On the rate of convergence of solutions in singular perturbation problems (Q1177031)

The pointwise convergence for solutions of the singular perturbation problem \[ u^ \varepsilon+F_ \varepsilon(x,u^ \varepsilon,\allowbreak Du^ \varepsilon,D^ 2u^ \varepsilon)= f,\qquad \hbox{ in }\Omega \tag{*} \] is studied. A simple form of (*) is the well-known obstacle problem. A new technique arising from the theory of viscosity solutions, which differs from the traditional technique of variational inequalities, is exploited to obtain an optimal rate of pointwise convergence estimates for the solutions of (*). This result is better than the \(L^ 2\) estimates derived from the variational method in the case of the obstacle problem.