Boundary regularity for minimizing biharmonic maps

TL;DR

This paper establishes full boundary regularity for minimizing biharmonic maps with smooth boundary conditions, extending techniques from harmonic maps and utilizing recent boundary monotonicity formulas.

Contribution

It proves full boundary regularity for biharmonic maps by combining boundary monotonicity formulas, compactness arguments, and partial regularity results, advancing understanding of biharmonic map regularity.

Findings

Full boundary regularity for minimizing biharmonic maps.

Nonexistence of nonconstant boundary tangent maps.

Application of boundary monotonicity formula.

Abstract

We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Scheven's interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong--Lamm--Wang.