Regularity Results for an Optimal Design Problem with a Volume Constraint

TL;DR

This paper establishes regularity results for optimal design problems with volume constraints, proving Hoelder continuity of solutions and partial/full regularity of minimal set boundaries under certain conditions.

Contribution

It provides new regularity results for minimal configurations involving convex bulk energies with p-growth, including boundary regularity under eigenvalue closeness assumptions.

Findings

Hoelder continuity of the function u in minimal configurations

Partial regularity of the boundary of the minimal set E

Full regularity of the boundary under eigenvalue closeness conditions

Abstract

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with p-power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u;E), Hoelder continuity of the function u is proved as well as partial regularity of the boundary of the minimal set E. Moreover, full regularity of the boundary of the minimal set is obtained under suitable closeness assumptions on the eigenvalues of the bulk energies.