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 é of n-dimensional Euclidean space R<sup>n</sup>, acting on suitable functions f by
is said to be semi-elliptic if all the eigenvalues û<sub>i</sub>(x), 1 ⤠i ⤠n, of the matrix a(x) = (a<sub>ij</sub>(x)) are non-negative. (By way of contrast, P is said to be elliptic if û<sub>i</sub>(x) > 0 for all x â é 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 â é.