Isotropic p -harmonic systems in 2D Jacobian estimates and univalent solutions

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3 Scopus citations

Abstract

The core result of this paper is an inequality (rather tricky) for the Jacobian determinant of solutions of nonlinear elliptic systems in the plane. The model case is the isotropic (rotationally invariant) p -harmonic system div |Dh|p-2Dh = 0, h= (u, v) ε W1,p(ω,ℝ2) , 1 <p <∞, as opposed to a pair of scalar p -harmonic equations: (equations presented) Rotational invariance of the systems in question makes them meaningful, both physically and geometrically. An issue is to overcome the nonlinear coupling between δu and δv . In the extensive literature dealing with coupled systems various differential expressions of the form φ(δuv) were subjected to thorough analysis. But the Jacobian determinant detDh = uxvy .uyvx was never successfully incorporated into such analysis. We present here new nonlinear differential expressions of the form φ(|Dh|, detDh) and show they are superharmonic, which yields much needed lower bounds for detDh. To illustrate the utility of such bounds we extend the celebrated univalence theorem of Radó-Kneser-Choquet on harmonic mappings ( p = 2 ) to the solutions of the coupled p -harmonic system.

Original languageEnglish (US)
Pages (from-to)57-77
Number of pages21
JournalRevista Matematica Iberoamericana
Volume32
Issue number1
DOIs
StatePublished - 2016

Keywords

  • Energy-minimal deformations
  • Jacobian determinants
  • Nonlinear systems of PDEs
  • P-harmonic mappings
  • Variational integrals

ASJC Scopus subject areas

  • Mathematics(all)

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