Abstract:
Motivated by applications in image processing, we study asymptotic behavior for power curvature flow as the exponent tends to infinity. When the initial value satisfies a convexity assumption, we can characterize the limit equation as a stationary obstacle problem involving 1-Laplacian. We show that the large exponent limit of the solution to power curvature flow equation instantaneously converges to the minimal solution of the obstacle problem.
The situation is more complicated when the convexity condition on the initial value is removed. In this case, we can still obtain the convergence by examining the initial layer. We shall mainly discuss a simplified problem and describe applications related to a math model describing unstable sandpiles.
Part of this talk is based on joint work with Professor Naoki Yamada at Fukuoka University.
Biography:
Qing Liu is an Assistant Professor in the Department of Applied Mathematics, Fukuoka University. He received a B.S. from Fudan University in 2005. He obtained his Ph.D. in Mathematical Sciences In 2011 from University of Tokyo under the supervision of Professor Yoshikazu Giga. Prior to joining Fukuoka University, he was working as a Postdoctoral Associate in the Department of Mathematics, University of Pittsburgh from 2011 to 2015.
Qing Liu's research interests lie in nonlinear PDEs with emphasis on viscosity solution theory and its applications in optimal control, differential games, image processing and crystal growth. He is also interested in geometric analysis on sub-Riemannian manifolds and general metric spaces.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai