The paper derives explicit sup-norm gradient estimates for bounded positive Allen-Cahn solutions on Riemannian manifolds, and a Liouville theorem under nonnegative Ricci curvature.
De Giorgi, Convergence problems for functionals and operato rs, In Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rom e, 1978), pages 131-
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Gradient estimates for the Allen-Cahn equation on Riemannian manifolds
The paper derives explicit sup-norm gradient estimates for bounded positive Allen-Cahn solutions on Riemannian manifolds, and a Liouville theorem under nonnegative Ricci curvature.