Dirichlet problem for weakly harmonic maps with rough data

TL;DR

This paper proves the existence and smoothness of weakly harmonic maps with rough boundary data in certain domains, using a nonvariational approach, and explores boundary data perturbations for stable harmonic maps.

Contribution

It introduces a nonvariational method to solve Dirichlet problems for weakly harmonic maps with rough boundary data and demonstrates local smoothness of solutions.

Findings

Solutions exist for boundary data in L^{ abla}(oundary ext{Omega}) or BMO(oundary ext{Omega})

Solutions are locally smooth (C^{ ext{infty}}_{loc})

Boundary data can be large if domain is smooth and boundary maps are perturbed from stable harmonic maps

Abstract

Weakly harmonic maps from a domain $\Omega$ (the upper half-space $\Rd$ or a bounded $C^{1,\alpha}$ domain, $\alpha\in (0,1]$) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes $L^{\infty}(\partial\Omega)$ or $BMO(\partial\Omega)$, we establish solvability of the resulting boundary value problems by means of a nonvariational method. As a by-product, solutions are shown to be locally smooth, $C^{\infty}_{loc}$. Moreover, we show that boundary data can be chosen large in the underlying topologies if $\Omega$ is smooth and bounded by perturbing strictly stable smooth harmonic maps.