Discontinuous implicit elliptic boundary value problems.
An implicity given elliptic differential equation with homogeneous Dirichlet boundary conditions on a bounded domain \(\Omega \subset \mathbb R^N\) with a smooth boundary is considered. The problem in question is described by the equation \(f(x,u,Lu)=0\), where \(f\) can be discontinuous in all its arguments, \(L\) is a semilinear elliptic operator of the form \(Lu = -\Delta u + a(x,u)\) and \(a\) is nondecreasing in the last argument. For \(u,v \in W^{2,2} (\Omega )\) the authors first prove the following comparison principle: if \(Lu \leq Lv\) in \(\Omega \) and \(u\leq v\) on \(\partial \Omega \), then \(u\leq v\) in \(\Omega \). The main theorem establishes the existence of the extremal solutions. A one-dimensional illustrative example is presented.
scientific article; zbMATH DE number 1858727