Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fine regularity of fractional harmonic maps and applications

Kyeongbae Kim, Simon Nowak, Yannick Sire

Abstract

In this paper, we derive several regularity results for harmonic mappings into Euclidean spheres associated with rather general energies related to fractional Sobolev spaces. These maps generalize families of maps introduced by Da Lio, Rivière and Schikorra and are related to harmonic maps with free boundaries. In our context, there is in general no monotonicity formula, which prevents the use of some classical methods. Despite this limitation, under natural assumptions on a Gagliardo-type energy, we succeed in proving a variety of small energy regularity results and improve on known results, even in the isotropic case for which some monotonicity formula is available. To this end, we exploit recent developments in the regularity theory of nonlocal equations and as a by-product, we explain how these results apply to classes of harmonic maps with free boundary and lead to new potential-theoretic estimates. As another application, we obtain higher differentiability results for the fractional harmonic map heat flow.

Fine regularity of fractional harmonic maps and applications

Abstract

In this paper, we derive several regularity results for harmonic mappings into Euclidean spheres associated with rather general energies related to fractional Sobolev spaces. These maps generalize families of maps introduced by Da Lio, Rivière and Schikorra and are related to harmonic maps with free boundaries. In our context, there is in general no monotonicity formula, which prevents the use of some classical methods. Despite this limitation, under natural assumptions on a Gagliardo-type energy, we succeed in proving a variety of small energy regularity results and improve on known results, even in the isotropic case for which some monotonicity formula is available. To this end, we exploit recent developments in the regularity theory of nonlocal equations and as a by-product, we explain how these results apply to classes of harmonic maps with free boundary and lead to new potential-theoretic estimates. As another application, we obtain higher differentiability results for the fractional harmonic map heat flow.
Paper Structure (13 sections, 26 theorems, 286 equations)

This paper contains 13 sections, 26 theorems, 286 equations.

Table of Contents

  1. Introduction
  2. Main results in the critical case
  3. Main results in the supercritical case
  4. Results for time-dependent equations
  5. Applications to harmonic maps with free boundary.
  6. Preliminaries
  7. Fractional Sobolev and BMO spaces
  8. Basic inequalities
  9. Div-curl lemma
  10. Decay estimates for homogeneous systems
  11. Decay estimates
  12. Regularity for $p=2$
  13. Applications to harmonic maps with free boundaries and geometric flows of Plateau-type

Key Result

theorem 1

Under the above assumptions, let $u$ be a weak solution to with $p=\frac{n}{s}$. Suppose $B_{2R}(x_0)\Subset \Omega$. Then there is a small constant $\delta=\delta(n,s,N,\Lambda)$ such that if $\mu\in L^{{\gamma}}(B_{2R}(x_0))$ with $\frac{1}{p-1}\left(sp-\frac{n}{\gamma}\right)<\delta$, we have $u\in C^{\frac{1}{p-1}\left(sp-\frac{n}{\gamma}\right)}(B_R(x_

Theorems & Definitions (55)

  • definition 1
  • definition 2
  • theorem 1
  • remark 1
  • theorem 2
  • theorem 3
  • theorem 4
  • remark 2
  • theorem 5
  • remark 3
  • ...and 45 more