Runge type approximation results for spaces of smooth Whitney jets

TL;DR

This paper establishes Runge type approximation results for solutions of constant coefficient linear PDEs on spaces of smooth Whitney jets, providing geometric characterizations and applications to complex analysis and wave operators.

Contribution

It introduces new criteria for density of restricted solutions of PDEs in Whitney jet spaces, including geometric conditions for elliptic and parabolic operators, and applies these to complex domains and wave equations.

Findings

Characterization of density conditions for elliptic operators.

Geometric criteria for parabolic operators.

Application to density of polynomials in complex domains.

Abstract

We prove Runge type approximation results for linear partial differential operators with constant coefficients on spaces of smooth Whitney jets. Among others, we characterize when for a constant coefficient linear partial differential operator $P(D)$ and for closed subsets $F_1\subset F_2$ of $\mathbb{R}^d$ the restrictions to $F_1$ of smooth Whitney jets $f$ on $F_2$ satisfying $P(D)f=0$ on $F_2$ are dense in the space of smooth Whitney jets on $F_1$ satisfying the same partial differential equation on $F_1$. For elliptic operators we give a geometric evaluation of this characterization. Additionally, for differential operators with a single characteristic direction, like parabolic operators, we give a sufficient geometric condition for the above density to hold. Under mild additional assumptions on $\partial F_1$ and for $F_2=\mathbb{R}^d$ this sufficient conditions is also necessary. As an application of our work, we characterize those open subsets $\Omega$ of the complex plane satisfying $\Omega=\operatorname{int}\overline{\Omega}$ for which the set of holomorphic polynomials are dense in $A^\infty(\Omega)$, under the mild additional hypothesis that $\overline{\Omega}$ satisfies the strong regularity condition. Furthermore, for the wave operator in one spatial variable, a simple sufficient geometric condition on $F_1, F_2\subset\mathbb{R}^2$ is given for the above density to hold. For the special case of $F_2=\mathbb{R}^2$ this sufficient condition is also necessary under mild additional hypotheses on $F_1$.