Bernstein and De Giorgi type problems: new results via a geometric approach
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 4, p. 741-791

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form $\phantom{\rule{0.166667em}{0ex}}\mathrm{div}\phantom{\rule{0.166667em}{0ex}}\left(a\left(|\nabla u\left(x\right)|\right)\nabla u\left(x\right)\right)+f\left(u\left(x\right)\right)=0.$ Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in ${ℝ}^{2}$ and ${ℝ}^{3}$ and of the Bernstein problem on the flatness of minimal area graphs in ${ℝ}^{3}$. A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach is also flexible to very degenerate operators: as an application, we prove one-dimensional symmetry for $1$-Laplacian type operators.

Classification:  32H02,  30C45
@article{ASNSP_2008_5_7_4_741_0,
author = {Farina, Alberto and Sciunzi, Berardino and Valdinoci, Enrico},
title = {Bernstein and De Giorgi type problems: new results via a geometric approach},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 7},
number = {4},
year = {2008},
pages = {741-791},
zbl = {1180.35251},
mrnumber = {2483642},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2008_5_7_4_741_0}
}

Farina, Alberto; Sciunzi, Berardino; Valdinoci, Enrico. Bernstein and De Giorgi type problems: new results via a geometric approach. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 4, pp. 741-791. http://www.numdam.org/item/ASNSP_2008_5_7_4_741_0/

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