A strong convergence of the weak gradient to A-harmonic type operators with [formula omitted] data

Abstract

We prove uk→u strongly in Wloc1,q(Ω) with 1≤q<p by Lipschitz truncation argument if u∈W1,p(Ω) is a weak solution of A-harmonic type equations −divA(x,Du)=f(x) with f∈L1(Ω), and uk is a sequence of their weak solutions with uk⇀u weakly in W1,p(Ω) and fk⇀f weakly in L1(Ω). As an application, we obtain a compactness property for p-harmonic maps defined from L∞-metric Riemannian manifold.