Inhomogeneous minimization problems for the p(x)-Laplacian

Abstract

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 (|∇u(x)|^ 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 ^⁎ )){Δ _p(x) u:=div(|∇u(x)|^ p(x)−2 ∇u)=∫ in{u>0} u=0,|∇u|=λ ^⁎ (x)on ∂{u>0} with λ^ ⁎ (x)=(p(x)/p(x)-1+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(|∇u|^ pε+B ε (v)+f ε v)dx, where B ε (s)=∫_ 0 ^ s β ε (τ)dτ ε>0, β ε (s)=1/3β(s/ε), 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)=(p(x)/p(x)-1M)^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.