Editing Elliptic operator (section)

==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.