CONICET | Buscador de Institutos y Recursos Humanos

This paper is devoted to the study of inhomogeneous minimization problems associated to the p(x)-Laplacian. We make a thorough analysis of the essential properties of their minimizers and we establish a relationship with a suitable free boundary problem. On the one hand, we study the problem of minimizing the functional J(v)=∫ Ω ([Formula presented]+λ(x)χ {v>0} +fv)dx. We show that nonnegative local minimizers u are solutions to the free boundary problem: u≥0 and (P(f,p,λ ⁎ )){Δ p(x) u:=div(|∇u(x)| p(x)−2 ∇u)=fin {u>0}u=0,|∇u|=λ ⁎ (x)on ∂{u>0} with λ ⁎ (x)=([Formula presented]λ(x)) 1/p(x) and that the free boundary is a C 1,α surface with the exception of a subset of H N−1 -measure zero. On the other hand, we study the problem of minimizing the functional J ε (v)=∫Ω([Formula presented]+B ε (v)+f ε v)dx, where B ε (s)=∫ 0 s β ε (τ)dτ ε>0, β ε (s)=[Formula presented]β([Formula presented]), with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1). We prove that if u ε are nonnegative local minimizers, then u ε are solutions to (P ε (f ε ,p ε ))Δ p ε (x) u ε =β ε (u ε )+f ε ,u ε ≥0. Moreover, if the functions u ε , f ε and p ε are uniformly bounded, we show that limit functions u (ε→0) are solutions to the free boundary problem P(f,p,λ ⁎ ) with λ ⁎ (x)=([Formula presented]M) 1/p(x) , M=∫β(s)ds, p=lim⁡p ε , f=lim⁡f ε , and that the free boundary is a C 1,α surface with the exception of a subset of H N−1 -measure zero. In order to obtain our results we need to overcome deep technical difficulties and develop new strategies, not present in the previous literature for this type of problems.