[Paper Review] Horizontal $\alpha$-Harmonic Maps

Francesca Da Lio, Tristan Rivière|arXiv (Cornell University)|Apr 19, 2016

Geometric Analysis and Curvature Flows13 references3 citations

TL;DR

This paper introduces horizontal α-harmonic maps into a distribution of planes on ℝ^m, generalizing classical harmonic maps to non-integrable distributions. For α = 1/2 in 1D and α = 2 in 2D, it establishes regularity by showing solutions satisfy a Schrödinger-type system with antisymmetric potential, enabling use of prior conservation law techniques to prove Hölder continuity and partial regularity.

ABSTRACT

Given a $C^1$ planes distribution $P_T$ on all ${\\mathbb R}^m$ we consider {\\em horizontal $\\alpha$-harmonic maps}, $\\alpha\\ge 1/2$, with respect to such a distribution. These are maps $u\\in H^{\\alpha}({{\\mathbb R}}^k,{{\\mathbb R}}^m)$ satisfying $P_T\ abla u=\ abla u$ and $P_T(u)(-\\Delta)^{\\alpha}u=0$ in ${\\mathcal D}'({{\\mathbb R}}^k).$ If the distribution of planes is integrable then we recover the classical case of $\\alpha$-harmonic maps with values into a manifold. In this paper we shall focus our attention to the case $\\alpha=1/2$ in dimension $1$ and $\\alpha=2$ in dimension $2$ and we investigate the regularity of the {\\em horizontal $\\alpha$-harmonic maps}. In both cases we show that such maps satisfy a Schr\\"odinger type system with an antisymmetric potential, that permits us to apply the previous results obtained by the authors. Finally we study the regularity of {\\em variational $\\alpha$-harmonic} maps which are critical points of $\\|(-\\Delta)^{\\alpha/2} u\\|^2_{L^2}$ under the constraint to be tangent (horizontal) to a given planes distribution. We produce a convexification of this variational problem which permits to write it's Euler Lagrange equations.

Motivation & Objective

• To generalize α-harmonic maps to non-integrable distributions of planes in ℝ^m, extending the classical theory beyond submanifolds.
• To investigate the regularity of horizontal α-harmonic maps when the distribution is non-integrable, particularly for α = 1/2 in 1D and α = 2 in 2D.
• To establish a variational framework for α-harmonic maps constrained to horizontal distributions, using convexification to derive Euler-Lagrange equations.
• To show that the resulting equations admit a Schrödinger-type structure with antisymmetric potential, enabling use of existing conservation law machinery.
• To prove partial Hölder regularity of weak solutions using commutator estimates and pseudo-differential operator theory.

Proposed method

• Define horizontal α-harmonic maps as u ∈ H^α(ℝ^k, ℝ^m) satisfying P_T∇u = ∇u and P_T(u)(−Δ)^αu = 0 in 𝒟′(ℝ^k), where P_T is a C¹ field of orthogonal projections.
• For α = 1/2 in 1D and α = 2 in 2D, derive a system of pseudo-differential equations with antisymmetric potential by analyzing commutators involving (−Δ)^α/2 and projections.
• Use the structure of 3-commutators and integrability-by-compensation properties to relate the non-local equation to a local system of Schrödinger-type PDEs.
• Apply prior results on conservation laws for systems with antisymmetric potentials (from [13] and [5]) to deduce Hölder continuity of solutions.
• Reformulate the variational problem for α-harmonic maps under horizontal constraints by convexifying the energy functional to derive Euler-Lagrange equations.
• Express the Euler-Lagrange equations as a matrix system involving pseudo-differential operators, with coefficients derived from commutators of P_T, P_N, (−Δ)^α/2, and the Riesz transform.

Experimental results

Research questions

• RQ1 Can the regularity theory for α-harmonic maps be extended to non-integrable distributions of planes, rather than just submanifolds?
• RQ2 What is the structure of the Euler-Lagrange equation for horizontal α-harmonic maps when the distribution is non-integrable?
• RQ3 How do commutator estimates and pseudo-differential operator theory enable regularity results in the absence of integrability?
• RQ4 Can the non-local equation P_T(u)(−Δ)^αu = 0 be transformed into a local system with antisymmetric potential to leverage existing conservation laws?
• RQ5 What is the variational structure of horizontal α-harmonic maps, and how can convexification yield a tractable Euler-Lagrange formulation?

Key findings

• For α = 1/2 in 1D, horizontal 1/2-harmonic maps satisfy a Schrödinger-type system with antisymmetric potential, enabling application of conservation law techniques to prove Hölder continuity.
• For α = 2 in 2D, the system reduces to a similar structure with antisymmetric potential, and the same conservation law machinery yields partial regularity results.
• The commutator terms involving (−Δ)^α/2 and projections P_T, P_N are shown to be bounded in H^1(ℝ) when P_T ∈ H^{1/2}(ℝ) ∩ L^∞(ℝ), enabling control of non-local terms.
• The Euler-Lagrange equation for the variational problem is reformulated as a matrix system of pseudo-differential operators, with coefficients expressed via commutators.
• The matrix of pseudo-differential operators C − 2D in the system (A.40) is shown to be bounded on L^2, ensuring the system is well-posed under suitable regularity of P_T.
• The key technical advance is the convexification of the constrained variational problem, allowing derivation of the Euler-Lagrange equations in a form amenable to regularity analysis.