A Quantitative Second Order Sobolev Regularity for (inhomogeneous) 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 −Δ ^N_p(·) u = 0 in Ω as well as the corresponding inhomogeneous equation −Δ ^N_p(·) u =f in Ω with f ∈ C⁰(Ω). In particular, given any viscosity solution u to −Δ ^N_p(·) u = 0 in Ω, we prove the following:

1. (i)

   in dimension n = 2, for any subdomain U ⋐ Ω and any β ≥ 0, one has ∣Du∣^βDu ∈ L ^2+δ_loc (U) with a quantitative upper bound, and moreover, the map \((x_{1},x_{2})\rightarrow\vert Du\vert^{\beta}(u_{x_{1}},-u_{x_{2}})\) is quasiregular in U in the sense that

   $$\vert D[\vert Du\vert^{\beta}\;Du]\vert^{2}\leq-C\;\text{det}\;D[\vert Du\vert^{\beta}\;Du]\;\;\;\;\;\text{a.e.}\;\text{in}\;U.$$
2. (ii)

   in dimension n ≥ 3, for any subdomain U ⋐ Ω with inf_U p(x) > 1 and \(\text{sup}_{U}\;p(x)<3+{2\over{n-2}}\), one has D²u ∈ L ^2+δ_loc (U) with a quantitative upper bound, and also with a pointwise upper bound

   $$\vert D^{2}u\vert^{2}\leq-C\sum_{1\leq i<j\leq n}[u_{x_{i}x_{j}}u_{x_{j}x_{i}}-u_{x_{i}x_{i}}u_{x_{j}x_{j}}]\;\;\;\;\;\text{a.e}\;\text{in}\;U.$$

Here constants δ > 0 and C ≥ 1 are independent of u. These extend the related results obtaind by Adamowicz–Hästö [Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, 10, 1940–1965 (2010)] when n = 2 and β = 0.