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On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property

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Abstract

This paper studies a conjecture made by De Giorgi in 1978 concerning the one-dimensional character (or symmetry) of bounded, monotone in one direction, solutions of semilinear elliptic equations Δu=F′(u) in all of R n. We extend to all nonlinearities F∈C 2 the symmetry result in dimension n=3 previously established by the second and third authors for a class of nonlinearities F which included the model case F′(u)=u 3−u. The extension of the present paper is based on new energy estimates which follow from a local minimality property of u.

In addition, we prove a symmetry result for semilinear equations in the halfspace R + 4. Finally, we establish that an asymptotic version of the conjecture of De Giorgi is true when n≤8, namely that the level sets of u are flat at infinity.

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Authors and Affiliations

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

    Giovanni Alberti

  2. Scuola Normale Superiore, Piazza Cavalieri 7, 56100, Pisa, Italy

    Luigi Ambrosio

  3. Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Diagonal, 647. 08028, Barcelona, Spain

    Xavier Cabré

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  1. Giovanni Alberti
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  2. Luigi Ambrosio
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  3. Xavier Cabré
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Alberti, G., Ambrosio, L. & Cabré, X. On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property. Acta Applicandae Mathematicae 65, 9–33 (2001). https://doi.org/10.1023/A:1010602715526

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