[Paper Review] Beltrami operators, non--symmetric elliptic equations and quantitative Jacobian bounds

Giovanni Alessandrini, Vincenzo Nesi|arXiv (Cornell University)|Jul 5, 2007

Analytic and geometric function theory24 references3 citations

TL;DR

This paper establishes a constructive method for generating primary pairs of quasiconformal mappings associated with non-symmetric elliptic operators, proving the pointwise positivity of the imaginary part of the product of their complex derivatives. It extends previous results on symmetric coefficients to the non-symmetric case and provides a quantitative lower bound for the Jacobian determinant of periodic σ-harmonic sense-preserving homeomorphisms of ℂ, resolving a conjecture on G-compactness for all K ≥ 1.

ABSTRACT

In recent studies on the G-convergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D'Onofrio, Iwaniec and Sbordone formulated a conjecture based on the existence of a so-called primary pair. Very recently, Bojarski proved the existence of one such pair. We provide a general, constructive, procedure for obtaining a new rich class of such primary pairs. This proof is obtained as a slight adaptation of previous work by the authors concerning the nonvanishing of the Jacobian of pairs of solutions of elliptic equations in divergence form in the plane. It is proven here that the results previously obtained when the coefficient matrix is symmetric also extend to the non-symmetric case. We also prove a much stronger result giving a quantitative bound for the Jacobian determinant of the so-called \emph{periodic} $σ$-harmonic sense preserving homeomorphisms of $\mathbb C$ onto itself.

Motivation & Objective

• To resolve the conjecture that the imaginary part of the product of complex derivatives of solutions to Beltrami equations remains positive almost everywhere.
• To extend results on Jacobian non-vanishing and higher integrability from symmetric to non-symmetric coefficient matrices in divergence-form elliptic equations.
• To establish a quantitative lower bound for the Jacobian determinant of periodic σ-harmonic homeomorphisms of the complex plane.
• To provide a constructive procedure for generating primary pairs in the non-symmetric case, building on earlier work on non-vanishing Jacobians.
• To demonstrate that G-compactness of Beltrami operators holds for all K ≥ 1, not just K ≤ 3 as previously known.

Proposed method

• Adapting a prior method for proving non-vanishing Jacobians of solutions to divergence-form elliptic equations in the plane to the non-symmetric case.
• Using the transformation between non-symmetric σ-harmonic functions and quasiregular mappings via the complex Beltrami equation.
• Applying the theory of H-convergence and correctors in the context of non-symmetric coefficient matrices in L∞(Ω;ℝ²ˣ²).
• Establishing a link between correctors in H-convergence and the Jacobian matrices of σ-harmonic mappings via boundary value problems.
• Deriving a quantitative exponent of higher integrability for σ-harmonic functions using the optimal K-quasiregular mapping framework.
• Proving that the optimal exponent of higher integrability is algebraically determined by the condition number of the coefficient matrix σ, even in the non-symmetric case.

Experimental results

Research questions

• RQ1 Does the imaginary part of the product of the complex derivatives of solutions to the Beltrami equation remain positive almost everywhere for non-symmetric coefficient matrices?
• RQ2 Can the non-vanishing Jacobian property for solutions of symmetric elliptic equations be extended to the non-symmetric case?
• RQ3 What is the quantitative lower bound for the Jacobian determinant of periodic σ-harmonic sense-preserving homeomorphisms of ℂ?
• RQ4 Is the G-compactness of Beltrami operators valid for all K ≥ 1, not just K ≤ 3?
• RQ5 Can the optimal exponent of higher integrability for σ-harmonic functions be algebraically characterized in the non-symmetric case?

Key findings

• The conjecture that Im(ΦzΨ̄z) > 0 a.e. in Ω holds for all K ≥ 1, resolving a key open problem in G-convergence of Beltrami operators.
• The G-compactness of the family of Beltrami operators is established for all K ∈ [1, ∞), extending the previous result valid only for K ≤ 3.
• A constructive procedure is provided for generating a rich class of primary pairs in the non-symmetric case, generalizing earlier results.
• A quantitative lower bound is proven for the Jacobian determinant of periodic σ-harmonic sense-preserving homeomorphisms of ℂ, valid for all K ≥ 1.
• The exponent of higher integrability for σ-harmonic functions is shown to be algebraically optimal and given by p < 2K/(K−1), where K = √(β/α) + √((β−α)/α).
• The optimality of this exponent in the non-symmetric case remains an open problem, as extremal matrices cannot be symmetric a.e.