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

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

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

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...

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

{\displaystyle u:U\cup \partial U\rightarrow \mathbb {R} } and the elliptic operator L {\displaystyle L} is of the divergence form: L u ( x ) = − ∑ i ...

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...

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...

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

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...

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...

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...

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

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...

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

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

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...

winding number. Any elliptic operator on a closed manifold can be extended to a Fredholm operator. The use of Fredholm operators in partial differential...

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

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...

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...

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...

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...

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...

energy functionals in the calculus of variations. Solutions to a uniformly elliptic partial differential equation with divergence form ∇ ⋅ ( A ∇ u ) = 0 {\displaystyle...

elliptic operators on a manifold with the same principal symbol. Usually Weitzenböck formulae are implemented for G-invariant self-adjoint operators between...

_{i}\nabla _{j}f-(\Delta f)g_{ij}-fR_{ij},} and it is an overdetermined elliptic operator in the case of a Riemannian metric. It is a straightforward consequence...