Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis (Ph. D.)--University of Chicago, 1912. / Vita. "Reprinted from American journal of mathematics, vol. XXXV, no. 4." Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, 1916. / Vita. "A Private Edition Distributed by the University of Chicago Libraries, 1916." "Reprinted from the Transactions of the American Mathematical Society, Volume 17, Number 4, October, 1916." Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, 1910. / From the Transactions of The American mathematics society, v. 13, 1912. Also available on the Internet.

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In this thesis the solution to the variational problem of Signorini is studied, namely: (i) Δv = 0 in Ω; (ii) v ≥ ѱ on əΩ; (iii) əv/əѵ ≤ g on əΩ; (iv) (v- ѱ) (əv/əѵ – g) = 0 on əΩ where Ω is a domain in R[sup n], and v is the unit inner normal vector to əΩ. In the case n = 2 a regularity theorem is proved. It is shown that if ѱ Є C[sup 1,α] (əΩ), g Є Lip α(əΩ) then v Є C[sup 1,α] (əΩ) if α < 1/2 . An example is given to shown that this result is optimal. The method of proof relies on techniques of complex analysis and therefore does not extend to higher dimensions. For n > 2 the case where Ω, is unbounded, or equivalently, where ѱ is unbounded in a neighbourhood of some point of əΩ is considered. This is a situation where known existence theorems do not apply. Some sufficient conditions for the pair (ѱ,g) are derived that will ensure the existence of a solution in this case, thereby extending some results obtained by A. Beurling and P. Malliavin in the two dimensional case. The proof involves a variational problem in a Hilbert space analogous to the one considered by Beurling and Malliavin, and some pointwise estimates of Riesz transforms. / Science, Faculty of / Mathematics, Department of / Graduate

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by Xiao-Qing Jin. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1986 / Bibliography: leaves 47-48

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The main focus of this PhD thesis is the study of microstructures and geometric patterns in materials, in the framework of the Calculus of Variations. My PhD research, carried out in collaboration with my supervisor Mariapia Palombaro and Marcello Ponsiglione, led to the production of three papers [21, 22, 23]. Papers [21, 22] have already been published, while [23] is currently in preparation. This thesis is divided into two main parts. In the first part we present the results obtained in [22, 23]. In these two works geometric patterns have to be understood as patterns of dislocations in crystals. The second part is devoted to [21], where suitable microgeometries are needed as a mean to produce gradients that display critical integrability properties.