Elliptic Operator
   HOME

TheInfoList



OR:

In the theory of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
, elliptic operators are differential operators 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 In mathematics, the symbol of a linear differential operator is a polynomial representing a differential operator, which is obtained, roughly speaking, by replacing each partial derivative by a new variable. The symbol of a differential operat ...
is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
, and they appear frequently in electrostatics and
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
.
Elliptic regularity In the theory of partial differential equations, a partial differential operator P defined on an open subset :U \subset^n is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty (smooth ...
implies that their solutions tend to be
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s (if the coefficients in the operator are smooth). Steady-state solutions to
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
and parabolic equations generally solve elliptic equations.


Definitions

Let L be
linear 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 that accepts a function and retur ...
of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^\alpha u where \alpha = (\alpha_1, \dots, \alpha_n) denotes a
multi-index Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
, and \partial^\alpha u = \partial^_1 \cdots \partial_n^u denotes the partial derivative of order \alpha_i in x_i. Then L is called ''elliptic'' if for every ''x'' in \Omega and every non-zero \xi in R''n'', \sum_ a_\alpha(x)\xi^\alpha \neq 0, where \xi^\alpha = \xi_1^ \cdots \xi_n^. In many applications, this condition is not strong enough, and instead a ''uniform ellipticity condition'' may be imposed for operators of order ''m'' = 2''k'': (-1)^k\sum_ a_\alpha(x) \xi^\alpha > C , \xi, ^, where ''C'' is a positive constant. Note that ellipticity only depends on the highest-order terms.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. A nonlinear operator L(u) = F\left(x, u, \left(\partial^\alpha u\right)_\right) is elliptic if its
linearization In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, lineari ...
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 mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
in R''d'' given by - \Delta u = - \sum_^d \partial_i^2 u 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 - \Delta \Phi = 4\pi\rho. ; Example 2: Given a matrix-valued function ''A''(''x'') which is symmetric and positive definite for every ''x'', having components ''a''''ij'', the operator Lu = -\partial_i\left(a^(x)\partial_ju\right) + b^j(x)\partial_ju + cu 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 L(u) = -\sum_^d\partial_i\left(, \nabla u, ^\partial_i u\right). A similar nonlinear operator occurs in glacier mechanics. The
Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
of ice, according to Glen's flow law, is given by \tau_ = B\left(\sum_^3\left(\partial_lu_k\right)^2\right)^ \cdot \frac \left(\partial_ju_i + \partial_iu_j\right) for some constant ''B''. The velocity of an ice sheet in steady state will then solve the nonlinear elliptic system \sum_^3\partial_j\tau_ + \rho g_i - \partial_ip = Q, 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 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 and the
Lax–Milgram lemma Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or c ...
, only guarantees that a
weak solution In mathematics, a weak solution (also called a generalized 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 precis ...
''u'' exists in the
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 together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
''H''''k''. 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''. Any differential operator exhibiting this property is called a
hypoelliptic operator In the theory of partial differential equations, a partial differential operator P defined on an open subset :U \subset^n is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty (sm ...
; thus, every elliptic operator is hypoelliptic. The property also means that every
fundamental solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ad ...
of an elliptic operator is infinitely differentiable in any neighborhood not containing 0. As an application, suppose a function f satisfies the
Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differenti ...
. Since the Cauchy-Riemann equations form an elliptic operator, it follows that f is smooth.


General definition

Let D be a (possibly nonlinear) differential operator between vector bundles of any rank. Take its
principal symbol In mathematics, the symbol of a linear differential operator is a polynomial representing a differential operator, which is obtained, roughly speaking, by replacing each partial derivative by a new variable. The symbol of a differential operat ...
\sigma_\xi(D) with respect to a one-form \xi. (Basically, what we are doing is replacing the highest order covariant derivatives \nabla by vector fields \xi.) We say D is ''weakly elliptic'' if \sigma_\xi(D) is a linear
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
for every non-zero \xi. We say D is (uniformly) ''strongly elliptic'' if for some constant c > 0, \left( sigma_\xi(D)v), v\right) \geq c\, v\, ^2 for all \, \xi\, =1 and all v. It is important to note that the definition of ellipticity in the previous part of the article is ''strong ellipticity''. Here (\cdot,\cdot) is an inner product. Notice that the \xi are covector fields or one-forms, but the v are elements of the vector bundle upon which D acts. The quintessential example of a (strongly) elliptic operator is the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
(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 \xi and its negative. On the other hand, a weakly elliptic first-order operator, such as the
Dirac 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. The original case which concerned Paul Dirac was to factorise forma ...
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 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 ...
, Schauder estimates, and the Atiyah–Singer index theorem. On the other hand, we need strong ellipticity for the
maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...
, and to guarantee that the eigenvalues are discrete, and their only limit point is infinity.


See also

* Elliptic partial differential equation *
Hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...
* Parabolic partial differential equation *
Hopf maximum principle The 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. Generalizing the maximum principle for harmonic functi ...
* Elliptic complex *
Ultrahyperbolic wave equation In the mathematical field of differential equations, the ultrahyperbolic equation is a partial differential equation (PDE) for an unknown scalar function of variables of the form \frac + \cdots + \frac - \frac - \cdots - \frac = 0. More ...
*
Semi-elliptic operator In 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. Ev ...
* Weyl's lemma


Notes


References

*
Review:
* *


External links


Linear Elliptic Equations
at EqWorld: The World of Mathematical Equations.
Nonlinear Elliptic Equations
at EqWorld: The World of Mathematical Equations. {{Authority control Differential operators