An approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators

TL;DR

This paper characterizes when certain open subsets of Euclidean space form a Runge pair for specific non-elliptic PDEs, providing conditions independent of boundary regularity and constructing solutions with support in narrow slabs.

Contribution

It introduces a boundary-regularity-free criterion for $P$-Runge pairs for non-elliptic operators and constructs solutions supported in arbitrarily narrow slabs.

Findings

Characterization of $P$-Runge pairs for non-elliptic PDEs.

Existence of solutions supported in narrow slabs between characteristic hyperplanes.

No boundary regularity assumptions needed for the criteria.

Abstract

For a constant coefficient partial differential operator $P(D)$ with a single characteristic direction such as the time-dependent free Schr\"odinger operator as well as non-degenerate parabolic differential operators like the heat operator we characterize when open subsets $X_1\subseteq X_2$ of $\mathbb{R}^d$ form a $P$-Runge pair. The presented condition does not require any kind of regularity of the boundaries of $X_1$ nor $X_2$. As part of our result we prove that for a large class of non-elliptic operators $P(D)$ there are smooth solutions $u$ to the equation $P(D)u=0$ on $\mathbb{R}^d$ with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for $P(D)$.