[Paper Review] On solvability of nonlinear partial differential systems of any order in the complex plane

Yifei Pan|arXiv (Cornell University)|Sep 15, 2011

Geometric Analysis and Curvature Flows7 references3 citations

TL;DR

This paper establishes a general existence theorem for nonlinear partial differential systems of any order in one complex variable, extending the Nijenhuis-Woolf theorem on J-holomorphic curves. It proves local and global solvability for systems with m-Laplace operators as principal parts, showing solutions exist for any jet order up to 2m−1 and vanishing order 2m at the origin, with applications to harmonic maps and complex analysis.

ABSTRACT

We prove a general existence theorem for nonlinear partial differential systems of any order in one complex variable. A special case of first order contains a well-known theorem of Nijenhuis and Woolf concerning local existence of J-holomorphic curves on almost complex manifolds. As an application to differential geometry, we prove that non constant harmonic map always exists locally from a Riemann surface to a Riemannian manifold with a prescribed tangent vector at a given point.

Motivation & Objective

• To establish a general existence theorem for nonlinear partial differential systems of arbitrary order in one complex variable.
• To extend the classical Nijenhuis-Woolf theorem on J-holomorphic curves to higher-order systems.
• To prove local solvability for m-Laplace systems with prescribed jet data up to order 2m−1.
• To establish global solvability for autonomous m-Laplace systems with vanishing jet data of order 2m at the origin.
• To apply the results to differential geometry, proving local existence of non-constant harmonic maps from Riemann surfaces to Riemannian manifolds with prescribed initial data.

Proposed method

• Uses complex analysis and the Cauchy-Riemann operator framework to analyze systems involving higher-order derivatives in one complex variable.
• Applies a constructive method based on power series expansions and jet data to solve systems with m-Laplace principal parts.
• Employs the complex derivatives ∂ = ½(∂x − i∂y) and ̄∂ = ½(∂x + i∂y) to express higher-order partial derivatives in terms of ∂ and ̄∂ operators.
• Implements a recursive technique to solve the system by controlling the jet data up to order 2m−1 near the origin.
• Uses the unique continuation property to ensure solution uniqueness up to jet order, despite non-uniqueness in general.
• Applies the method to the ∂̄^m f = F system, proving existence of C^{m+k} solutions near the origin with prescribed initial jet conditions.

Experimental results

Research questions

• RQ1 Can a general existence theorem be established for nonlinear PDE systems of any order in one complex variable?
• RQ2 To what extent does the Nijenhuis-Woolf theorem on first-order J-holomorphic curves extend to higher-order systems?
• RQ3 Under what conditions does the m-Laplace operator Δ^m admit local solutions for arbitrary jet data of order 2m−1?
• RQ4 Can global solutions be constructed for autonomous m-Laplace systems with vanishing jet data of order 2m at the origin?
• RQ5 Does the existence of local harmonic maps from Riemann surfaces to Riemannian manifolds with prescribed initial values follow from the PDE solvability result?

Key findings

• Theorem A establishes local solvability of the m-Laplace system Δ^m u = A(z, u, ∇u, ..., ∇^{2m−1}u) for C^k functions A, with solutions in C^{2m+k} class in a small disk |z| ≤ R.
• Solutions satisfy u = p(z) + O(|z|^{2m}) near the origin, where p(z) is a polynomial of degree ≤ 2m−1.
• Theorem B proves global existence of solutions for autonomous m-Laplace systems Δ^m u = A(u, ∇u, ..., ∇^{2m}u) with A(0) = 0 and ∇A(0) = 0, vanishing to order 2m at the origin.
• The radius R in Theorem A cannot be arbitrarily large, as shown by counterexamples like Δu = e^{2u} having no global solution.
• Theorem C proves the existence of a C^2 harmonic map φ from a neighborhood of z₀ ∈ S to a C^3 Riemannian manifold N with φ(z₀) = p and dφ(z₀)(∂/∂x) = v.
• The method applies to m-harmonic maps and bi-harmonic maps, and enables the definition of a Kobayashi metric on the tangent bundle of a Riemannian manifold via local harmonic maps.