Derivation of a boundary monotonicity inequality for variationally biharmonic maps

TL;DR

This paper establishes a boundary monotonicity formula for biharmonic maps with Dirichlet conditions, enabling partial and full boundary regularity results in super-critical dimensions.

Contribution

It introduces a novel boundary monotonicity formula for variationally biharmonic maps, advancing the understanding of their regularity properties.

Findings

Partial regularity for variationally biharmonic maps

Full boundary regularity for minimizing biharmonic maps

Boundary monotonicity formula as a key tool

Abstract

We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in super-critical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.