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Global Stable Solutions to the Free Boundary Allen--Cahn and Bernoulli Problems in 3D are One-Dimensional
A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen--Cahn equation in $\mathbb{R}^{n+1}$ is one-dimensional if $n \leq 7$. A stronger version of the conjecture, also widely studied and often called ``the stable De Giorgi conjecture'', proposes that every stable
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