25:
2502:
90:
1617:
2582:
being used to mean that an upper bound exists on the symbol of the operator as well. It is important to check the definitions the author is using, as conventions may differ. See, e.g., Evans, Chapter 6, for a use of the first definition, and
Gilbarg and Trudinger, Chapter 3, for a use of the
1429:
451:
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1412:
989:
879:
1723:
738:
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1083:
364:
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is a uniformly elliptic operator. The
Laplace operator occurs frequently in electrostatics. If ρ is the charge density within some region Ω, the potential Φ must satisfy the equation
1612:{\displaystyle \tau _{ij}=B\left(\sum _{k,l=1}^{3}\left(\partial _{l}u_{k}\right)^{2}\right)^{-{\frac {1}{3}}}\cdot {\frac {1}{2}}\left(\partial _{j}u_{i}+\partial _{i}u_{j}\right)}
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is elliptic. This is the most general form of a second-order divergence form linear elliptic differential operator. The
Laplace operator is obtained by taking
585:
666:
2748:
1017:
125:. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the
54:
2471:
can square to become a strongly elliptic operator, such as the
Laplacian. The composition of weakly elliptic operators is weakly elliptic.
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1937:
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2675:, Grundlehren der Mathematischen Wissenschaften, vol. 224 (2nd ed.), Berlin, New York: Springer-Verlag,
2447:
needs to be of even order for strong ellipticity to even be an option. Otherwise, just consider plugging in both
2743:
2605:
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has the appropriate boundary values and normal derivatives. The existence theory for elliptic operators, using
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1881:
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2068:
1942:
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2331:
130:
37:
446:{\displaystyle \partial ^{\alpha }u=\partial _{1}^{\alpha _{1}}\cdots \partial _{n}^{\alpha _{n}}u}
41:
33:
456:
2272:
2550:
2535:
1268:{\displaystyle Lu=-\partial _{i}\left(a^{ij}(x)\partial _{j}u\right)+b^{j}(x)\partial _{j}u+cu}
58:
2490:, and to guarantee that the eigenvalues are discrete, and their only limit point is infinity.
1407:{\displaystyle L(u)=-\sum _{i=1}^{d}\partial _{i}\left(|\nabla u|^{p-2}\partial _{i}u\right).}
2515:
2001:
1850:
1791:
1623:. The velocity of an ice sheet in steady state will then solve the nonlinear elliptic system
541:
203:
193:
118:
98:
94:
2155:
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1996:
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of an elliptic operator is infinitely differentiable in any neighborhood not containing 0.
1854:
1419:
483:
367:
2721:
2467:
and its negative. On the other hand, a weakly elliptic first-order operator, such as the
2450:
2363:
2108:
2021:
1978:
984:{\displaystyle L(u)=F\left(x,u,\left(\partial ^{\alpha }u\right)_{|\alpha |\leq m}\right)}
561:
8:
874:{\displaystyle (-1)^{k}\sum _{|\alpha |=2k}a_{\alpha }(x)\xi ^{\alpha }>C|\xi |^{2k},}
149:
145:
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2131:
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is a positive constant. Note that ellipticity only depends on the highest-order terms.
513:
175:
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2613:
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2501:
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1763:
89:
1853:; thus, every elliptic operator is hypoelliptic. The property also means that every
2651:
2540:
1718:{\displaystyle \sum _{j=1}^{3}\partial _{j}\tau _{ij}+\rho g_{i}-\partial _{i}p=Q,}
1293:
a non-negative number, the p-Laplacian is a nonlinear elliptic operator defined by
137:
126:
122:
102:
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733:{\displaystyle \xi ^{\alpha }=\xi _{1}^{\alpha _{1}}\cdots \xi _{n}^{\alpha _{n}}}
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2623:
153:
1884:. Since the Cauchy-Riemann equations form an elliptic operator, it follows that
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2468:
141:
2737:
1933:
1802:
1795:
992:
156:(if the coefficients in the operator are smooth). Steady-state solutions to
301:{\displaystyle Lu=\sum _{|\alpha |\leq m}a_{\alpha }(x)\partial ^{\alpha }u}
2427:(or its negative, depending upon convention). It is not hard to see that
2102:
743:
In many applications, this condition is not strong enough, and instead a
656:{\displaystyle \sum _{|\alpha |=m}a_{\alpha }(x)\xi ^{\alpha }\neq 0,}
2424:
2324:
The definition of ellipticity in the previous part of the article is
1007:
2423:
The quintessential example of a (strongly) elliptic operator is the
1811:
This situation is ultimately unsatisfactory, as the weak solution
1283:. These operators also occur in electrostatics in polarized media.
1995:. (Basically, what we are doing is replacing the highest order
1849:
Any differential operator exhibiting this property is called a
999:
and its derivatives about any point is an elliptic operator.
1078:{\displaystyle -\Delta u=-\sum _{i=1}^{d}\partial _{i}^{2}u}
995:
is; i.e. the first-order Taylor expansion with respect to
2486:. On the other hand, we need strong ellipticity for the
1932:
be a (possibly nonlinear) differential operator between
359:{\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})}
2672:
Elliptic partial differential equations of second order
2474:
Weak ellipticity is nevertheless strong enough for the
1838:
square-integrable weak derivatives. In particular, if
129:
is invertible, or equivalently that there are no real
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might not have enough derivatives for the expression
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1139:) which is symmetric and positive definite for every
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is the most famous example of an elliptic operator.
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2260:{\displaystyle \left((v),v\right)\geq c\|v\|^{2}}
2735:
2730:at EqWorld: The World of Mathematical Equations.
2724:at EqWorld: The World of Mathematical Equations.
2664:
2637:"Partial differential equations, by L. C. Evans"
1778:and some appropriate boundary values, such that
46:but its sources remain unclear because it lacks
2608:, vol. 19 (2nd ed.), Providence, RI:
1842:is infinitely-often differentiable, then so is
1745:
164:equations generally solve elliptic equations.
2400:are elements of the vector bundle upon which
2644:Journal of the American Mathematical Society
2282:
2276:
2248:
2241:
1819:to be well-defined in the classical sense.
2380:are covector fields or one-forms, but the
1733:is the gravitational acceleration vector,
2655:
2634:
1119:{\displaystyle -\Delta \Phi =4\pi \rho .}
77:Learn how and when to remove this message
2526:Hyperbolic partial differential equation
453:denotes the partial derivative of order
152:implies that their solutions tend to be
88:
2749:Elliptic partial differential equations
2531:Parabolic partial differential equation
1414:A similar nonlinear operator occurs in
2736:
2696:
2521:Elliptic partial differential equation
1860:As an application, suppose a function
747:may be imposed for operators of order
2596:
2360:is an inner product. Notice that the
1907:
18:
2574:Note that this is sometimes called
13:
2005:
1754:be an elliptic operator of order 2
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136:Elliptic operators are typical of
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2094:{\displaystyle \sigma _{\xi }(D)}
1968:{\displaystyle \sigma _{\xi }(D)}
2500:
140:, and they appear frequently in
23:
2606:Graduate Studies in Mathematics
2353:{\displaystyle (\cdot ,\cdot )}
1131:Given a matrix-valued function
2602:Partial differential equations
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111:partial differential equations
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2657:10.1090/s0273-0979-00-00868-5
2610:American Mathematical Society
2590:
2546:Ultrahyperbolic wave equation
745:uniform ellipticity condition
16:Type of differential operator
2728:Nonlinear Elliptic Equations
1762:continuous derivatives. The
194:linear differential operator
7:
2704:Encyclopedia of Mathematics
2493:
2484:Atiyah–Singer index theorem
1975:with respect to a one-form
1824:elliptic regularity theorem
1746:Elliptic regularity theorem
473:{\displaystyle \alpha _{i}}
10:
2765:
2294:{\displaystyle \|\xi \|=1}
1826:guarantees that, provided
1758:with coefficients having 2
2722:Linear Elliptic Equations
1794:, only guarantees that a
2561:
1882:Cauchy–Riemann equations
32:This article includes a
2697:Shubin, M. A. (2001) ,
2011:{\displaystyle \nabla }
1936:of any rank. Take its
551:{\displaystyle \Omega }
213:{\displaystyle \Omega }
61:more precise citations.
2744:Differential operators
2551:Semi-elliptic operator
2536:Hopf maximum principle
2461:
2441:
2414:
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2374:
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2315:
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2171:{\displaystyle c>0}
2142:
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2012:
1989:
1969:
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1830:is square-integrable,
1770:is to find a function
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119:differential operators
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2516:Hypoelliptic operator
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2152:if for some constant
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2013:
1997:covariant derivatives
1990:
1970:
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1851:hypoelliptic operator
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1422:of ice, according to
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888:A nonlinear operator
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500:{\displaystyle x_{i}}
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2476:Fredholm alternative
2460:{\displaystyle \xi }
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2373:{\displaystyle \xi }
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2031:{\displaystyle \xi }
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1988:{\displaystyle \xi }
1979:
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1888:
1864:
1855:fundamental solution
1788:Gårding's inequality
1737:is the pressure and
1729:is the ice density,
1627:
1430:
1420:Cauchy stress tensor
1297:
1151:
1143:, having components
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1018:
1006:The negative of the
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571:{\displaystyle \xi }
562:
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121:that generalize the
2699:"Elliptic operator"
2580:uniform ellipticity
2105:for every non-zero
1774:, given a function
1071:
991:is elliptic if its
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704:
558:and every non-zero
439:
414:
150:Elliptic regularity
146:continuum mechanics
2635:Rauch, J. (2000).
2576:strict ellipticity
2508:Mathematics portal
2480:Schauder estimates
2457:
2437:
2410:
2390:
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2326:strong ellipticity
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2008:
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1908:General definition
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1834:will in fact have
1741:is a forcing term.
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115:elliptic operators
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95:Laplace's equation
34:list of references
2682:978-3-540-13025-3
2619:978-0-8218-4974-3
2488:maximum principle
2440:{\displaystyle D}
2413:{\displaystyle D}
2393:{\displaystyle v}
2314:{\displaystyle v}
2150:strongly elliptic
2141:{\displaystyle D}
2054:{\displaystyle D}
2018:by vector fields
1925:{\displaystyle D}
1897:{\displaystyle f}
1873:{\displaystyle f}
1792:Lax–Milgram lemma
1764:Dirichlet problem
1554:
1539:
1416:glacier mechanics
781:
589:
523:{\displaystyle L}
240:
185:{\displaystyle L}
109:In the theory of
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2667:Trudinger, N. S.
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2101:is a linear
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53:Please help
45:
2103:isomorphism
1904:is smooth.
368:multi-index
168:Definitions
59:introducing
2738:Categories
2591:References
2482:, and the
530:is called
366:denotes a
158:hyperbolic
2709:EMS Press
2669:(1983) ,
2600:(2010) ,
2455:ξ
2425:Laplacian
2368:ξ
2345:⋅
2339:⋅
2283:‖
2280:ξ
2277:‖
2249:‖
2242:‖
2236:≥
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2195:σ
2113:ξ
2078:ξ
2074:σ
2026:ξ
2006:∇
1983:ξ
1952:ξ
1948:σ
1695:∂
1691:−
1678:ρ
1663:τ
1653:∂
1632:∑
1586:∂
1563:∂
1544:⋅
1529:−
1493:∂
1460:∑
1435:τ
1385:∂
1376:−
1360:∇
1341:∂
1320:∑
1316:−
1286:Example 3
1245:∂
1205:∂
1168:∂
1164:−
1128:Example 2
1111:ρ
1108:π
1099:Φ
1096:Δ
1093:−
1059:∂
1038:∑
1034:−
1025:Δ
1022:−
1014:given by
1008:Laplacian
1003:Example 1
969:≤
961:α
942:α
938:∂
851:ξ
835:α
831:ξ
816:α
792:α
783:∑
766:−
751: = 2
720:α
710:ξ
706:⋯
695:α
685:ξ
676:α
672:ξ
645:≠
640:α
636:ξ
621:α
600:α
591:∑
566:ξ
546:Ω
462:α
430:α
420:∂
416:⋯
405:α
395:∂
383:α
379:∂
345:α
338:…
326:α
316:α
291:α
287:∂
272:α
259:≤
251:α
242:∑
224:given by
208:Ω
196:of order
162:parabolic
2494:See also
2301:and all
2269:for all
1790:and the
532:elliptic
2691:0737190
2632:Review:
2628:2597943
2583:second.
2578:, with
2328:. Here
2128:We say
2041:We say
99:annulus
55:improve
2689:
2679:
2626:
2616:
2420:acts.
1780:Lu = f
1725:where
1418:. The
881:where
663:where
370:, and
308:where
101:. The
2640:(PDF)
2562:Notes
510:Then
192:be a
117:are
40:, or
2677:ISBN
2614:ISBN
2163:>
1912:Let
1822:The
1766:for
1750:Let
1289:For
840:>
172:Let
160:and
144:and
2652:doi
2065:if
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1010:in
578:in
538:in
480:in
220:in
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