Partial regularity for manifold constrained p(x)-harmonic maps

TL;DR

This paper proves partial regularity results for manifold constrained p(x)-harmonic maps, showing they are smooth outside a small singular set and estimating its Hausdorff dimension.

Contribution

It establishes the regularity of p(x)-harmonic maps constrained to manifolds and bounds the size of their singular set.

Findings

Maps are $C^{1,eta}$ outside a measure-zero set

Provides an upper bound for the Hausdorff dimension of the singular set

Advances understanding of regularity in variable exponent harmonic maps

Abstract

We prove that manifold constrained $p(x)$-harmonic maps are $C^{1,\beta}$-regular outside a set of zero $n$-dimensional Lebesgue's measure, for some $\beta \in (0,1)$. We also provide an estimate from above of the Hausdorff dimension of the singular set.