Some free boundary problems in Orlicz spaces

MARTÍNEZ, SANDRA RITA; NOEMI WOLANSKI

Pavia, Italia

Seminario; Seminari del Dipartemento di Matematica; 2006

We consider the  optimization problem of minimizing$int_{Omega}G(| abla u|)+lambda chi_{{u>0}}, dx$ in theclass of functions  $W^{1,G}(Omega)$ with $u-arphi_0inW_0^{1,G}(Omega)$, for a given $arphi_0geq 0$ and bounded.$W^{1,G}(Omega)$ is the class of weakly differentiable functionswith $int_Omega G(| abla u|),dx0}$, is a regular surface.Also, we introduce the notion of weak solution to the free boundary problemsolved by the minimizers and prove the Lipschitz regularity of the weaksolutions and the $C^{1,alpha}$ regularity of their free boundaries near``flat´´ free boundary points.