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. Every elliptic operator is also semi-elliptic, and semi-elliptic operators share many of the nice properties of elliptic operators: for example, much of the same existence and uniqueness theory is applicable, and semi-elliptic Dirichlet problems can be solved using the methods of stochastic analysis.

Definition

A second-order partial differential operator P defined on an open subset Ω of n-dimensional Euclidean space Rn, acting on suitable functions f by

P f(x) = \sum_{i, j = 1}^{n} a_{ij} (x) \frac{\partial^{2} f}{\partial x_{i} \, \partial x_{j}}(x) + \sum_{i = 1}^{n} b_{i} (x) \frac{\partial f}{\partial x_{i}} (x) + c(x) f(x),

is said to be semi-elliptic if all the eigenvalues λi(x), 1  i  n, of the matrix a(x) = (aij(x)) are non-negative. (By way of contrast, P is said to be elliptic if λi(x) > 0 for all x  Ω and 1  i  n, and uniformly elliptic if the eigenvalues are uniformly bounded away from zero, uniformly in i and x.) Equivalently, P is semi-elliptic if the matrix a(x) is positive semi-definite for each x  Ω.

References

This article is issued from Wikipedia - version of the Thursday, August 06, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.