## Elliptic Systems in the Plane |

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### Contents

Review of the RiemannHilbertProblem in Hilberts Normal | 3 |

The Normal Form of Elliptic Systems with Two Unknowns | 39 |

Normal Forms of General FirstOrder Systems | 75 |

Copyright | |

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### Common terms and phrases

adjoint analytic functions approximation Berlin Bojarski boundary conditions boundary integrals boundary value problems bounded Bramble C1+a Carleman's theorem Cauchy compact complex computations conformal mapping convergence corresponding defined denote derivatives difference equations Dirichlet problem domain eigenvalues eigenvectors elliptic equations elliptic systems elliptische error estimates exists a constant finite element method Fredholm integral equations Fredholm operator fundamental solution Galerkin Galerkin method Gilbert given Green function Heidelberg Hence holds holomorphic function homeomorphism homogeneous hyperanalytic functions hyperanalytic systems hypercomplex implies independent of h inequality inverse kernels Lemma let us consider linear Lopatinski condition Math matrix Neumann functions Neumann problem nonlinear norm normal form numerical partial differential equations plane polynomials priori estimate Proc Proof properties prove quadrature regular representation formula right hand side satisfies Section semilinear singular integral equations solving space Springer suitable theory tions transformation uniformly vanish variables vector Vekua Wendland yields zero