Orienting method for obstacle problems (Q1607613)

Let \(u\) be a minimizer of \(\int_\Omega F(x,v,\nabla v) dx\), where \(\Omega\subset\mathbb R^n\) is an open bounded set and \(v\in W^{1,p}(\Omega)\) satisfies \(v\geq r\) in \(\Omega\), \(v=g\) on \(\partial\Omega\). If \(\widetilde\Omega\subset\Omega\), \(\widetilde g=r\) on \(\partial\widetilde\Omega\cap\Omega\), \(r\leq\widetilde g\leq g\) on \(\partial\widetilde\Omega\cap\partial\Omega\), and the minimizer \(\widetilde u\) of \(\int_{\widetilde\Omega}F(x,v,\nabla v) dx\) in \(\{v\in W^{1,p}(\widetilde\Omega) : v=\widetilde g\text{ on }\partial\widetilde\Omega\}\) is unique, then the authors show the lower estimate \(u\geq\widetilde u\) in \(\widetilde\Omega\). In particular, the set \(\{x\in\widetilde\Omega : \widetilde u(x)>r(x)\}\) is a subset of the non-coincidence set \(\{x\in\Omega : u(x)>r(x)\}\). Similar upper estimates are also derived and the results are applied to the problem with \(n=1\), \(\Omega=(-2,3)\), \(F=4v+|v'|^2\), \(r(x)=|x|\), \(g(-2)=4\), \(g(3)=5\).