A minimum problem with free boundary in Orlicz spaces (original) (raw)

2008, Advances in Mathematics

We consider the optimization problem of minimizing R Ω G(|∇u|) + λχ {u>0} dx in the class of functions W 1,G (Ω) with u − ϕ0 ∈ W 1,G 0 (Ω), for a given ϕ0 ≥ 0 and bounded. W 1,G (Ω) is the class of weakly differentiable functions with R Ω G(|∇u|) dx < ∞. 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. Ω G(|∇u|) + λχ {u>0} dx in the class of functions K = v ∈ L 1 (Ω) : Ω G(|∇v|) dx < ∞, v = ϕ 0 on ∂Ω. This kind of optimization problem has been widely studied for different functions G. In fact, the first paper in which this problem was studied is [3]. The authors considered the case G(t) = t 2. They proved that minimizers are weak solutions to the free boundary problem (1.2) ∆u = 0 in {u > 0} u = 0, |∇u| = λ on ∂{u > 0} and proved the Lipschitz regularity of the solutions and the C 1,α regularity of the free boundaries. This free boundary problem appears in several applications. A very important one is that of fluid flow. In that context, the free boundary condition is known as Bernoulli's condition. The results of [3] have been generalized to several cases. For instance, in [5] the authors consider problem (1.1) for a convex function G such that ct < G ′ (t) < Ct for some positive constants c and C. Recently, in the article [7] the authors considered the case G(t) = t p with 1 < p < ∞. In these two papers only minimizers are studied. Minimizers satisfy very good properties like nondegeneracy at the free boundary and uniform positive density of the set {u = 0} at free boundary points. On the other hand, the free boundary problem (1.2) and its