Projection Method for Steady states of Cahn-Hilliard Equation with the Dynamic Boundary Condition

The paper’s energy-stability proof for the time-discrete projected AC scheme follows the standard “test-by-increment + boundary cancellation + Taylor remainder” strategy and is technically sound in its core manipulations (see the scheme (19) and Theorem 4, together with the inner-product estimates leading to (22)–(25) and the final inequality) . However, Lemma 3 asserts convexity for E_e under the global conditions S1 ≥ 1/2 max_{ϕ∈R} F''(ϕ), S2 ≥ 1/2 max_{ψ∈R} G''(ψ), which is impossible for the quartic double-well since F''(ϕ)=3ϕ^2−1 is unbounded, and the proposed split places −(1/4)ϕ^4 into E_e, making E_e non-convex regardless of S1 . The theorem’s conclusion explicitly relies on these impossible global bounds, so as stated it is incorrect. By contrast, the candidate solution executes the same energy estimate correctly and notes the needed hypothesis: either assume bounded second derivatives or take S1,S2 to dominate F'',G'' on the range attained by the iterates. The cancellation of boundary couplings via ϕ|_Γ=ψ and the use of the projection properties are consistent with the paper’s framework .