从具有部分有界支撑的零解到偏微分方程的可解性与龙格逼近
From zero solutions with partially bounded supports to solvability and Runge approximation for partial differential equations
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中文总结 AI 辅助
研究齐次偏微分方程具规定支撑性质非平凡解的存在性，为常系数偏微分算子建立有部分有界支撑的光滑零解存在性。通过多种方法获偏微分方程可解性与逼近相关结果，包括\(P\)-凸性特征、龙格逼近等，还开发龙格逼近框架并应用于相关系统。
中文摘要 AI 辅助
齐次偏微分方程具有规定支撑性质的非平凡解的存在性在线性偏微分算子理论中起着基础性作用。本文为一大类常系数偏微分算子建立了具有部分有界支撑的光滑零解的存在性。作为应用，得到了关于偏微分方程可解性和逼近的新几何结果。特别地，导出了一大类非椭圆算子支撑的\(P\)-凸性的几何特征，扩展了霍尔曼德的经典几何思想。通过马尔格兰奇定理，支撑的\(P\)-凸性等同于微分算子在光滑函数空间以及更一般地在有限阶分布的局部子空间上的满射性。进一步证明了每个满射半椭圆算子的核满足条件\((\Omega)\)，这意味着参数依赖结果。还研究了龙格型逼近现象，得到了光滑函数、分布和光滑惠特尼射流空间的龙格对的几何特征。最后，为常系数偏微分方程的平方系统开发了一个龙格逼近的一般框架。作为应用，用替代方法恢复了贝尔特拉米场和三维非定常斯托克斯系统的已知龙格逼近结果，并在光滑惠特尼射流空间中用相应的逼近定理对其进行了补充。特别是，这为具有赫尔德连续边界的合适域上的解提供了直到边界的逼近。
英文摘要
The existence of non-trivial solutions of homogeneous partial differential equations with prescribed support properties plays a fundamental role in the theory of linear partial differential operators. In this article, we establish the existence of smooth zero solutions with partially bounded supports for a broad class of constant coefficient partial differential operators. As applications, we obtain new geometric results concerning solvability and approximation for partial differential equations. In particular, we derive geometric characterizations of $P$-convexity for supports for a large class of non-elliptic operators, thereby extending classical geometric ideas of Hörmander. By a theorem of Malgrange, $P$-convexity for supports is equivalent to the surjectivity of the differential operator on spaces of smooth functions and, more generally, on local subspaces of distributions of finite order. As a further consequence, we prove that the kernel of every surjective semi-elliptic operator satisfies condition ($Ω$) implying parameter dependence results. We also investigate Runge-type approximation phenomena. We obtain geometric characterizations of Runge pairs for smooth functions, distributions, and spaces of smooth Whitney jets. Finally, we develop a general framework for Runge approximation for square systems of constant coefficient partial differential equations. As applications, we recover by alternative methods known Runge approximation results for Beltrami fields and for the three-dimensional unsteady Stokes system, and we complement them by corresponding approximation theorems in spaces of smooth Whitney jets. In particular, this yields approximation up to the boundary for solutions on suitable domains with Hölder continuous boundary.
