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.
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
Pf(x)=
| n | |
| \sum | |
| i,j=1 |
aij(x)
| \partial2f | |
| \partialxi\partialxj |
(x)+
| n | |
| \sum | |
| i=1 |
bi(x)
| \partialf | |
| \partialxi |
(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.
. Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications . Sixth. Springer. Berlin . 2003 . 3-540-04758-1. (See Section 9)