Von Neumann's theorem

2013-12-12T21:13:43Z

194.78.125.34: typo

In [[mathematics]] — specifically, in the theory of [[partial differential equation]]s — 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 problem]]s can be solved using [[Stochastic processes and boundary value problems|the methods of stochastic analysis]].

==Definition==

A second-order [[partial differential operator]] ''P'' defined on an [[open subset]] Ω of ''n''-[[dimension]]al [[Euclidean space]] '''R'''''n'', acting on suitable functions ''f'' by

:<math>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),</math>

is said to be '''semi-elliptic''' if all the [[eigenvalues]] ''λ''''i''(''x''), 1 ≤ ''i'' ≤ ''n'', of the [[matrix (mathematics)|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, ''P'' is semi-elliptic if the matrix ''a''(''x'') is [[Definite bilinear form|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)

[[Category:Differential operators]]
[[Category:Partial differential equations]]

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Von Neumann's theorem