
100:u
95:2
74:)
72:y
70:,
68:x
66:(
64:u
52:D
44:D
27:.
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