On approximating minimizers of convex functionals with a convexity constraint by singular Abreu equations without uniform convexity

We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.

Generalized radial inversion of 2D potential‐field data

We introduce a new 2D method for inverting potential‐field data with model constraints designed by the interpreter. Our method uses an interpretation model consisting of a source with polygonal cross‐section whose vertices are described by polar coordinates with an origininside the source. With this coordinate system, constraints in an inversion are easier to develop and apply. Our inversion method assumes a known physical property contrast for the source and estimates the radii associated with the polygon vertices for a fixed number of equally spaced angles from 0° to 360°. A wide variety of constraints may be used to stabilize the solutions by introducing information about the source shape. The method recovers stable solutions whose shapes range from almost circular or pear‐shaped to elongated in one or more directions. The convexity constraint applied to the source shape, despite requiring no quantitative information, is more versatile than the other constraints. The convexity constraint efficiently recovers source geometries that are either isometric or elongated in one direction.