Elliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highestorder derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real 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 functions (if the coefficients in the operator are smooth). Steadystate solutions to hyperbolic and parabolic equations generally solve elliptic equations.
Contents
Definitions
A linear differential operator L of order m on a domain in R^{n} given by
(where is a multiindex, and ) is called elliptic if for every x in and every nonzero in R^{n},
where .
In many applications, this condition is not strong enough, and instead a uniform ellipticity condition may be imposed for operators of degree m = 2k:
where C is a positive constant. Note that ellipticity only depends on the highestorder terms.^{[1]}
A nonlinear operator
is elliptic if its firstorder 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^{d} given by
 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
 Example 2
 Given a matrixvalued function A(x) which is symmetric and positive definite for every x, having components a^{ij}, the operator
 is elliptic. This is the most general form of a secondorder 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 nonnegative number, the pLaplacian is a nonlinear elliptic operator defined by
 A similar nonlinear operator occurs in glacier mechanics. The Cauchy stress tensor of ice, according to Glen's flow law, is given by
 for some constant B. The velocity of an ice sheet in steady state will then solve the nonlinear elliptic system
 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^{k}.
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 squareintegrable, u will in fact have 2k squareintegrable weak derivatives. In particular, if f is infinitelyoften 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 satisfies the Cauchy–Riemann equations. Since the CauchyRiemann equations form an elliptic operator, it follows that is smooth.
General definition
Let be a (possibly nonlinear) differential operator between vector bundles of any rank. Take its principal symbol with respect to a oneform . (Basically, what we are doing is replacing the highest order covariant derivatives by vector fields .)
We say is weakly elliptic if is a linear isomorphism for every nonzero .
We say is (uniformly) strongly elliptic if for some constant ,
for all and all . It is important to note that the definition of ellipticity in the previous part of the article is strong ellipticity. Here is an inner product. Notice that the are covector fields or oneforms, but the are elements of the vector bundle upon which 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 needs to be of even order for strong ellipticity to even be an option. Otherwise, just consider plugging in both and its negative. On the other hand, a weakly elliptic firstorder 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
 Hopf maximum principle
 Elliptic complex
 Hyperbolic partial differential equation
 Ultrahyperbolic wave equation
 Parabolic partial differential equation
 Semielliptic operator
 Weyl's lemma
Notes
 ^ 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.
References

Evans, L. C. (2010) [1998], Partial differential equations, Graduate Studies in Mathematics, 19 (2nd ed.), Providence, RI: American Mathematical Society, ISBN 9780821849743, MR 2597943
Review:
Rauch, J. (2000). "Partial differential equations, by L. C. Evans" (pdf). Journal of the American Mathematical Society. 37 (3): 363–367. doi:10.1090/s0273097900008685.  Gilbarg, D.; Trudinger, N. S. (1983) [1977], Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, 224 (2nd ed.), Berlin, New York: SpringerVerlag, ISBN 9783540130253, MR 0737190
 Shubin, M. A. (2001) [1994], "Elliptic operator", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
External links
 Linear Elliptic Equations at EqWorld: The World of Mathematical Equations.
 Nonlinear Elliptic Equations at EqWorld: The World of Mathematical Equations.