On Second Order Beltrami Systems 2 Aleksis Koski In analogy to the Beltrami equation in the plane, we consider second-order elliptic systems of the form $(1+ab)f_{z\bar{z}}-af_{zz}-bf_{\bar{z}\bar{z}}=0$, known to classify all complex linear equations in the homotopy component of the two-dimensional Laplacian. We prove existence and uniqueness of solutions for given boundary values in the unit disc for $b=0$ and $a$ antiholomorphic, and give a new proof for constant coefficients $a$ and $b$. In addition, we prove new results for the Beurling transform for the Dirichlet Problem, such as finding its $L^2$-operator norm and finding a bound for its spectral radius that also yields a Fredholm-type result. We also state some results connecting harmonic mappings in the plane and second-order equations of the form mentioned above. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5607 10.1512/iumj.2015.64.5607 en Indiana Univ. Math. J. 64 (2015) 1059 - 1101 state-of-the-art mathematics http://iumj.org/access/