Stationary non-radial localized patterns in the planar Swift-Hohenberg PDE: constructive proofs of existence

The paper’s Theorem 3.2 states a Newton–Kantorovich fixed-point criterion for the planar Swift–Hohenberg problem in the D2-symmetric Hilbert space H^l, using bounds Y0, Z1, and Z2(r) together with the radii/contractivity conditions 1/2 Z2(r) r^2 − (1 − Z1) r + Y0 < 0 and Z1 + Z2(r) r < 1 to ensure a unique zero of F in B_r(u0). It defines ||u||_l = ||Lu||_2 with L an isometric isomorphism, and sets up T(u) = u − A F(u), where A is a bounded linear operator later constructed concretely as A = L^{-1} B (not merely L^{-1}) for the computer-assisted proof framework . The candidate solution reproduces the standard radii-polynomial argument: T maps the ball into itself and is a contraction, whence a unique fixed point ũ ∈ B_r(u0), and hence a unique zero of F there. The proof is self-contained and correct in this framework. Two differences: (i) the candidate specializes to A = L^{-1} and leverages the isometric norm identity ||Id_{L^2} − DF(v)A||_{L^2→L^2} = ||Id_{H^l} − A DF(v)||_{H^l→H^l}, which relies on A = L^{-1} and the definition ||u||_l = ||Lu||_2; this identity does not hold for the paper’s constructed A = L^{-1}B, but the paper does not need it . (ii) in the last step, the candidate incorrectly invokes injectivity of DF(ũ)A to conclude F(ũ)=0; the correct justification is simply injectivity of A (here A = L^{-1}), already established. With this minor correction, the model’s proof aligns with the paper’s theorem statement. The paper’s proof sketch cites standard results instead of rederiving them, but the argument is complete at the stated level of detail .