Von Neumann's theorem: Difference between revisions

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In [[mathematics]] &mdash; specifically, in the theory of [[partial differential equation]]s &mdash; 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]].


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==Definition==
 
A second-order [[partial differential operator]] ''P'' defined on an [[open subset]] Ω of ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>, 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''<sub>''i''</sub>(''x''), 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''n'', of the [[matrix (mathematics)|matrix]] ''a''(''x'')&nbsp;=&nbsp;(''a''<sub>''ij''</sub>(''x'')) are non-negative. (By way of contrast, ''P'' is said to be elliptic if ''λ''<sub>''i''</sub>(''x'')&nbsp;&gt;&nbsp;0 for all ''x''&nbsp;∈&nbsp;Ω and 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''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''&nbsp;∈&nbsp;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]]

Revision as of 22:13, 12 December 2013

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

Pf(x)=i,j=1naij(x)2fxixj(x)+i=1nbi(x)fxi(x)+c(x)f(x),

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

References

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