Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

© 2016, Springer Science+Business Media Dordrecht. We study nonlinear elliptic equations in divergence form div A(x, Du) = divG When A has linear growth in Du, and assuming that x↦ A(x, ξ) enjoys (Formula presented.) smoothness, local well-posedness is found in (Formula presented.) for certain values of p∈[2,n/α) and q∈ [1 , ∞]. In the particular case A(x, ξ) = A(x) ξ, G = 0 and (Formula presented.), 1 ≤ q≤ ∞, we obtain Du ∈ (Formula presented.) for each p<n/α. Our main tool in the proof is a more general result, that holds also if A has growth s−1 in Du, 2 ≤ s ≤ n, and asserts local well-posedness in L q for each q > s, provided that x↦ A(x, ξ) satisfies a locally uniform VMO condition.