Some sharp Sobolev regularity for inhomogeneous infinity Laplace equation in plane

Abstract

Suppose Ω⋐R2 and f∈BVloc(Ω)∩C0(Ω) with |f|>0 in Ω. Let u∈C0(Ω) be a viscosity solution to the inhomogeneous ∞-Laplace equation−Δ∞u:=−12∑i=12(|Du|2)iui=−∑i,j=12uiujuij=finΩ. The following are proved in this paper:(i)For α>3/2, we have |Du|α∈Wloc1,2(Ω), which is (asymptotic) sharp when α→3/2. Indeed, the function w(x1,x2)=−x14/3 is a viscosity solution to −Δ∞w=4334 in R2. For any p>2, we have |Dw|α∉Wloc1,p(R2) whenever α∈(3/2,3−3/p).(ii)For α∈(0,3/2] and p∈[1,3/(3−α)), we have |Du|α∈Wloc1,p(Ω), which is sharp when p→3/(3−α). Indeed, |Dw|α∉Wloc1,3/(3−α)(R2).(iii)For ϵ>0, we have |Du|−3+ϵ∈Lloc1(Ω), which is sharp when ϵ→0. Indeed, |Dw|−3∉Lloc1(R2).(iv)For α>0, we have −(|Du|α)iui=2α|Du|α−2f almost everywhere in Ω. Some quantative bounds are also given.