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Rendiconti Lincei - Matematica e Applicazioni

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Approximating the inverse matrix of the G-limit through changes of variables in the plane

Gioconda Moscariello (1), Carlo Sbordone (2) and François Murat (3)

(1) Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo 4, 80126, NAPOLI, ITALY
(2) Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo 4, 80126, NAPOLI, ITALY
(3) Laboratoire J.L. Lions, B.C. 187, Université Pierre et Marie Curie, 4, Place Jussieu, 75013, PARIS, FRANCE

Let $A_j$ be a sequence of coercive symmetric matrices of $L^\infty(\mathbb{R}^2)^{2\times 2}$ with $det \, A_j=1$ which $G$-converges to $A$. We prove that there exists a sequence of $K$-quasiconformal mappings $F_j$ which converge locally uniformly to a $K$-quasiconformal mapping $F$ such that $A_j^{-1}\circ F_j^{-1}$ $G$-converges to $A^{-1}\circ F^{-1}$. The result is specific to the two dimensional case but a similar result holds in dimension $1$.

Keywords: G-convergence, quasiconformal mappings, Beltrami operators, elliptic equations