|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In the theory of [[partial differential equations]], '''elliptic operators''' are [[differential operator]]s that generalize the [[Laplace operator]]. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the [[principal symbol]] is invertible, or equivalently that there are no real [[Method of characteristics|characteristic]] directions.
| | The person who wrote the text is called Eusebio. His friends say it's not good for him but [http://www.wired.com/search?query=precisely precisely] he loves doing typically is acting and he's been doing doing it for many years. Filing has been his profession for a short time. Massachusetts has always been his lifestyle place and his spouse and [http://Www.Adobe.com/cfusion/search/index.cfm?term=&children&loc=en_us&siteSection=home children] members loves it. Go to his website to discover a out more: http://prometeu.net<br><br>my page - how to hack clash of clans [[http://prometeu.net click through the up coming document]] |
| | |
| Elliptic operators are typical of [[potential theory]], and they appear frequently in [[electrostatics]] and [[continuum mechanics]]. [[Elliptic regularity]] implies that their solutions tend to be [[smooth function]]s (if the coefficients in the operator are smooth). Steady-state solutions to [[Hyperbolic partial differential equation|hyperbolic]] and [[Parabolic partial differential equation|parabolic]] equations generally solve elliptic equations.
| |
| | |
| ==Definitions ==
| |
| | |
| A linear differential operator ''L'' of order ''m'' on a domain <math>\Omega</math> in '''R'''<sup>''d''</sup> given by
| |
| | |
| :<math> Lu = \sum_{|\alpha| \le m} a_\alpha(x)\partial^\alpha u\, </math> | |
| | |
| (where <math>\alpha</math> is a [[Multi-index notation|multi-index]]) is called ''elliptic'' if for every ''x'' in <math>\Omega</math> and every non-zero <math>\xi</math> in '''R'''<sup>''d''</sup>,
| |
| | |
| :<math> \sum_{|\alpha| = m} a_\alpha(x)\xi^\alpha \neq 0.\,</math>
| |
| | |
| In many applications, this condition is not strong enough, and instead a ''uniform ellipticity condition'' may be imposed for operators of degree ''m = 2k'':
| |
| | |
| :<math> (-1)^k\sum_{|\alpha| = 2k} a_\alpha(x) \xi^\alpha > C |\xi|^{2k},\,</math>
| |
| | |
| where ''C'' is a positive constant. Note that ellipticity only depends on the highest-order terms.<ref>Note that this is sometimes called ''strict ellipticity'', with ''uniform ellipticity'' 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 6, for a use of the second.</ref>
| |
| | |
| A nonlinear operator
| |
| | |
| :<math> L(u) = F(x, u, (\partial^\alpha u)_{|\alpha| \le 2k})\,</math>
| |
| | |
| is elliptic if its first-order Taylor expansion with respect to ''u'' and its derivatives about any point is a linear elliptic operator.
| |
| | |
| ;Example 1
| |
| :The negative of the [[Laplacian]] in '''R'''<sup>''d''</sup> given by | |
| | |
| ::<math> - \Delta u = -\sum_{i=1}^d \partial_i^2u\, </math>
| |
| | |
| :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
| |
| | |
| ::<math> - \Delta \Phi = 4\pi\rho.\,</math>
| |
| | |
| ;Example 2
| |
| :Given a matrix-valued function ''A(x)'' which is symmetric and positive definite for every ''x'', having components ''a''<sup>''ij''</sup>, the operator
| |
| | |
| ::<math> Lu = -\partial_i(a^{ij}(x)\partial_ju) + b^j(x)\partial_ju + cu\, </math>
| |
| | |
| :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 ''A = I''. These operators also occur in electrostatics in polarized media.
| |
| | |
| ;Example 3
| |
| :For ''p'' a non-negative number, the p-Laplacian is a nonlinear elliptic operator defined by
| |
| | |
| ::<math> L(u) = -\sum_{i = 1}^d\partial_i (|\nabla u|^{p - 2}\partial_i u).\,</math>
| |
| | |
| :A similar nonlinear operator occurs in [[Ice sheet dynamics|glacier mechanics]]. The [[Cauchy stress tensor]] of ice, according to Glen's flow law, is given by
| |
| | |
| ::<math>\tau_{ij} = B\left(\sum_{k,l = 1}^3(\partial_lu_k)^2\right)^{-\frac{1}{3}}\cdot\frac{1}{2}(\partial_ju_i + \partial_iu_j)\,</math>
| |
| | |
| :for some constant ''B''. The velocity of an ice sheet in steady state will then solve the nonlinear elliptic system
| |
| | |
| ::<math>\sum_{j = 1}^3\partial_j\tau_{ij} + \rho g_i - \partial_ip = Q,\,</math>
| |
| | |
| :where ρ is the ice density, ''g'' is the gravitational acceleration vector, ''p'' is the pressure and ''Q'' is a forcing term.
| |
| | |
| == Elliptic regularity theorem == | |
| | |
| Let ''L'' be an elliptic operator of order ''2k'' with coefficients having ''2k'' continuous derivatives. The Dirichlet problem for ''L'' is to find a function ''u'', given a function ''f'' and some appropriate boundary values, such that ''Lu = f'' and such that ''u'' has the appropriate boundary values and normal derivatives. The existence theory for elliptic operators, using [[Gårding's inequality]] and the [[Lax–Milgram lemma]], only guarantees that a [[weak solution]] ''u'' exists in the [[Sobolev space]] ''H''<sup>''k''</sup>.
| |
| | |
| This situation is ultimately unsatisfactory, as the weak solution ''u'' might not have enough derivatives for the expression ''Lu'' to even make sense.
| |
| | |
| The ''elliptic regularity theorem'' guarantees that, provided ''f'' is square-integrable, ''u'' will in fact have ''2k'' square-integrable weak derivatives. In particular, if ''f'' is infinitely-often differentiable, then so is ''u''.
| |
| | |
| Any differential operator exhibiting this property is called a [[hypoelliptic operator]]; thus, every elliptic operator is hypoelliptic. The property also means that every [[fundamental solution]] of an elliptic operator is infinitely differentiable in any neighborhood not containing 0.
| |
| | |
| As an application, suppose a function <math>f</math> satisfies the [[Cauchy-Riemann equations]]. Since the Cauchy-Riemann equations form an elliptic operator, it follows that <math>f</math> is smooth.
| |
| | |
| ==General definition==
| |
| | |
| Let <math>D</math> be a (possibly nonlinear) differential operator between vector bundles of any rank. Take its [[Symbol of a differential operator|principal symbol]] <math>\sigma_\xi(D)</math> with respect to a one-form <math>\xi</math>. (Basically, what we are doing is replacing the highest order [[covariant derivative]]s <math>\nabla</math> by vector fields <math>\xi</math>.)
| |
| | |
| We say <math>D</math> is ''weakly elliptic'' if <math>\sigma_\xi(D)</math> is a linear [[isomorphism]] for every non-zero <math>\xi</math>.
| |
| | |
| We say <math>D</math> is (uniformly) ''strongly elliptic'' if for some constant <math>c>0</math>,
| |
| | |
| :<math>([\sigma_\xi(D)](v),v) \geq c\|v\|^2 </math> | |
| | |
| for all <math>\|\xi\|=1</math> and all <math>v</math>. It is important to note that the definition of ellipticity in the previous part of the article is ''strong ellipticity''. Here <math>(\cdot,\cdot)</math> is an inner product. Notice that the <math>\xi</math> are covector fields or one-forms, but the <math>v</math> are elements of the vector bundle upon which <math>D</math> acts.
| |
| | |
| The quintessential example of a (strongly) elliptic operator is the [[Laplacian]] (or its negative, depending upon convention). It is not hard to see that <math>D</math> needs to be of even order for strong ellipticity to even be an option. Otherwise, just consider plugging in both <math>\xi</math> and its negative. On the other hand, a weakly elliptic first-order operator, such as the [[Dirac operator]] can square to become a strongly elliptic operator, such as the Laplacian. The composition of weakly elliptic operators is weakly elliptic.
| |
| | |
| Weak ellipticity is nevertheless strong enough for the [[Fredholm alternative]], [[Schauder estimate]]s, and the [[Atiyah–Singer index theorem]]. On the other hand, we need strong ellipticity for the [[maximum principle]], and to guarantee that the eigenvalues are discrete, and their only limit point is infinity.
| |
| | |
| == See also ==
| |
| | |
| * [[Hopf maximum principle]]
| |
| * [[Elliptic complex]]
| |
| * [[Hyperbolic partial differential equation]]
| |
| * [[Ultrahyperbolic wave equation]]
| |
| * [[Parabolic partial differential equation]]
| |
| * [[Semi-elliptic operator]]
| |
| * [[Weyl's lemma (Laplace equation)|Weyl's lemma]]
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| == References ==
| |
| | |
| *{{Citation | last1 = Evans | first1 = L. C. | title = Partial differential equations | origyear = 1998 | publisher = [[American Mathematical Society]] | location = Providence, RI | edition = 2nd | series = Graduate Studies in Mathematics | isbn = 978-0-8218-4974-3 | mr = 2597943 | year = 2010 | volume = 19 }}<br> Review:<br> {{ cite journal | author = Rauch, J. | title = Partial differential equations, by L. C. Evans | journal = Journal of the American Mathematical Society | year = 2000 | volume = 37 | issue = 3 | pages = 363–367 | url = http://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00868-5/S0273-0979-00-00868-5.pdf | format = pdf }}
| |
| *{{Citation | last1 = Gilbarg | first1 = D. | last2 = Trudinger | first2 = N. S. | author2-link = Neil Trudinger | title = Elliptic partial differential equations of second order | origyear = 1977 | url = http://www.springer.com/mathematics/dyn.+systems/book/978-3-540-41160-4 | publisher = [[Springer-Verlag]] | location = Berlin, New York | edition = 2nd | series = Grundlehren der Mathematischen Wissenschaften | isbn = 978-3-540-13025-3 | mr = 737190 | year = 1983 | volume = 224 }}
| |
| *{{eom | first = M. A. | last = Shubin | id = Elliptic_operator | title = Elliptic operator }}
| |
| | |
| ==External links==
| |
| * [http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc3.pdf Linear Elliptic Equations] at EqWorld: The World of Mathematical Equations.
| |
| * [http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc3.pdf Nonlinear Elliptic Equations] at EqWorld: The World of Mathematical Equations.
| |
| | |
| [[Category:Differential operators]]
| |
| [[Category:Elliptic partial differential equations| ]]
| |