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 R^"n", 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 "λ"_"i"("x"), 1 ≤ "i" ≤ "n", of the matrix "a"("x") = ("a"_"ij"("x")) are non-negative. (By way of contrast, "P" is said to be elliptic if "λ"_"i"("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, the matrix "a"("x") is positive semi-definite for each "x" ∈ Ω.

References

* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
isbn = 3-540-04758-1 (See Section 9)