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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n plus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is one-dimensional if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to 7"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>7</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \leq 7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A stronger version of the conjecture, also widely studied and often called “the stable De Giorgi conjecture”, proposes that every stable solution in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be one-dimensional for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to 7"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>7</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \leq 7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . To this date, both conjectures remain open for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 less-than-or-equal-to n less-than-or-equal-to 7"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>n</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>7</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">3 \leq n \leq 7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . An elegant variant of this problem, advocated by Caffarelli, Córdoba, and Jerison since the 1990s, considers a free boundary version of the Allen–Cahn equation. This variant features a step-like double-well potential, leading to multiple free boundaries. Locally, near each free boundary, the solution satisfies the Bernoulli free boundary problem. However, the interaction of the free boundaries causes the global behavior of the solution to resemble that of the Allen–Cahn equation. In this paper, we establish the validity of the stable De Giorgi conjecture in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the free boundary Allen–Cahn equation and, as a corollary, we prove the corresponding De Giorgi conjecture for monotone solutions in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . To obtain these results, a key aspect of our work is to address a classical open problem in free boundary theory of independent interest: the classification of global stable solutions to the one-phase Bernoulli problem in three dimensions. This result, which is the core of our paper, implies universal curvature estimates for local stable solutions to Bernoulli, and serves as a foundation for adapting some classical ideas from minimal surface theory—after significant refinements—to the free boundary Allen–Cahn equation.