Blowup rate control for solution of Jang's equation and its application on Penrose inequality

We prove that the blowup term of a blowup solution of Jang's equation on an initial data set (M,g,k) near an arbitrary strictly stable MOTS $\Sigma$ is exactly $-\frac{1}{\sqrt{\lambda}}\log \tau$, where $\tau$ is the distance from $\Sigma$ and $\lambda$ is the principal eigenvalue of the MOTS stability operator of $\Sigma$. We also prove that the gradient of the solution is of order $\tau^{-1}$. Moreover, we apply these results to get a Penrose-like inequality under additional assumptions.