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Runge type approximation results for spaces of smooth Whitney jets

Tomasz Ciaś, Thomas Kalmes

TL;DR

This work develops Runge-type approximation results for kernels of constant-coefficient PDEs on spaces of smooth Whitney jets over closed subsets. It provides a unified duality framework to characterize when restrictions of jet-solutions on a larger set are dense in jet-solutions on a smaller set, with explicit criteria for both elliptic and certain non-elliptic operators. The elliptic case yields a sharp geometric condition: a Runge pair $(F_1,F_2)$ holds iff $F_2$ contains no bounded component of the complement of $F_1$, which leads to density results for holomorphic/harmonic spaces and, under strong regularity, density of polynomials in $A^ ty(Ω)$. For non-elliptic operators, the authors develop geometric sufficient conditions tied to a single characteristic direction and prove density statements (with partial converses under extra assumptions), also applying to the wave operator in 1D and related settings. Overall, the paper extends Runge-type approximation to Whitney-jet spaces on closed sets, linking PDE, complex analysis, and geometric boundary regularity in a broad framework.

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 $Ω$ 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 $F_1$.

Runge type approximation results for spaces of smooth Whitney jets

TL;DR

This work develops Runge-type approximation results for kernels of constant-coefficient PDEs on spaces of smooth Whitney jets over closed subsets. It provides a unified duality framework to characterize when restrictions of jet-solutions on a larger set are dense in jet-solutions on a smaller set, with explicit criteria for both elliptic and certain non-elliptic operators. The elliptic case yields a sharp geometric condition: a Runge pair holds iff contains no bounded component of the complement of , which leads to density results for holomorphic/harmonic spaces and, under strong regularity, density of polynomials in . For non-elliptic operators, the authors develop geometric sufficient conditions tied to a single characteristic direction and prove density statements (with partial converses under extra assumptions), also applying to the wave operator in 1D and related settings. Overall, the paper extends Runge-type approximation to Whitney-jet spaces on closed sets, linking PDE, complex analysis, and geometric boundary regularity in a broad framework.

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 and for closed subsets of the restrictions to of smooth Whitney jets on satisfying on are dense in the space of smooth Whitney jets on satisfying the same partial differential equation on . 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 and for this sufficient conditions is also necessary. As an application of our work, we characterize those open subsets of the complex plane satisfying for which the set of holomorphic polynomials are dense in , under the mild additional hypothesis that satisfies the strong regularity condition. Furthermore, for the wave operator in one spatial variable, a simple sufficient geometric condition on is given for the above density to hold. For the special case of this sufficient condition is also necessary under mild additional hypotheses on .

Paper Structure

This paper contains 10 sections, 20 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Runge pairs for smooth Whitney jets
  4. Runge pairs for elliptic partial differential operators
  5. Runge pairs for certain non-elliptic partial differential operators
  6. Appendix
  7. Case 1: $x,y\in\overline{\Omega}\cap B(x_0,\varepsilon)$ and $0\leq\delta(x),\delta(y)\leq2C|x'-y'|^\theta$.
  8. Case 2: $x,y\in\Omega\cap B(x_0,\varepsilon)$ and $\delta(x),\delta(y)\geq2C|x'-y'|^\theta$.
  9. Case 3: $x,y\in\Omega\cap B(x_0,\varepsilon)$, $0\leq\delta(x)\leq2C|x'-y'|^\theta$ and $\delta(y)\geq2C|x'-y'|^\theta$.
  10. Case 4: $x,y\in\Omega\cap B(x_0,\varepsilon)$, $\delta(x)\geq2C|x'-y'|^\theta$ and $0\leq\delta(y)\leq2C|x'-y'|^\theta$.

Key Result

Proposition 2

Let $E_1,E_2$ be Fréchet spaces and let $T_1\colon E_1\to E_1$, $T_2\colon E_2\to E_2$ and $r\colon E_2\to E_1$ be continuous linear maps. Assume moreover that $T_2$ is surjective and $r\circ T_2=T_1\circ r$. Then the following assertions are equivalent:

Theorems & Definitions (44)

  • Definition 1
  • Proposition 2
  • proof
  • Remark 3
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • Corollary 7
  • ...and 34 more