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
is said to be semi-elliptic if all the eigenvalues λi, 1 ≤ i ≤ n, of the matrix a = are non-negative. Equivalently, P is semi-elliptic if the matrix a is positive semi-definite for each x ∈ Ω.
