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

In this paper, we look at quasiconformal solutions φ : C → C of Beltrami equations

No abstract

No abstract

We study the distributional solutions to the (generalized) Beltrami equation under Sobolev assumptions on the Beltrami coefficients. In this setting, we prove that these distributional solutions are true quasiregular maps and they are smoother than expected, that is, they have second order derivatives in L 1+ε loc , for some ε > 0.1 K −1 ) −1 shows that for any p < 2K K+1 there exists a weakly Kquasiregular map in W 1,p loc which is not K-quasiregular. Later, Petermichl and Volberg [13] proved that the weaker assumption p ≥ 2K K+1 is already sufficient for weakly K-quasiregular maps to be K-quasiregular. (See the monograph [3] for a complete and detailed information).