A minimum problem with free boundary in Orlicz spaces

Abstract

We consider the optimization problem of minimizing ∫ Ω G ( | ∇ u | ) + λ χ { u > 0 } d x in the class of functions W 1 , G ( Ω ) with u − φ 0 ∈ W 0 1 , G ( Ω ) , for a given φ 0 ⩾ 0 and bounded. W 1 , G ( Ω ) is the class of weakly differentiable functions with ∫ Ω G ( | ∇ u | ) d x < ∞ . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω ∩ ∂ { u > 0 } , is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C 1 , α regularity of their free boundaries near “flat” free boundary points.