Continuous dependence on obstacles for the double obstacle problem on metric spaces

Let XX be a complete metric space equipped with a doubling Borel measure supporting a pp-Poincaré inequality. We obtain various convergence results for solutions of double obstacle problems on open subsets of XX. In particular, we consider a sequence of double obstacle problems with converging obstacles and show that the corresponding solutions converge as well. We use the convergence properties to define and study a generalized solution of the double obstacle problem.