Year: 2011
Type: preprint
Abstract: We consider the problem of finding a real number lambda and a function u satisfying the PDE max{lambda -Δu -f,|Du|-1}=0, for all x in R^n. Here f is a convex, superlinear function. We prove that there is a unique lambda* such that the above PDE has a viscosity solution u satisfying u(x)/|x|->1 as |x| tends to infinity. Moreover, we show that associated to lambda^* is a convex solution u^* with D^2u^* uniformly bounded and give two min-max formulae for lambda^*. lambda^* has a probabilistic interpretation as being the least, long-time averaged ("ergodic") cost for a singular control problem involving f.
Source: arXiv (Cornell University)
Author: Hynd, Ryan
Institution: Courant Institute of Mathematical Sciences
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W2952914697