Inhomogeneous minimization problems for the p(x)-Laplacian

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(|∇v|p(x)p(x)+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⁎)){Δp(x)u:=div(|∇u(x)|p(x)−2∇u)=fin {u>0}u=0,|∇u|=λ⁎(x)on ∂{u>0} with λ⁎(x)=(p(x)p(x)−1λ(x))1/p(x) and that the free boundary is a C1,α surface with the exception of a subset of HN−1-measure zero. On the other hand, we study the problem of minimizing the functional Jε(v)=∫Ω(|∇v|pε(x)pε(x)+Bε(v)+fεv)dx, where Bε(s)=∫0sβε(τ)dτ, ε>0, βε(s)=1εβ(sε), with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1). We prove that if uε are nonnegative local minimizers, then uε are solutions to(Pε(fε,pε))Δpε(x)uε=βε(uε)+fε,uε≥0. Moreover, if the functions uε, fε and pε 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)=(p(x)p(x)−1M)1/p(x), M=∫β(s)ds, p=lim⁡pε, f=lim⁡fε, and that the free boundary is a C1,α surface with the exception of a subset of HN−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.