A second-order Sobolev regularity for p(x)-Laplace equations

Abstract

Given a Hölder regular variable exponent p(x), under the assumption Dp∈Lloc2, we establish some quantative interior Sobolev Wloc1,2-regularity of |Du|βDu for weak solutions u to the quasi-linear elliptic equations div(|Du|p(x)−2Du)=0 anddiv(p(x)|Du|p(x)−2Du)=0 in domains of Euclidean space Rn, where β>−1 in dimension 2 and β depends on p(x) and n in dimension n≥3. If Dp∈Lloc2+μ for some μ>0, we obtain Wloc1,2+δ-regularity of |Du|βDu for some δ>0 depending on n,β,p(x) and μ.