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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 ( | u ( x ) | ) 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  2 and  3 and of the Bernstein problem on the flatness of minimal area graphs in  3 . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

Boundaries of prescribed mean curvature

Eduardo H. A. Gonzales, Umberto Massari, Italo Tamanini (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The existence of a singular curve in R 2 is proven, whose curvature can be extended to an L 2 function. The curve is the boundary of a two dimensional set, minimizing the length plus the integral over the set of the extension of the curvature. The existence of such a curve was conjectured by E. De Giorgi, during a conference held in Trento in July 1992.

Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents

Manuel del Pino, Fethi Mahmudi, Monica Musso (2014)

Journal of the European Mathematical Society

Let Ω be a bounded domain in n with smooth boundary Ω . We consider the equation d 2 Δ u - u + u n - k + 2 n - k - 2 = 0 in Ω , under zero Neumann boundary conditions, where Ω is open, smooth and bounded and d is a small positive parameter. We assume that there is a k -dimensional closed, embedded minimal submanifold K of Ω , which is non-degenerate, and certain weighted average of sectional curvatures of Ω is positive along K . Then we prove the existence of a sequence d = d j 0 and a positive solution u d such that d 2 | u d | 2 S δ K as d 0 in the sense of measures, where δ K ...

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