Quasiharmonic fields and Beltrami operators. (English) Zbl 1069.35024

A quasiharmonic field is a pair \([B,E]\) of vector fields on a domain \(\Omega \subset \mathbb R^n\) satisfying \(\text{div}\,B=0\), \(\text{curl}\,E=0\) and \(| B(x)| ^2 +| E(x)| ^2 \leq 2 c(x) \langle B(x),E(x) \rangle \) for a.a.\(x\in \Omega \), for some measurable function \(c(x)\geq 1\) on \(\Omega \). T. Iwaniec and C. Sbordone [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18, 519–572 (2001; Zbl 1068.30011)]. The author gives a simple construction of a field \(A\) of symmetric positive-definite matrices such that \(A(x)E(x)=B(x)\) for a.a.\(x\in \Omega \). An optimality property of the field \(A\) is also established, and applications to the study of regularity of solutions to elliptic PDE’s and to some questions of \(G\)-convergence of S. Spagnolo [Ann. Sc. Norm. Super. Pisa 22, 571–597 (1968; Zbl 0174.42101)] are given.


35J60 Nonlinear elliptic equations
47F05 General theory of partial differential operators
35D10 Regularity of generalized solutions of PDE (MSC2000)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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