Title: Flow by Gauss curvature to the $L_p$-Gaussian Minkowski problem Show Chinese title

Title: 通过高斯曲率流求解$L_p$-高斯闵可夫斯基问题

Abstract: In this paper, we study the $L_p$-Gaussian Minkowski problem, which arises in the $L_p$-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the logarithmic Gauss Minkowski problem. We construct a suitable Gauss curvature flow of closed, convex hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$, and prove its long-time existence and converges smoothly to a smooth solution of the normalized $L_p$ Gaussian Minkowski problem in cases of $p>0$ and $-n-1<p\leq 0$ with even prescribed function respectively. We also provide a parabolic proof in the smooth category to the $L_p$-Gaussian Minkowski problem in cases of $p\geq n+1$ and $0<p<n+1$ with even prescribed function, respectively. Show Chinese abstract

Abstract: 在本文中，我们研究了$L_p$-高斯闵可夫斯基问题，该问题出现在高斯概率空间中的$L_p$-布伦-闵可夫斯基理论中。 我们使用 Aleksandrov 的变分方法结合拉格朗日乘数法来证明对数高斯闵可夫斯基问题的存在性。 我们在欧几里得空间$\mathbb{R}^{n+1}$中构造了一个合适的闭合凸超曲面的高斯曲率流，并证明了其长时间存在性，并在$p>0$和$-n-1<p\leq 0$的情况下分别光滑收敛到归一化的$L_p$高斯闵可夫斯基问题的光滑解，且给定函数为偶函数。 我们还在光滑范畴中提供了关于$L_p$-高斯曲率闵可夫斯基问题的抛物型证明，在$p\geq n+1$和$0<p<n+1$的情况下，分别具有偶数的预设函数。