UC Irvine

In 1978, the physicist P.S. Jang introduced a quasilinear elliptic equation in an attempt to generalize Geroch's approach to the positive mass conjecture of general relativity. The first existence and regularity result of Jang's equation was obtained by R. Schoen and S.-T. Yau through the capillary regularization procedure and stability-based a priori estimates. Yet, the solutions produced by this procedure may blow up in some black hole regions.

Schoen--Yau showed that the graph of a blowup solution to Jang's equation is asymptotic to cylinders over apparent horizons. J. Metzger showed that such cylindrical asymptotics are exponential, and he estimated the asymptotic rate by certain spectral properties of apparent horizons, followed by Q. Han, M. Khuri, and W. Yu. Their estimates involve delicate barrier construction and require the assistance of regularized solutions. We provide a simple proof of the sharp estimates that also apply to general blowup solutions.

We prove the first analytic and geometric result of regularized solutions to Jang's equation in black hole regions by applying two natural geometric treatments: translation and dilation. First, we show that the graphs of properly translated solutions converge subsequentially to constant expansion surfaces. Second, we characterize the limits of properly rescaled solutions. Third, we investigate the structure of black hole regions that arise in the Schoen--Yau regularization procedure. Finally, we discuss a special case of low-speed blowup behavior.