论文标题
均质$ k $ - hessian方程的外部差异问题
The exterior Dirichlet problem for the homogeneous $k$-Hessian equation
论文作者
论文摘要
我们研究了均质$ k $ hessian方程的外部差异问题。如果$ k <\ frac {n} {2} $，则解决方案无限的规定渐近行为为零，它是$ \ log | x |+o（1）$如果$ k = \ frac {n} {n} {2} {2} $ $ k> \ frac {n} {2} $。通过构建具有均匀$ c^{1,1} $的近似非分级$ k $ -Hessian方程的平滑解决方案 - 我们证明存在部分。唯一性来自比较定理，因此证明了外部域中同质$ k $ -Hessian方程的$ c^{1,1} $。我们还证明了梯度的统一正限制。作为$ c^{1,1} $估计值的含义，我们沿近似解决方案的级别进行了几乎单调的公式。特别是，我们得到了加权的几何不平等，这是$ k = 1 $案例的自然概括。
We study the exterior Dirichlet problem for the homogeneous $k$-Hessian equation. The prescribed asymptotic behavior at infinity of the solution is zero if $k<\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and it is $|x|^{\frac{2k-n}{n}}+O(1)$ if $k>\frac{n}{2}$. By constructing smooth solutions of approximating non-degenerate $k$-Hessian equations with uniform $C^{1,1}$-estimates, we prove the existence part. The uniqueness follows from the comparison theorem and thus the $C^{1,1}$ regularity of the solution of the homogeneous $k$-Hessian equation in the exterior domain is proved. We also prove a uniform positive lower bound of the gradient. As an implication of the $C^{1,1}$ estimates, we derive an almost monotonicity formula along the level set of the approximating solution. In particular, we get an weighted geometric inequality which is a natural generalization of the $k=1$ case.
