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Runge type approximation results for spaces of smooth Whitney jets
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DOI：10.1007/s10231-026-01655-7.png)
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Abstract
En 中文
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 of F-2 of Rd the restrictions to F1 of smooth Whitney jets f on F(2 )satisfy-ing 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 derivative F1 and for F2=R(d)this sufficient conditions is also necessary. As an application of our work, we character-ize those open subsets ohm of the complex plane satisfying ohm=int ohm for which the set of holomorphic polynomials are dense in A infinity(ohm), under the additional hypothesis that ohm 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 of R-2 is given for the above density to hold. For the special case of F-2=R-2 this sufficient condition is also necessary under mild additional hypotheses on F1
Keywords:
Runge type approximation theorem
Lax-Malgrange theorem
Partial differential operators
Smooth Whitney jets
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0.9
Papers:
75
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