We study nonlinear elliptic equations in divergence form(Formula presented.) When (Formula presented.) has linear growth in Du, and assuming that (Formula presented.) enjoys (Formula presented.) smoothness, local well-posedness is found in (Formula presented.) for certain values of (Formula presented.) and (Formula presented.). In the particular case (Formula presented.), G = 0 and (Formula presented.), (Formula presented.), we obtain (Formula presented.) for each (Formula presented.). Our main tool in the proof is a more general result, that holds also if (Formula presented.) has growth s−1 in Du, 2 ≤ s ≤ n, and asserts local well-posedness in Lq for each q > s, provided that (Formula presented.) satisfies a locally uniform VMO condition

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We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., {\operatorname{div}\mathcal{A}(x,Du)=\operatorname{div}F,} when {\mathcal{A}} is a p-harmonic type operator, and under the assumption that {x\mapsto\mathcal{A}(x,\xi\/)} belongs to the critical Besov–Lipschitz space {B^{\alpha}_{{n/\alpha},q}} . We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability. When {\operatorname{div}F=0} , we show that an analogous extra differentiability property for Du holds true under a Triebel–Lizorkin assumption on the partial map {x\mapsto\mathcal{A}(x,\xi\/)} .

We prove the higher differentiability and the higher integrability of the a priori bounded local minimizers of integral functionals of the form\ud \ud \ud F(v,Ω)=∫ Ω f(x,Dv(x))dx, \ud \ud \ud with convex integrand satisfying p-growth conditions with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x-variable belongs to a suitable Sobolev space. The a priori boundedness of the minimizers allows us to obtain the higher differentiability under a Sobolev assumption which is independent on the dimension n and that, in the case p≤n−2 , improves previous known results. We also deal with solutions of elliptic systems with discontinuous coefficients under the so-called Uhlenbeck structure. In this case, it is well known that the solutions are locally bounded and therefore we obtain analogous regularity results without the a priori boundedness assumptio

We provide a higher differentiability result for the finite energy solutions u ∈ W 1,1 ( , R N ) of a degenerate elliptic system of the formwhere is a bounded open set in R n and A(x, ξ) is a degenerate elliptic operator. The function which measures the degree of degeneracy of the problem is assumed to belong to the Sobolev class W 1,n ( ).