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Topological Aspects of Critical Points and Level Sets in Elliptic PDEs

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Geometry of PDEs and Related Problems

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2220))

Abstract

In this paper we discuss the emergence of topological structures through the critical points and level sets of solutions to elliptic PDEs. First, we analyze the local and global properties of the critical points of Green’s functions on complete Riemannian manifolds, showing that the number of critical points on a surface of finite type admits a topological upper bound, a property that does not hold in higher dimensions. Second, we introduce two technical tools, Thom’s isotopy theorem and a Runge-type global approximation theorem, which will allow us to construct solutions to a wide range of elliptic PDEs with level sets of complicated (sometimes bizarre) topologies. The model elliptic equation that we consider to illustrate the power of these techniques is the Helmholtz equation (monochromatic waves). These ideas are used in two seemingly unrelated applications: a 2001 conjecture of Sir Michael Berry about the existence of Schrödinger operators in Euclidean space with eigenfunctions having nodal lines of arbitrary knot type, and the construction of bounded solutions to the Allen-Cahn equation with level sets of any compact topology. The aim of these notes is not to provide detailed proofs of all the stated results but to introduce the main ideas and methods behind certain selected topics in the study of topological structures in PDEs. This is the set of lecture notes the second named author gave at the CIME Summer School held in Cetraro (Italy) during June 2017.

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Acknowledgements

The second named author is very grateful to the CIME Foundation and the organizers of the Summer School “Geometry of PDEs and Related Problems” for their kind invitation to deliver one of the courses. The authors are supported by the ERC Starting Grants 633152 (A.E.) and 335079 (D.P.-S.). This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554.

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain

    Alberto Enciso & Daniel Peralta-Salas

Authors
  1. Alberto Enciso
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  2. Daniel Peralta-Salas
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Correspondence to Daniel Peralta-Salas .

Editor information

Editors and Affiliations

  1. Dipt di Matematica e Informatica Ulisse Dini, Università degli Studi di Firenze, Firenze, Italy

    Chiara Bianchini

  2. Institut Elie Cartan – Faculté des Sciences et Technologies, Université de Lorraine, Vandoeuvre-les-Nancy, France

    Antoine Henrot

  3. Dipt di Matematica e Informatica Ulisse Dini, Università degli Studi di Firenze, Firenze, Italy

    Rolando Magnanini

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Enciso, A., Peralta-Salas, D. (2018). Topological Aspects of Critical Points and Level Sets in Elliptic PDEs. In: Bianchini, C., Henrot, A., Magnanini, R. (eds) Geometry of PDEs and Related Problems. Lecture Notes in Mathematics(), vol 2220. Springer, Cham. https://doi.org/10.1007/978-3-319-95186-7_3

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