Hessian算子、零拉格朗日量与积分比较原理
Hessian operators, null Lagrangians and the integral comparison principle
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中文总结 AI 辅助
本文研究Hessian算子满足积分比较原理的条件，发现其与零拉格朗日量相关，并完整分类得出仅有k-Hessian线性组合满足该原理，复情形亦类似，揭示多复势理论障碍。
中文摘要 AI 辅助
我们研究了Hessian算子满足积分比较原理的条件。我们建立了其与变分法中零拉格朗日量概念之间一个相当意外的联系。在温和条件下，我们获得了完整分类，表明满足积分比较原理的仅有算子为$k$-Hessian算子的线性组合，且在齐次情形下，恰好就是$k$-Hessian算子本身。对于复Hessian算子，我们建立了类似的结果，特别揭示了发展$\mathcal{J}$-方程真正的多复势理论的实质性障碍。
英文摘要
We investigate the conditions under which a Hessian operator satisfies the integral comparison principle. We establish a rather unexpected connection with the notion of a null Lagrangian from the calculus of variations. Under mild conditions we obtain a complete classification, showing that the only operators for which the integral comparison principle holds are linear combinations of $k$-Hessians and, in the homogeneous case, precisely the $k$-Hessians themselves. An analogous result is established for the complex Hessian operator, revealing in particular substantial obstructions to the development of a genuine pluripotential theory for the $\mathcal{J}$-equation.
发表机构
- Jagiellonian University（雅盖隆大学）
- Gdańsk University（格但斯克大学）
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