Regular solutions to a monodimensional model with discontinuous elliptic operator

Abstract

The note examines qualitative behavior of solutions to a monodimensional nonlinear elliptic equation $\frac{d}{dx}(u_x+\mathrm {sgn} \ u_x)=f$ with Dirichlet boundary data. This simple example explains the phenomenon of facets – flat regions of solutions, characteristic for models arising from theories of crystal growth and image prossessing.