## An Axiomatic Treatment of Pairs of Elliptic Differential Equations |

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### Contents

PROPERTIES OF A SINGLE HARMONIC CLASS OF FUNCTIONS | 4 |

Regular Open Subsets of W | 17 |

3 The Superharmonic and Subharmonic Classes | 27 |

4 other sections not shown

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3C is parabolic 3C-extension of f 3C-measure 3C-modification Ascoli theorem assume Axiom D-III barrier for ft boundary bounded function Cauchy sequence classes with 3C compact subset comparable harmonic classes constant function continuous function converges Corollary Daniell integral denote directed by decreasing Dirichlet problem excluded set f is integrable f on dft ft is regular function in 3C function on dft H(dA H(dW h(xq increasing sequence integrable function isometric isomorphism K(dW Let 3C Let f Let h lim inf lim sup f lower semicontinuous lower semicontinuous function maximum principle non-negative function open subset outer-regular compact set outer-regular with respect point in ft point xq positive function Proof Proposition 6.3 real number regular compact region regular for 3C regular open set respect to 3C Riemann surface satisfies Axioms seminorm sequence of functions Theorem upper semicontinuous v e 3C v(xq whence x e ft xeft