The adjoint method, recently introduced by Evans, is used to study obstacle problems,weakly coupled systems, cell problems for weakly coupled systems of Hamilton − Jacobiequations, and weakly coupled systems of obstacle type. In particular, new results aboutthe speed of convergence of some approximation procedures are derived.

For solving linear complementarity problems LCP more attention has recently been paid on a class of iterative methods called the matrix-splitting. But up to now, no paper has discussed the effect of preconditioning technique for matrix-splitting methods in LCP. So, this paper is planning to fill in this gap and we use a class of preconditioners with generalized Accelerated Overrelaxation (GAOR) methods and analyze the convergence of these methods for LCP. Furthermore, Comparison between our methods and other non-preconditioned methods for the studied problem shows a remarkable agreement and reveals that our models are superior in point of view of convergence rate and computing efficiency. Besides, by choosing the appropriate parameters of these methods, we derive same results as the other iterative methods such as AOR, JOR, SOR etc. Finally the method is tested by some numerical experiments.

Given a Borel function ψ defined on a bounded open set Ω with Lipschitz boundary and $\varphi\in L^1(\partial\Omega,{\mathcal H}^{n-1})$ , we prove an explicit representation formula for the L 1 lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint $u^+\ge\psi$ ${\mathcal H}^{n-1}$ a.e. on Ω and the Dirichlet boundary condition $u=\varphi$ on $\partial\Omega$ .