Connectivity of Phase Boundaries in Strictly Convex Domains

Sternberg, Peter; Zumbrun, Kevin

doi:10.1007/s002050050081

Cited by 131 publications

(155 citation statements)

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“…The proofs of our main results are based on a geometric formula, which will be given in Theorem 6 below, and which can be considered as a weighted Poincaré inequality. The use of such a type of formula in the Euclidean setting was started in [SZ98a,SZ98b] and its importance for symmetry results was explained in [Far02]. Further applications to PDEs have been given in [FSV08,SV08,FV08].…”

Section: Resultsmentioning

confidence: 99%

Stable Solutions of Elliptic Equations on Riemannian Manifolds

2011

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Abstract. This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincaré inequality which allows to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.

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Section: Resultsmentioning

confidence: 99%

Stable Solutions of Elliptic Equations on Riemannian Manifolds

2011

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“…Next we state a lemma from [61], which allows us to estimate the size of certain sets. In what follows given a set E and s > 0 we define the set…”

Section: Proposition 48 Assume That W Satisfies (24)-(27) and Thamentioning

confidence: 99%

Second-Order Γ-limit for the Cahn–Hilliard Functional

Leoni

Murray

2015

Arch Rational Mech Anal

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“…Since then, the existence of solutions with a single interface intersecting the boundary has been established and studied by many authors. See [Alikakos et al 2000;Bronsard and Stoth 1996;Flores et al 2001;Kowalczyk 2005;Padilla and Tonegawa 1998;Sternberg and Zumbrun 1998] and the references therein. However, the existence of multiple interfaces is only proved, in the one-dimensional case, for the AllenCahn equation (with inhomogeneous terms)…”

Section: Introductionmentioning

confidence: 99%