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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...

Nonlinear boundary value problems involving the extrinsic mean curvature operator

Jean Mawhin (2014)

Mathematica Bohemica

The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type · v 1 - | v | 2 = f ( | x | , v ) in B R , u = 0 on B R , where B R is the open ball of center 0 and radius R in n , and f is continuous. Comparison is made with similar results for the Laplacian. Topological and variational methods are used and the case of positive solutions is emphasized. The paper ends with the case of a general domain.

Regularity of stable solutions of p -Laplace equations through geometric Sobolev type inequalities

Daniele Castorina, Manel Sanchón (2015)

Journal of the European Mathematical Society

We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of Δ p u = g ( u ) in a smooth bounded domain Ω n . In particular, we obtain new L r and W 1 , r bounds for the extremal solution u when the domain is strictly convex. More precisely, we prove that u L ( Ω ) if n p + 2 and u L n p n - p - 2 ( Ω ) W 0 1 , p ( Ω ) if n > p + 2 .

The mean curvature measure

Quiyi Dai, Neil S. Trudinger, Xu-Jia Wang (2012)

Journal of the European Mathematical Society

We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the...

Two-dimensional curvature functionals with superquadratic growth

Ernst Kuwert, Tobias Lamm, Yuxiang Li (2015)

Journal of the European Mathematical Society

For two-dimensional, immersed closed surfaces f : Σ n , we study the curvature functionals p ( f ) and 𝒲 p ( f ) with integrands ( 1 + | A | 2 ) p / 2 and ( 1 + | H | 2 ) p / 2 , respectively. Here A is the second fundamental form, H is the mean curvature and we assume p > 2 . Our main result asserts that W 2 , p critical points are smooth in both cases. We also prove a compactness theorem for 𝒲 p -bounded sequences. In the case of p this is just Langer’s theorem [16], while for 𝒲 p we have to impose a bound for the Willmore energy strictly below 8 π as an additional condition....

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