Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity

Wenjia Jing; Panagiotis E. Souganidis; Hung V. Tran

doi:10.3934/dcdss.2018055

Article Contents

2018, Volume 11, Issue 5: 915-939. Doi: 10.3934/dcdss.2018055

This issue Previous Article One-dimensional, forward-forward mean-field games with congestion Next Article A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations

Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity

1.
Yau Mathematical Sciences Center, Tsinghua University, No 1. Tsinghua Yuan, Beijing 100084, China
2.
Department of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA
3.
Department of Mathematics, University of Wisconsin at Madison, 480 Lincoln Drive, Madison, WI 53706, USA

* Corresponding author: Wenjia Jing
* Corresponding author: Wenjia Jing

Received: January 31, 2017

Revised: June 30, 2017

Published: June 2018

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Abstract

We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

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Mathematics Subject Classification: Primary: 35B27, 70H20; Secondary: 49L25.

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Figure 1. The construction of admissible paths

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