In mathematics — specifically, in the theory of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

s — a semi-elliptic operator is a

partial differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

satisfying a positivity condition slightly weaker than that of being an

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which im ...

. 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 problem In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet pro ...

s can be solved using the methods of stochastic analysis.

Definition

A second-order

''P'' defined on an

open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...

Ω of ''n''-

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

R^''n'', acting on suitable functions ''f'' by :

P f(x) = \sum_^ a_ (x) \frac(x) + \sum_^ b_ (x) \frac (x) + c(x) f(x),

is said to be semi-elliptic if all the

eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

''λ''_''i''(''x''), 1 ≤ ''i'' ≤ ''n'', of the

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

''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 In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the famil ...

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

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