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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.03815 |
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Table of Contents:
- 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 $Ω$ of the complex plane satisfying $Ω=\operatorname{int}\overlineΩ$ for which the set of holomorphic polynomials are dense in $A^\infty(Ω)$, under the mild additional hypothesis that $\overlineΩ$ 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 $F_1$.