Flattening Results for Elliptic PDEs in Unbounded Domains with Applications to Overdetermined Problems

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Geometry and Topology of some overdetermined elliptic problems (2012, background): “2 of [9], even though this is only a part of the original theorem): if f is locally Lipschitz, n = 2, and u ∈ C(2)(Ω) is a solution of (3) increasing in the second variable and with bounded gradient, then Ω is a half-plane.”
A Rigidity Result for Overdetermined Elliptic Problems in the Plane (2015, methodology): “The same result is proved in [16], but under the hypothesis that the domain is of class C (3).”
Monotonicity and one-dimensional symmetry for solutions of −Δpu = f(u) in half-spaces (2011, result): “Also we point out some interesting results recently obtained in [6] and in [18].”
Overdetermined problems with fractional Laplacian (2013, background): “Since a complete list of references cannot be given, we only quote: [1, 4, 7, 8, 11, 10, 12, 19, 20, 22, 23, 25, 28, 31, 32, 34, 33, 39, 38].”
Serrin’s overdetermined problem and constant mean curvature surfaces (2013, methodology): “[10] A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems.”