On the \(\Gamma\)-convergence of Laplace-Beltrami operators in the plane. (English) Zbl 0955.30016

A mapping \(f:\Omega\to\mathbb{R}^2\) from a plane domain has finite distortion if \(f\in W^{1,2}(\Omega)\) and \(|f'(x)|^2\leq K(x) J(x,f)\) a.e. where \(K(x)\leq K<\infty\) a.e., then \(f\) is called quasiregular. It is shown that if \(f_i\) is a sequence of mappings with finite distortion \(K_i\) and the \(K_i\)’s are bounded in a special exponential norm and if \(f_i\to f\) weakly in \(W^{1,2}(\Omega)\), then the matrices \(A(x,f_i)= G(x,f_i)^{-1}\) \(\Gamma\)-converge to the matrix \(A(x,f)= G(x,f)^{-1}\). Here \(G(x,f)= Df(x)^* Df(x)/J(x,f)\), \(J(x,f)\neq 0\), refers to the dilatation tensor of \(f\). The \(\Gamma\)-converge, in the sense of De Giorgi, is also specially adapted to the situation. The use of \(\Gamma\)-convergence for the Bers-Bojarskii-Strebel convergence theorem is due to S. Spagnolo [S. Spagnolo, Sympos. Math. 18, 391-398 (1976; Zbl 0332.46020)] and the result of the author extends this theorem. Recently [V. Gutlyanskij, O. Martio, V. I. Ryazanov and M. Vuorinen, Forum. Math. 10, No. 3, 353-375 (1998; Zbl 0907.30024)] this convergence theorem has been extended to space quasiregular mappings.


30C62 Quasiconformal mappings in the complex plane
35B40 Asymptotic behavior of solutions to PDEs
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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