Entire solutions of the minimal surface equation

A Hölder Estimate for Entire Solutions to the Two-valued Minimal Surface Equation

We prove a Hölder estimate near infinity for solutions to the twovalued minimal surface equation over R2 \ {0}, and give a Bernstein-type theorem in case the solution can be extended continuously across the origin. The main results follow by modifying methods used to study exterior solutions to equations of minimal surface type.

Entire Solutions of the Allen-cahn Equation and Complete Embedded Minimal Surfaces

We review some recent results on construction of entire solutions to the classical semilinear elliptic equation ∆u+u−u = 0 in R . In various cases, large dilations of an embedded, complete minimal surface approximate the transition set of a solution that connects the equilibria ±1. In particular, our construction answers negatively a celebrated conjecture by E. De Giorgi in dimensions N ≥ 9.

Entire Solutions of a Fully Nonlinear Equation

Entire Solutions of the Allen-cahn Equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in R

We consider minimal surfaces M which are complete, embedded and have finite total curvature in R, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation ∆u+ f(u) = 0 in R. Here f = −W ′ with W bistable and balanced, for instance W (u) = 1 4 (1 − u). We assume that M has m ≥ 2 ends, and additionally that M is non-degenerate, in the sense that its bounded Jacobi fields a...

ENTIRE SOLUTIONS OF THE ALLEN-CAHN EQUATION AND COMPLETE EMBEDDED MINIMAL SURFACES OF FINITE TOTAL CURVATURE IN R3 by

— We consider minimal surfaces M which are complete, embedded and have finite total curvature in R3, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation ∆u+f(u) = 0 in R3. Here f = −W ′ with W bistable and balanced, for instance W (u) = 1 4 (1− u2)2. We assume that M has m ≥ 2 ends, and additionally that M is non-degenerate, in the sense that its bounded Jacobi fiel...