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25994 results in Abstract analysis

Page 953 of 1300

  • Infiltrations in immiscible fluids systems

  • Gian Paolo Leonardi
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 131 / Issue 2 / April 2001
    Published online by Cambridge University Press:
    11 July 2007, pp. 425-436
    Print publication:
    April 2001

  • In this paper we prove a certain regularity property of configurations of immiscible fluids, filling a bounded container Ω and locally minimizing the interface energy ∑i<jcij‖Sij‖, where Sij represents the interface between fluid i and fluid j, ‖·‖ stands for area or more general area-type functional, and cij is a positive coefficient. More precisely, we show that, under strict triangularity of the cij, no infiltrations of other fluids are allowed between two main ones. A remarkable consequence of this fact is the almost-everywhere regularity of the interfaces. Our analysis is performed in general dimension n ≥ 2 and with volume constraints on fluids.

  • Preface

  • Ravi P. Agarwal, National University of Singapore, Maria Meehan, Dublin City University, Donal O'Regan, National University of Ireland, Galway
  • Book:
    Fixed Point Theory and Applications
    Published online:
    04 September 2009
    Print publication:
    22 March 2001, pp vii-x

  • Summary

    Perhaps the most well known result in the theory of fixed points is Banach's contraction mapping principle. It is therefore fitting that we commence this book with a discussion of contractions and a proof of this result. In addition in Chapter 1, a local version and a generalisation of Banach's contraction theorem are presented. We choose the problem of existence and uniqueness of solutions of certain first order initial value problems to demonstrate the results detailed in the chapter.

    It is inevitable that any discussion on contractive maps will lead naturally to another on nonexpansive maps, which is why we choose this as the topic of Chapter 2. Schauder's theorem for nonexpansive maps is presented but the main theorem discussed is a result proved independently in 1965 by Browder, Göhde and Kirk which shows that each nonexpansive map F : C → C, where C is a particular set in a Hilbert space, has at least one fixed point. As a natural lead in to the next chapter, we close Chapter 2 with a nonlinear alternative of Leray–Schauder type for nonexpansive maps.

    Chapter 3 is concerned with continuation methods for contractive and nonexpansive maps. We show initially that the property of having a fixed point is invariant by homotopy for contractions. Using this result a nonlinear alternative of Leray–Schauder type is presented for contractive maps and subsequently generalised for nonexpansive maps. An application of the nonlinear alternative for contractions is demonstrated with a second order homogeneous Dirichlet problem.

Page 953 of 1300