Energy decay for Dirichlet problems in irregular domains with quadratic Hamiltonian (Q1198723)

Let \(u\) be a solution of the formal Dirichlet problem \[ Lu\equiv - \partial_ j(a_{ij}\partial_ i u)=f(x,u,Du)\;\text{ in }\Omega-E, \qquad u=h\;\text{ in }E \] with \(h\in H^ 1\), \(E\) a bounded Borel subset of \(\mathbb{R}^ n\), \(f\) a Carathéodory function growing quadratically in the gradient variable and \(L\) a uniformly elliptic operator. The author estimates the energy decay and gives sufficient conditions on \(h\) and \(E\) to guarantee the continuity of \(u\) in \(E\) through the investigation of two-obstacle problems. Similar results are also proved for the one- obstacle case.