Abstract:In this short note we present new results on a higher-dimensional generalization of De~Giorgi's conjecture for Allen--Cahn type equations, focusing on dimensions $n \ge 9$. Although counterexamples are known in this regime, our goal is to identify assumptions on solutions that still enforce one-dimensional symmetry. We prove an analogue of Savin's theorem in arbitrary dimension: for energy-minimizing solutions whose level sets enjoy n-7 directions of monotonicity, we deduce one-dimensional symmetry. In the same spirit, we extend these ideas to nonlocal phase transitions, and we discuss an application to free boundary problems Finally, we establish a counterpart of the Ambrosio--Cabré theorem for solutions that are not necessarily energy minimizers and may lack bounded energy density, assuming instead n-2 directions of monotonicity everywhere. Overall, this note aims to further strengthen the connection between phase transition models and minimal surface theory.
From: Gabriele Ferla [view email]
[v1]
Mon, 30 Mar 2026 22:53:05 UTC (12 KB)
[v2]
Tue, 16 Jun 2026 13:41:25 UTC (12 KB)
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