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;
2001-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
