发表机构
Harish-Chandra Research Institute; Homi Bhabha National Institute（哈立什-昌德拉研究所; 霍米·巴巴国立研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对二维空间中预定Jacobian方程，为紧支撑及快速衰减数据显式构造全局解，证明解属于适当Sobolev空间，并推广到部分可测数据。
AI 中文摘要
本文中，我们针对一类数据，给出了预定Jacobian方程\\[ \det(\nabla u)=f\mbox{ in }\mathbb{R}^2,\\] 的全局解的显式构造。对于每个$f\in C_c^1(\mathbb{R}^2)$和$p>1$，我们构造了一个解$u\in \dot W^{1,p}(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$。特别地，对紧支撑数据不施加符号条件或积分约束。对于$f\in C_c^\infty(\mathbb{R}^2)$，该构造产生光滑解。我们还考虑了快速衰减的数据，并证明了对于每个$f\in S(\mathbb{R}^2)$，存在具有有界梯度的全局解。最后，我们将构造推广到在一个变量上可测、在另一个变量上为$C^1$的紧支撑数据。
英文摘要
In this note, we give an explicit construction of global solutions to the prescribed Jacobian equation \[ \det(\nabla u)=f\mbox{ in }\mathbb{R}^2, \] for a class of data. For every $f\in C_c^1(\mathbb{R}^2)$ and $p>1$, we construct a solution $u\in \dot W^{1,2p}(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$. In particular, no sign condition or integral constraint is imposed on the compactly supported data. For $f\in C_c^\infty(\mathbb{R}^2)$, the construction yields a smooth solution. We also consider rapidly decaying data and prove the existence of global solutions with bounded gradient for every $f\in\mathcal{S}(\mathbb{R}^2)$. Finally, we extend the construction to compactly supported data that are measurable in one variable and $C^1$ in the other. Our proof relies on a similar idea as in [Moser, Trans. Amer. Math. Soc. 1965].
