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

TL;DR

The paper develops a rigorous regularity theory for a fourth-order Alt-Caffarelli-type free boundary problem in two dimensions, motivated by minimizing the critical buckling load. It builds a Weiss-type monotonicity formula for the biharmonic energy and proves an epiperimetric inequality to obtain convergence toward 2-homogeneous blow-ups, classifying them into four canonical types with explicit angular openings. This framework yields detailed boundary regularity results, including C1,α regularity near flat and angular points and a structured description of boundary point types (flat, angular, nodal, junction, explosion). The methods combine spectral (buckling) basis constructions, inner-variation techniques, and conformal/hodograph analyses, and are extended to quasi-minimizers with applications to buckling eigenvalue optimization under area constraints.

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 $\approx 1.43π$) 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.

Theorems & Definitions (130)

• Remark 1
• Remark 2
• Theorem 3
• Theorem 4: Existence and uniqueness of blow-ups
• Theorem 5: $\epsilon$-regularity : flat case
• Theorem 6: $\epsilon$-regularity : angular case
• Theorem 7: $\epsilon$-regularity : nodal and isolated case
• Theorem 8
• Lemma 9
• proof