Abstract. We obtain a priori estimates in L p (ω) for the generalized Beltrami equation, provided that the coefficients are compactly supported V M O functions with the expected ellipticity condition, and the weight ω lies in the Muckenhoupt class A p . As an application, we obtain improved regularity for the jacobian of certain quasiconformal mappings.

We show that the Dirichlet to Neumann map for the equation ∇·σ∇u = 0 in a two-dimensional domain uniquely determines the bounded measurable conductivity σ. This gives a positive answer to a question of A. P. Calderón from 1980. Earlier the result has been shown only for conductivities that are sufficiently smooth. In higher dimensions the problem remains open.

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

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 settle the problem of the uniqueness of normalized homeomorphic solutions to nonlinear Beltrami equations ∂f (z) = H(z, ∂f (z)). It turns out that the uniqueness holds under definite and explicit bounds on the ellipticity at infinity, but not in general.