Rosinger nowhere-dense generalized function algebra

In the general framework of generalized function algebras developed by E.E. Rosinger [a3], [a4], [a5], [a6], a distinguished role is played by ideals in the sequence algebra $\mathcal{C} ^ { \infty } ( \Omega ) ^ { \text{N} }$, $\Omega$ an open subset of ${\bf R} ^ { n }$, which are defined by vanishing properties. Given a family $S$ of subsets of $\Omega$, stable under finite unions, one considers the ideal $\mathcal{I} _ { S }$ in $\mathcal{C} ^ { \infty } ( \Omega ) ^ { \text{N} }$ determined by those $( u_j )_{ j \in \mathbf{N}}$ for which there is a $\Gamma \in S$ such that for all $x \in \Omega \backslash \Gamma$, $( u_j )_{ j \in \mathbf{N}}$ vanishes near $x$ eventually, that is, there are a $j _0$ and a neighbourhood $V \subset \Omega \backslash \Gamma$ of $x$ such that $u _ { j } | _ { V } \equiv 0$ for $j \geq j_0$. The nowhere-dense generalized function algebra $\mathcal{R} _ { \text{nd} } ( \Omega ) = \mathcal{C} ^ { \infty } ( \Omega ) ^ { \text{N} } / \mathcal{I} _ { \text{nd} }$ is obtained when $S$ is the class of nowhere-dense, closed subsets of $\Omega$. The space $\mathcal{R} _ { \text{nd} } ( \Omega )$ contains $\mathcal{C} ^ { \infty } ( \Omega )$ via the constant imbedding. It has two distinguishing features. First, the family $\{ \mathcal{R} _ { \text{nd} } ( \Omega ) : \Omega \, \text{open} \}$ forms a flabby sheaf, and in a certain sense the smallest flabby sheaf containing $\mathcal{C} ^ { \infty } ( \Omega )$, see [a2]. Secondly, the algebra $C _ { \text{nd} } ^ { \infty } ( \Omega )$ of (equivalence classes of) smooth functions defined off some nowhere-dense, closed subset of $\Omega$ can be imbedded into $\mathcal{R} _ { \text{nd} } ( \Omega )$.

In particular, solutions to partial differential equations defined piecewise off nowhere-dense closed sets $\Gamma$ (no growth restrictions near $\Gamma$) can be interpreted as global solutions in $\mathcal{R} _ { \text{nd} } ( \Omega )$ by means of a suitable regularization method. The space of distributions $\mathcal{D} ^ { \prime } ( \Omega )$ (cf. also Generalized functions, space of) is imbedded in any algebra of the form $\mathcal{C} ^ { \infty } ( \Omega ) / \mathcal{I} _ { S }$ by a general procedure [a4] using an algebraic basis.

Further generalizations of the ideal $\mathcal{I} _ {\operatorname{nd} }$ to include larger exceptional sets as well as applications to non-smooth differential geometry can be found in [a1]; non-linear Lie group actions on generalized functions using the framework of $\mathcal{R} _ { \text{nd} } ( \Omega )$ are studied in [a7].

Replacing the single differential algebra ${\cal R = C} ^ { \infty } ( \Omega ) / {\cal I} _ { S }$ with chains of algebras $\mathcal{R} ^ { \infty } \rightarrow \ldots \rightarrow \mathcal{R} ^ { m } \rightarrow \ldots \rightarrow \mathcal{R} ^ { 0 }$ using the spaces $\mathcal{C} ^ { m } ( \Omega )$ in the place of $\mathcal{C} ^ { \infty } ( \Omega )$ at each level, allows one to achieve consistency of the multiplication and derivation with the pointwise product of $\mathcal{C} ^ { m }$-functions as well as the derivative of $\mathcal{C} ^ { m + 1 } \rightarrow \mathcal{C} ^ { m }$ at each fixed level $m$.

See also Generalized function algebras.

References

[a1]	A. Mallios, E.E. Rosinger, "Space-time foam dense singularities and de Rham cohomology" Acta Applic. Math. (to appear) MR1847884 Zbl 1005.46020
[a2]	M. Oberguggenberger, E.E. Rosinger, "Solution of continuous nonlinear PDEs through order completion" , North-Holland (1994) MR1286940 Zbl 0821.35001
[a3]	E.E. Rosinger, "Distributions and nonlinear partial differential equations" , Springer (1978) MR0514014 Zbl 0469.35001
[a4]	E.E. Rosinger, "Nonlinear partial differential equations. Sequential and weak solutions" , North-Holland (1980) MR0590891 Zbl 0447.35001
[a5]	E.E. Rosinger, "Generalized solutions of nonlinear partial differential equations" , North-Holland (1987) MR0918145 Zbl 0635.46033
[a6]	E.E. Rosinger, "Nonlinear partial differential equations, an algebraic view of generalized solutions" , North-Holland (1990) Zbl 0717.35001
[a7]	E.E. Rosinger, "Parametric Lie group actions on global generalized solutions of nonlinear PDEs. Including a solution to Hilbert's fifth problem" , Kluwer Acad. Publ. (1998)