Boundary regularity of a fourth order Alt-Caffarelli problem and applications to the minimization of the critical buckling load

2025/12/21 by Jimmy Lamboley, Lamboley, Jimmy, Mickaël Nahon +1
Computer Science · Engineering · Mathematics · #31A30 #35A16 #35B65 #35N25 #35R35 #49Q10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Topology Optimization in Engineering

paper · doi:10.48550/arxiv.2512.18626

openalex publication_date 2025/12/21 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

Abstract

We study a higher order analogue to the Alt-Caffarelli functional that arises in several shape optimization problems, among which the minimization of the critical buckling load of a clamped plate of fixed area. We obtain several regularity results up to the boundary in two dimensions, in particular we prove the full regularity of the boundary (analytic outside angles of opening ≈ 1.43π) near any point of density less than 1 of the optimal shape. These results are based on the monotonicity formula discovered by Dipierro, Karakhanyan, and Valdinoci, which we improve with a new epiperimetric inequality.

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