Extremal periodic solutions for nonlinear parabolic equations with discontinuities (Q2785357)

The authors consider a periodic boundary value problem of the form NEWLINE\[NEWLINE{\partial x\over\partial t}- \sum^N_{k=1} D_ka_k(t,z,Dx)= f(x(t,z))\text{ in }T\times\Omega,\;x(0,z)= x(b,z)\text{ a.e. in }\Omega,\;x|_{T\times\partial\Omega}= 0,NEWLINE\]NEWLINE where \(T=[0,b]\), \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(a_k\) satisfies the Leray-Lions conditions, and \(f\) is discontinuous having locally bounded variation. Assuming that there exist an upper solution \(\varphi\) and a lower solution \(\psi\) with \(\psi\leq\varphi\) they prove the existence of a maximal and a minimal periodic solution within the order interval \([\psi,\varphi]\). The approach is based on a Jordan type decomposition for \(f\) and on a fixed point theorem for monotone maps in order structures.