A quantitative second order Sobolev regularity for (inhmogeneous) normalized p(·)-Laplace equations

Let Ω be a domain of Rⁿ with n≥ 2 and p(·) be a local Lipschitz funcion in Ω with 1<p(x)<∞ in Ω. We build up an interior quantitative second order Sobolev regularity for the normalized p(·)-Laplace equation -ΔNp(·)u=0 in Ω as well as the corresponding inhomogeneous equation -ΔNp(·)u=f in Ω with f∈ C⁰(Ω). In particular, given any viscosity solution u to -ΔNp(·)u=0 in Ω, we prove the following: (i) in dimension $n=2$, for any subdomain UΩ and any β≥ 0, one has |Du|^β Du∈ L2+δ(U) locally with a quantitative upper bound, and moreover, the map (x₁,x₂)→ |Du|^β(ux₁,-ux₂) is quasiregular in U in the sense that |D[|Du|^β Du]|²≤ -C D[|Du|^β Du] a.e. inU. (ii) in dimension n≥3, for any subdomain UΩ with infU p(x)>1 and Up(x)<3+2n-2, one has D²u∈ L2+δ(U) locally with a quantitative upper bound, and also with a pointwise upper bound |D²u|²≤ -C∑1≤ i<j≤ n[uxᵢxⱼuxⱼxᵢ-uxᵢxᵢuxⱼxⱼ] a.e. inU. Here constants δ>0 and C≥ 1 are independent of u. These extend the related results obtaind by Adamowicz-H\"ast\"o {AH2010} when $n=2$ and β=0.