Editing Elliptic operator

{{Short description|Type of differential operator}}
{{More footnotes needed|date=September 2024}}
[[File:Laplace's equation on an annulus.svg|right|thumb|300px|A solution to [[Laplace's equation]] defined on an [[Annulus (mathematics)|annulus]]. The [[Laplace operator]] is the most famous example of an elliptic operator.]]

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.

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==

Let <math>L</math> be a [[Differential operator|linear differential operator]] of order ''m'' on a domain <math>\Omega</math> in '''R'''<sup>''n''</sup> given by
<math display="block"> Lu = \sum_{|\alpha| \le m} a_\alpha(x)\partial^\alpha u </math>
where <math>\alpha = (\alpha_1, \dots, \alpha_n)</math> denotes a [[Multi-index notation|multi-index]], and <math>\partial^\alpha u = \partial^{\alpha_1}_1  \cdots \partial_n^{\alpha_n}u </math>  denotes the partial derivative of order <math>\alpha_i</math> in <math>x_i</math>.

Then <math>L</math> is called ''elliptic'' if for every ''x'' in <math>\Omega</math> and every non-zero <math>\xi</math> in '''R'''<sup>''n''</sup>,
<math display="block"> \sum_{|\alpha| = m} a_\alpha(x)\xi^\alpha \neq 0,</math>
where <math>\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n}</math>.

In many applications, this condition is not strong enough, and instead a ''uniform ellipticity condition'' may be imposed for operators of order ''m''&nbsp;= 2''k'':
<math display="block"> (-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 3, for a use of the second.</ref>

A nonlinear operator
<math display="block"> L(u) = F\left(x, u, \left(\partial^\alpha u\right)_{|\alpha| \le m}\right)</math>
is elliptic if its [[linearization]] is; i.e. the first-order Taylor expansion with respect to ''u'' and its derivatives about any point is an elliptic operator.

; Example 1: The negative of the [[Laplacian]] in '''R'''<sup>''d''</sup> given by <math display="block"> - \Delta u = - \sum_{i=1}^d \partial_i^2 u </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 display="block"> - \Delta \Phi = 4\pi\rho.</math>
; Example 2<ref>See Evans, Chapter 6-7, for details.</ref>: Given a matrix-valued function ''A''(''x'') which is uniformly positive definite for every ''x'', having components ''a''<sup>''ij''</sup>, the operator <math display="block"> Lu = -\partial_i\left(a^{ij}(x)\partial_ju\right) + 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 display="block"> L(u) = -\sum_{i = 1}^d\partial_i\left(|\nabla u|^{p - 2}\partial_i u\right).</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 display="block">\tau_{ij} = B\left(\sum_{k,l = 1}^3\left(\partial_lu_k\right)^2\right)^{-\frac{1}{3}} \cdot \frac{1}{2} \left(\partial_ju_i + \partial_iu_j\right)</math> for some constant ''B''. The velocity of an ice sheet in steady state will then solve the nonlinear elliptic system <math display="block">\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 theorems==

Let ''L'' be an elliptic operator of order 2''k'' with coefficients having 2''k'' 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]], [[Lax–Milgram lemma]] and [[Fredholm alternative]], states the sufficient condition for a [[weak solution]] ''u'' to exist in the [[Sobolev space]] ''H''<sup>''k''</sup>.

For example, for a Second-order Elliptic operator as in '''Example 2''', 
* There is a number ''γ>0'' such that for each ''μ>γ'', each <math>f\in L^2(U)</math>, there exists a unique solution <math>u\in H_{0}^{1}(U)</math> of the boundary value problem<br /><math>Lu+\mu u=f \text{ in }U, u=0\text{ on }\partial  U</math>, which is based on [[Lax-Milgram lemma]]. 
* Either (a) for any <math>f\in L^2(U)</math>, <math>Lu=f \text{ in }U, u=0\text{ on }\partial  U</math> (1) has a unique solution, or (b)<math>Lu=0 \text{ in }U, u=0\text{ on }\partial  U</math> has a solution <math>u\not\equiv 0</math>, which is based on the property of [[compact operator]]s and [[Fredholm alternative]].

This situation is ultimately unsatisfactory, as the weak solution ''u'' might not have enough derivatives for the expression ''Lu'' to be well-defined in the classical 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''.

For ''L'' as in '''Example 2''', 
* '''Interior regularity''': If ''m'' is a natural number, <math>a^{ij},b^{j},c \in C^{m+1}(U), f\in H^{m}(U)</math> (2) , <math>u\in H_{0}^{1}(U)</math> is a weak solution to (1), then for any open set ''V'' in ''U'' with compact closure, <math>\|u\|_{H^{m+2}(V)}\le C(\|f\|_{H^{m}(U)}+\|u\|_{L^2(U)})</math>(3), where ''C'' depends on ''U, V, L, m'', per se <math>u\in H_{loc}^{m+2}(U)</math>, which also holds if ''m'' is infinity by [[Sobolev inequality|Sobolev embedding theorem]].
* '''Boundary regularity''': (2) together with the assumption that <math>\partial U</math> is <math>C^{m+2}</math> indicates that (3) still holds after replacing ''V'' with ''U,'' i.e. <math>u\in H^{m+2}(U)</math>, which also holds if ''m'' is infinity.

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.

== Properties ==
For ''L'' as in '''Example 2''' on ''U'', which is an open domain with ''C<sup>1</sup>'' boundary, then there is a number ''γ''>0 such that for each ''μ''>''γ'', <math>L+\mu I: H_{0}^{1}(U)\rightarrow H_{0}^{1}(U)</math> satisfies the assumptions of  [[Lax–Milgram lemma]].

* Invertibility: For each ''μ''>''γ'', <math>L+\mu I:L^2(U)\rightarrow L^2(U)</math> admits a [[Compact operator|compact]] inverse.
* [[Eigenvalues and eigenvectors]]: If ''A'' is symmetric, ''b<sup>i</sup>,c'' are zero, then (1) Eigenvalues of ''L'',  are real, positive, countable, unbounded (2) There is an orthonormal basis of ''L<sup>2</sup>(U)'' composed of eigenvectors of ''L''. (See [[Spectral theorem]].)
* Generates a [[C0-semigroup|semigroup]] on ''L<sup>2</sup>(U)'': &minus;''L'' generates a semigroup <math>\{S(t);t\geq 0\}</math> of bounded linear operators on ''L<sup>2</sup>(U)'' s.t. <math>\frac{d}{dt}S(t)u_0=-LS(t)u_0, \|S(t)\|\leq e^{\gamma t}</math> in the norm of ''L<sup>2</sup>(U),'' for every <math>u_0\in L^2(U)</math>, by [[Hille–Yosida theorem]].

==General definition==

Let <math>D</math> be a (possibly nonlinear) differential operator between [[vector bundle]]s 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 display="block">\left([\sigma_\xi(D)](v), v\right) \geq c\|v\|^2 </math>

for all <math>\|\xi\|=1</math> and all <math>v</math>.

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 estimates]], 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==
{{Portal|Mathematics}}
* [[Sobolev space]]
* [[Hypoelliptic operator]]
* [[Elliptic partial differential equation]]
* [[Hyperbolic partial differential equation]]
* [[Parabolic partial differential equation]]
* [[Hopf maximum principle]]
* [[Elliptic complex]]
* [[Ultrahyperbolic wave equation]]
* [[Semi-elliptic operator]]
* [[Weyl's lemma (Laplace equation)|Weyl's lemma]]

==Notes==
{{reflist}}

==References==
*{{Citation | last1 = Evans | first1 = L. C. |author-link=Lawrence C. Evans | title = Partial differential equations | orig-year = 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 = https://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00868-5/S0273-0979-00-00868-5.pdf | doi=10.1090/s0273-0979-00-00868-5| doi-access = free }}
*{{Citation | last1 = Gilbarg | first1 = D. | last2 = Trudinger | first2 = N. S. | author2-link = Neil Trudinger | title = Elliptic partial differential equations of second order | orig-year = 1977 | url = https://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 }}
*{{SpringerEOM | 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.

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[[Category:Differential operators]]
[[Category:Elliptic partial differential equations| ]]