elliptic operator

From its association with a corresponding elliptic partial differential equation — a second-order linear partial differential equation with coefficients specified analogously to those of the equation of a conic section that generates an ellipse.

elliptic operator (plural elliptic operators)

1. (mathematical analysis) Any of a particular class of second-order linear differential operators.
   • 2001, L. Gross, Heat Kernel Analysis on Lie Groups, Laurent Decreusefond, Bernt Øksendal, Ali S. Üstünel (editors), Stochastic Analysis and Related Topics VII: Proceedings of the Seventh Silivri Workshop, Springer-Verlag (Birkhäuser), page 1,

     The study of second order elliptic operators over some of these infinite dimensional manifolds has been the subject of much work in the past forty years, some of it for the purposes of quantum field theory and some of it for the purposes of stochastic analysis or infinite dimensional differential topology.

   • 2008, Vladimir E. Nazaikinskii, Anton Yu. Savin, Boris Yu. Sternin, Elliptic Theory and Noncommutative Geometry: Nonlocal Elliptic Operators, Springer, page 37:

     Just as in the usual elliptic theory, in the theory of nonlocal elliptic operators we need not restrict ourselves to individual operators but can also consider families of nonlocal elliptic operators.

   • 2014, Lester L. Helms, Potential Theory, 2nd edition, Springer, page 412:

     The Dirichlet problem for an elliptic operator was solved in Chap. 10 by morphing a solution of the Dirichlet problem for the Laplacian on a ball into a solution of the Dirichlet problem for an elliptic operator on a ball.

• (second-order linear differential operator): Laplace operator

• hyperbolic operator, parabolic operator

• hypoelliptic operator

• elliptic operator on Wikipedia.Wikipedia
• Differential operator on Wikipedia.Wikipedia
• Elliptic partial differential equation on Wikipedia.Wikipedia
• Harmonic function on Wikipedia.Wikipedia
• Potential theory on Wikipedia.Wikipedia