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

Abstract

For a constant coefficient partial differential operator $P(D)$ with a single characteristic direction such as the time-dependent free Schrödinger 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 $P(D)$.

accepted for publication in Bulletin des Sciences Mathématiques

References in corpus (1)

• Power bounded weighted composition operators on function spaces defined by local properties

Cited by in corpus (2)

• Quantitative Runge type approximation theorems for zero solutions of certain partial differential operators
• Boundary values of zero solutions of hypoelliptic differential operators in ultradistribution spaces