Elliptic operator
Encyclopedia
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 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
and parabolic
equations generally solve elliptic equations.
in Rd given by
is called elliptic if for every x in
and every non-zero 
in Rd,
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 highest-order terms.
A nonlinear operator
is elliptic if its first-order Taylor expansion with respect to u and its derivatives about any point is a linear elliptic operator.
Example
Another example
Yet another example
and the Lax–Milgram lemma, only guarantees that a weak solution
u exists in the Sobolev space
Hk.
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
satisfies the Cauchy-Riemann equations
. Since the Cauchy-Riemann equations form an elliptic operator, it follows that
is smooth.
be a (possibly nonlinear) differential operator between vector bundles of any rank. Take its principal symbol
with respect to a one-form 
. (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 non-zero
.
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 one-forms, 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 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.
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
s that generalize the Laplace operator
Laplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
. 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 characteristic
Method of characteristics
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation...
directions.
Elliptic operators are typical of potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
, and they appear frequently in electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
and continuum mechanics
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
. Elliptic regularity implies that their solutions tend to be smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
and parabolic
Parabolic partial differential equation
A parabolic partial differential equation is a type of second-order partial differential equation , describing a wide family of problems in science including heat diffusion, ocean acoustic propagation, in physical or mathematical systems with a time variable, and which behave essentially like heat...
equations generally solve elliptic equations.
Definitions
A linear differential operator L of order m on a domain in Rd given byis called elliptic if for every x in
and every non-zero in Rd,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 highest-order terms.
A nonlinear operator
is elliptic if its first-order Taylor expansion with respect to u and its derivatives about any point is a linear elliptic operator.
Example
- The negative of the Laplacian in Rd 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
Another example
- Given a matrix-valued function A(x) which is symmetric and positive definite for every x, having components aij, the operator
- is elliptic. This is the most general form of a second-order linear elliptic differential operator. The Laplace operator is obtained by taking A = I. These operators also occur in electrostatics in polarized media.
Yet another example
- For p a non-negative number, the p-Laplacian is a nonlinear elliptic operator defined by
- A similar nonlinear operator occurs in glacier mechanicsIce sheet dynamicsIce sheet dynamics describe the motion within large bodies of ice, such those currently on Greenland and Antarctica. Ice motion is dominated by the movement of glaciers, whose gravity-driven activity is controlled by two main variable factors: the temperature and strength of their bases...
. The 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 inequalityGårding's inequality
In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator...
and the Lax–Milgram lemma, only guarantees that a weak solution
Weak solution
In mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for...
u exists in the Sobolev space
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
Hk.
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
Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
of an elliptic operator is infinitely differentiable in any neighborhood not containing 0.
As an application, suppose a function
satisfies the Cauchy-Riemann equationsCauchy-Riemann equations
In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable...
. Since the Cauchy-Riemann 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 symbolSymbol of a differential operator
In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this...
with respect to a one-form . (Basically, what we are doing is replacing the highest order covariant derivativesCovariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
by vector fields .)We say
is weakly elliptic if is a linear isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
for every non-zero
.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 one-forms, 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 first-order operator, such as the Dirac operatorDirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
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
Fredholm alternative
In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators...
, Schauder estimates, and the Atiyah–Singer index theorem
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
. On the other hand, we need strong ellipticity for the maximum principle
Maximum principle
In mathematics, the maximum principle is a property of solutions to certain partial differential equations, of the elliptic and parabolic types. Roughly speaking, it says that the maximum of a function in a domain is to be found on the boundary of that domain...
, and to guarantee that the eigenvalues are discrete, and their only limit point is infinity.
See also
- Hopf maximum principleHopf maximum principleThe Hopf maximum principle is a maximum principle in the theory of second order elliptic partial differential equations and has been described as the "classic and bedrock result" of that theory...
- Elliptic complexElliptic complexIn mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for...
- Hyperbolic partial differential equationHyperbolic partial differential equationIn mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
- Ultrahyperbolic wave equationUltrahyperbolic wave equationIn the mathematical field of partial differential equations, the ultrahyperbolic equation is a partial differential equation for an unknown scalar function u of 2n variables x1, ..., xn, y1, ..., yn of the form...
- Parabolic partial differential equationParabolic partial differential equationA parabolic partial differential equation is a type of second-order partial differential equation , describing a wide family of problems in science including heat diffusion, ocean acoustic propagation, in physical or mathematical systems with a time variable, and which behave essentially like heat...
- Semi-elliptic operatorSemi-elliptic operatorIn mathematics — specifically, in the theory of partial differential equations — a semi-elliptic operator is a partial differential operator satisfying a positivity condition slightly weaker than that of being an elliptic operator...
- Weyl's lemmaWeyl's lemma (Laplace equation)In mathematics, Weyl's lemma is a result that provides a "very weak" form of the Laplace equation. It is named after the German mathematician Hermann Weyl.-Statement of the lemma:...
External links
- Linear Elliptic Equations at EqWorld: The World of Mathematical Equations.
- Nonlinear Elliptic Equations at EqWorld: The World of Mathematical Equations.