2025/02/07 by Nicolas Beuvin, Alberto Farina, Beuvin, Nicolas +1
#math.AP
paper · pdf · doi:10.1007/s00208-026-03531-4
In this work we establish some rigidity results for Serrin's overdetermined problem \ - Δu=f(u) · in · Ω,\newline u > 0 · in · Ω,\newline u=0 · on · ∂ Ω,\newline \dfrac∂ u∂ η = \mathfrakc = const. · on · ∂ Ω, . when Ω⊂ ℝN is an epigraph (not necessarily globally Lipschitz-continuous) and u is a classical solution, possibly unbounded. In broad terms, our main results prove that Ω must be an affine half-space and u must be one-dimensional, provided the epigraph is bounded from below. These results hold when f is of Allen-Cahn type and N ≥ 2 or, alternatively, when f is locally Lipschitz-continuous (with no restriction on the sign of f(0)) and N ≤ 3. These results partially answer a question raised by Berestycki, Caffarelli and Nirenberg in [1]. Finally, when f(0) <0, we also prove a new monotonicity result, valid in any dimension N ≥ 2.