Von Neumann's theorem

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

$Pf(x)=\sum _{i,j=1}^{n}a_{ij}(x){\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x)+\sum _{i=1}^{n}b_{i}(x){\frac {\partial f}{\partial x_{i}}}(x)+c(x)f(x),$

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, P is semi-elliptic if the matrix a(x) is positive semi-definite for each x ∈ Ω.

References

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