2019/01/04 by Lederman, Claudia, Wolanski, Noemi
#35B65 #35J20 #35J60 #35J70 #35R35 #49K20 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1901.01165
We study an 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(\frac|∇ 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 \begincases Δp(x)u:=div(|∇ u(x)|p(x)-2∇ u)= f amp; in \ugt;0\
u=0, |∇ u| = λ^*(x) amp; on ∂\ugt;0\ \endcases with λ^*(x)=((p(x))/(p(x)-1) λ(x))1/p(x) and that the free boundary is a C1,α surface. On the other hand, we study the problem of minimizing the functional Jε(v)= ∫Ω(\frac|∇ v|pε(x)pε(x)+Bε(v)+fε v) dx, where Bε(s)=∫ 0sβε(τ) dτ, ε>0, βε(s)=1 \over ε β(s \over ε), 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 any limit function u (ε→ 0) is a solution to the free boundary problem P(f,p,λ^*) with λ^*(x)=((p(x))/(p(x)-1) M)1/p(x), M=∫ β(s) ds, p=lim pε, f=lim fε, and that the free boundary is a C1,α surface. 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.