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Global Structure for Nonlinear Operators in Differential and Integral Equations I. Folds

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Topological Nonlinear Analysis II

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 27))

Abstract

For the classic Dirichlet problem

$$\left\{ {\begin{array}{*{20}c} {\Delta u = g} & {\text{in}} & \Omega \\ {u = 0} & {\text{on}} & {\partial \Omega } \\ \end{array} } \right.$$

for Ω a bounded domain in ℝn, for each g there exists a unique solution u, where u and g are in appropriate spaces H and H′, respectively. In fact, Δ−1 : g→ u is a bounded linear isomorphism (onto) and so Δ and Δ−1 are homeomorphisms. In particular, for each \(\bar g\) there is a unique point \(\bar u\) in Δ−1 (\(\bar g\)), and if g is near \(\bar g\), then the corresponding solution u = Δ−1(g) is near \(\bar u\).

The authors were partially supported by NSERC Contract #A7357. P. T. Church is grateful to the University of Alberta for its hospitality during the summer of 1995.

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  1. Department of Mathematics, Syracuse University, 13244-1150, Syracuse, New York, USA

    P. T. Church

  2. Department of Mathematics, University of Alberta, T6G 2G11, Edmonton, Alberta, Canada

    J. G. Timourian

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  1. Department of Mathematics, University of Rome, Tor Vergata, 00133, Rome, Italy

    Michele Matzeu  & Alfonso Vignoli  & 

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Church, P.T., Timourian, J.G. (1997). Global Structure for Nonlinear Operators in Differential and Integral Equations I. Folds. In: Matzeu, M., Vignoli, A. (eds) Topological Nonlinear Analysis II. Progress in Nonlinear Differential Equations and Their Applications, vol 27. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4126-3_3

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