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Monotonicity of the Energy for Entire Solutions of Semilinear Elliptic Equations

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Partial Differential Equations and the Calculus of Variations

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 1))

Abstract

In the mathematical theory of phase transitions in Van der Waals fluids (see, for instance, Alikakos & Bates [AB], Caginalp [CA], Gurtin [GU], Hagau & Serrin [HS], Modica [M03]) the following problem arises: to study the asymptotic behaviour as ε → 0+ of solutions to the semilinear elliptic equations

$$ - {\varepsilon ^2}\Delta u(x) + f(u(x)) = 0\;(x \in \Omega )$$
((1))

on an open subset Ω of R n. The model case is f(u) = u 3 − u.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Via F.Buonarroti 2, I-56127, Pisa, Italy

    Luciano Modica

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  1. Luciano Modica
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Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Universita di Pisa, Via F. Buonarroti 2, 56100, Pisa, Italy

    Ferruccio Colombini , Antonio Marino , Luciano Modica  & Sergio Spagnolo , ,  & 

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Dedicated to Ennio De Giorgi on his sixtieth birthday

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© 1989 Springer Science+Business Media New York

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Modica, L. (1989). Monotonicity of the Energy for Entire Solutions of Semilinear Elliptic Equations. In: Colombini, F., Marino, A., Modica, L., Spagnolo, S. (eds) Partial Differential Equations and the Calculus of Variations. Progress in Nonlinear Differential Equations and Their Applications, vol 1. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4615-9831-2_14

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  • DOI: https://doi.org/10.1007/978-1-4615-9831-2_14

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4615-9833-6

  • Online ISBN: 978-1-4615-9831-2

  • eBook Packages: Springer Book Archive

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