### Abstract

The shape of a condenser of maximal cross-sectional area at a given capacity is derived in an analytical explicit form. Optimization is performed in the class of practically arbitrary curves by solution of the Dirichlet boundary-value problem for a complex coordinate and expansions of the Cauchy integral kernels in Chebyshev polynomials. The criterion (area) becomes a quadratic form of the Fourier coefficients and both the necessary and sufficient extremum conditions are rigorously satisfied such that a global and unique extremum is achieved. The resulting curve coincides with the Polubarinova-Kochina contour of a concrete dam of constant hydraulic gradient, which in its own term coincides with the Taylor-Saffman bubble. In the limit of high capacitance, the Polubarinova-Kochina contour tends to the Saffman-Taylor finger, which in its own turn coincides with the Morse-Feshbach condenser contour of constant field intensity. Thus the contour found is of a minimal breakdown danger in the dielectric between charged surfaces (non-isoperimetric optimum) and of maximal confined area (isoperimetric extremum).

Original language | English |
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Pages (from-to) | 485-494 |

Number of pages | 10 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 457 |

Issue number | 2006 |

DOIs | |

Publication status | Published - Feb 8 2001 |

### Keywords

- Dirichlet's boundary-value problem
- Global extremum
- Laplace's equation
- Optimization
- Shaping

### ASJC Scopus subject areas

- General