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Wikipedia, In mathematics, an elliptic operator is one of the major types of differential operator. It can be defined on spaces of complexvalued functions, or some more general functionlike objects. What is distinctive is that the coefficients of the highestorder derivatives satisfy a positivity condition. An important example of an elliptic operator is the Laplacian. Equations of the form are called elliptic partial differential equations if P is an elliptic operator. The usual partial differential equations involving time, such as the heat equation and the Schrödinger equation, also contain elliptic operators involving the spatial variables, as well as time derivatives. Elliptic operators are typical of potential theory. Their solutions (harmonic functions of a general kind) tend to be smooth functions (if the coefficients in the operator are continuous). More simply, steadystate solutions to hyperbolic and parabolic equations generally solve elliptic equations. Example: Second order operatorsFor expository purposes, we consider initially second order linear partial differential operators of the form where . Such an operator is called elliptic if for every x the matrix of coefficients of the highest order terms is a positivedefinite real symmetric matrix. In particular, for every nonzero vector the following ellipticity condition holds: In many applications, this condition is not strong enough, and instead a uniform ellipticity condition must be used: where C is a positive constant. Example. The negative of the Laplacian in R given by is a uniformly elliptic operator. Regularity propertiesLet P be an elliptic operator with infinitely differentiable coefficients. If f = P u is infinitely differentiable, then so is u. This is a very special case of a regularity theorem for a differential equation. Any differential operator (with constant coefficients) 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 off 0. As an application, suppose a function f satisfies the CauchyRiemann equations. Since f then satisfies the Laplacian in particular and since the Laplacian is an elliptic operator, it follows that f is infinitely differentiable. General DefinitionLet D be a (possibly nonlinear) differential operator between vector bundles of any rank. Take its principal symbol σ_{ξ}(D) with respect to a oneform ξ. (Basically, what we are doing is replacing the highest order covariant derivatives by vector fields ξ.) We say D is weakly elliptic if σ_{ξ}(D) is a linear isomorphism for every ξ. We say D is (uniformly) strongly elliptic if for some constant c > 0, for all and all v. 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 ξ are covector fields or oneforms, but the v are elements of the vector bundle upon which D 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 D needs to be of even order for strong ellipticity to even be an option. Otherwise, just consider plugging in both ξ 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 AtiyahSinger 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
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Published  July 2009
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