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Mean curvature properties for $p$ -Laplace phase transitions

Berardino Sciunzi; Enrico Valdinoci — 2005

Journal of the European Mathematical Society

This paper deals with phase transitions corresponding to an energy which is the sum of a kinetic part of $p$ -Laplacian type and a double well potential $h_{0}$ with suitable growth conditions. We prove that level sets of solutions of $Δ_{p} u = h_{0}^{'} (u)$ possessing a certain decay property satisfy a mean curvature equation in a suitable weak viscosity sense. From this, we show that, if the above level sets approach uniformly a hypersurface, the latter has zero mean curvature.

Bernstein and De Giorgi type problems: new results via a geometric approach

Alberto Farina; Berardino Sciunzi; Enrico Valdinoci — 2008

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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 $div (a (| \nabla u (x) |) \nabla u (x)) + f (u (x)) = 0 .$ 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...

Liouville results for $m$ -Laplace equations of Lane-Emden-Fowler type

Lucio Damascelli; Alberto Farina; Berardino Sciunzi; Enrico Valdinoci — 2009

Annales de l'I.H.P. Analyse non linéaire

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Currently displaying 1 – 3 of 3

Mean curvature properties for p -Laplace phase transitions

Bernstein and De Giorgi type problems: new results via a geometric approach

Liouville results for m -Laplace equations of Lane-Emden-Fowler type

Mean curvature properties for $p$ -Laplace phase transitions

Liouville results for $m$ -Laplace equations of Lane-Emden-Fowler type