This paper studies a conjecture made by E. De Giorgi in 1978 concerning the one-dimensional character (or symmetry) of bounded, monotone in one direction, solutions of semilinear elliptic equations $\Delta u=F'(u)$ in all of $R^n$. We extend to all nonlinearities $F\in C^2$ the symmetry result in dimension $n=3$ previously established by the second and the 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 a new energy estimates which follows 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 $n \le 8$, namely that the level sets of $u$ are flat at infinity.

On a long standing conjecture of E. De Giorgi: old and recent results

ALBERTI, GIOVANNI;
2001

Abstract

This paper studies a conjecture made by E. De Giorgi in 1978 concerning the one-dimensional character (or symmetry) of bounded, monotone in one direction, solutions of semilinear elliptic equations $\Delta u=F'(u)$ in all of $R^n$. We extend to all nonlinearities $F\in C^2$ the symmetry result in dimension $n=3$ previously established by the second and the 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 a new energy estimates which follows 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 $n \le 8$, namely that the level sets of $u$ are flat at infinity.
Alberti, Giovanni; Ambrosio, L.; Cabre', X.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11568/185396
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 146
  • ???jsp.display-item.citation.isi??? 145
social impact