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


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Article Info

DATE

2001-01

AUTHORS

Giovanni Alberti, Luigi Ambrosio, Xavier Cabré

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 Rn. We extend to all nonlinearities F∈C2 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)=u3−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. More... »

PAGES

9-33

References to SciGraph publications

  • 1962-01. On the oriented Plateau Problem in RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO SERIES 2
  • 2000-01. Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 1989. Monotonicity of the Energy for Entire Solutions of Semilinear Elliptic Equations in PARTIAL DIFFERENTIAL EQUATIONS AND THE CALCULUS OF VARIATIONS
  • 1969-09. Minimal cones and the Bernstein problem in INVENTIONES MATHEMATICAE
  • 1998-03. Symmetry or not? in THE MATHEMATICAL INTELLIGENCER
  • 1990. Calibrations and New Singularities in Area-minimizing Surfaces: A Survey in VARIATIONAL METHODS
  • 1999-04. Some remarks on a conjecture of De Giorgi in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 1998-07. On a conjecture of De Giorgi and some related problems in MATHEMATISCHE ANNALEN
  • 1985-03. Uniqueness of the solution of a semilinear boundary value problem in MATHEMATISCHE ANNALEN
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    http://scigraph.springernature.com/pub.10.1023/a:1010602715526

    DOI

    http://dx.doi.org/10.1023/a:1010602715526

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1039249404


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