## Dirichlet's Principle: Conformal Mapping, and Minimal Surfaces |

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### Contents

Introduction | 1 |

Semicontinuity of Dirichlets integral Dirichlets Principle for cir | 11 |

Proof of Dirichlets Principle for general domains | 23 |

Copyright | |

16 other sections not shown

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### Common terms and phrases

admissible function admissible vectors analytic function arbitrarily small arbitrary assume boundary curve boundary points boundary slit boundary value problem branch points catenoid cells Chapter circular closed curve closed subdomain condition conformal mapping consider constant continuous contour convergence coordinates corresponding defined degeneration denote derivatives Dirichlet's integral Dirichlet's Principle disk domain G doubly connected dx dy equation existence Figure finite number formula free boundary genus zero greatest lower bound Green's function harmonic function harmonic vectors Hence inequality infinity interior Jordan arc Jordan curves least area lemma length lim inf mapping theorem minimal surface minimizing sequence neighborhood obtain parameter domains piecewise smooth plane domain Plateau's problem polygon prescribed proof proved radius Riemann domains semicontinuity simply connected single-valued solution stationary streamlines sufficiently small surface G surface of least tends to zero tion transformation uniformly unit circle univalent unstable minimal surfaces vanishes variational problem w-plane