of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition...

the Laplacian operator has been used for various tasks, such as blob and edge detection. The Laplacian is the simplest elliptic operator and is at the...

Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the...

mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently...

semi-elliptic operator is a partial differential operator satisfying a positivity condition slightly weaker than that of being an elliptic operator. Every...

well-behaved comprises the pseudo-differential operators. The differential operator P {\displaystyle P} is elliptic if its symbol is invertible; that is for...

with an elliptic operator An elliptic partial differential equation This disambiguation page lists articles associated with the title Elliptic equation...

{\displaystyle P} is said to be analytically hypoelliptic. Every elliptic operator with C ∞ {\displaystyle C^{\infty }} coefficients is hypoelliptic...

papers from 1968 to 1971. Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space Y. In this case...

a pseudo-differential operator is a pseudo-differential operator. If a differential operator of order m is (uniformly) elliptic (of order m) and invertible...

are other ways to prove this.) Indeed, the operators Δ are elliptic, and the kernel of an elliptic operator on a closed manifold is always a finite-dimensional...

opposite of this winding number. Any elliptic operator can be extended to a Fredholm operator. The use of Fredholm operators in partial differential equations...

The zeta function of a mathematical operator O {\displaystyle {\mathcal {O}}} is a function defined as ζ O ( s ) = tr O − s {\displaystyle \zeta _{\mathcal...

equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features...

multi-dimensional parabolic PDE. Noting that − Δ {\displaystyle -\Delta } is an elliptic operator suggests a broader definition of a parabolic PDE: u t = − L u , {\displaystyle...

consider the negative of the Laplacian −Δ since as an operator it is non-negative; (see elliptic operator). Theorem — If n = 1, then −Δ has uniform multiplicity...

data. The argument goes as follows. A typical simple-to-understand elliptic operator L would be the Laplacian plus some lower order terms. Combined with...

of differential operator involved. For an elliptic operator, one discusses elliptic boundary value problems. For a hyperbolic operator, one discusses hyperbolic...

constraints in Hamiltonian mechanics Regularity of an elliptic operator Regularity theory of elliptic partial differential equations Regular algebra, or...

differential operator on sections of the bundle of differential forms on a pseudo-Riemannian manifold. On a Riemannian manifold it is an elliptic operator, while...

differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview...

inequality is a distributional inequality for the Laplace operator or certain elliptic operators. It was proven in 1972 by the Japanese mathematician Tosio...

frequently admits all of these interpretations, as follows. Given an elliptic operator L , {\displaystyle L,} the parabolic PDE u t = L u {\displaystyle...

Here, L stands for a linear differential operator. For example, one might take L to be an elliptic operator, such as L = d 2 d x 2 {\displaystyle L={\frac...

In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the steady state of an evolution...

of Hecke operators are called "Hecke algebras", and are commutative rings. In the classical elliptic modular form theory, the Hecke operators Tn with n...

that for the ordinary Poisson problem. In general, to each scalar elliptic operator L of order 2k, there is associated a bilinear form B on the Sobolev...

domain in R n {\displaystyle \mathbb {R} ^{n}} and consider the linear elliptic operator L u = ∑ i , j = 1 n a i j ( t , x ) ∂ 2 u ∂ x i ∂ x j + ∑ i = 1 n...

Markov process is a second-order elliptic operator. The infinitesimal generator of Brownian motion is the Laplace operator and the transition probability...

clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differential operators on free loop spaces. In...