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On the convergence of Laplace-Beltrami operators associated to quasiregular mappings. (English) Zbl 0646.30024
Let \(\Omega_ 0\) be a domain in \({\mathbb{R}}^ n.\) A mapping \(f\in W^{1,n}_{loc}(\Omega_ 0)\) is K-quasiregular in \(\Omega_ 0\) if \(| D(f,x)| \leq Kn^{n/2}J(f,x)\) a.e. \(x\in \Omega_ 0\). Define the matrix \[ a(f,x)=J(f,x)[D(f,x)\quad t D(f,x)]^{-1}. \] The operator \(\Delta_ fu=-div(a(f,x)\nabla u)\) is the associated “Laplace- Beltrami” operator to f. (When f is smooth and 1-1, i.e., quasiconformal \(\Delta_ f\) is indeed the Laplace-Beltrami operator in the Riemannian manifold \((f(\Omega_ 0),g)\) where g is the pull-back of the Euclidean metric in \(\Omega_ 0\) by \(f^{-1})\). Given a sequence of K- quasiregular mappings in \(\Omega_ 0\), \(f_ h\) converging in \(L^ 1_{loc}(\Omega_ 0)\) to a mapping f and bounded in \(W^{1,n}_{loc}(\Omega_ 0)\), Yu. G. Reshetnyak [Sib. Mat. Zh. 8, 629-658 (1967; Zbl 0162.381) and ibid. 9, 667-684 (1968; Zbl 0162.383)] had shown that f is still K-quasiregular. Assume that f is nonconstant (so \(J(x,f)>0\) a.e. \(x\in \Omega_ 0)\), the authors study in this paper whether and in what sense \(\Delta_{f_ n}\to \Delta_ f\). Their basic hypothesis is the following:
There exists a nonnegative function w in \(\Omega_ 0\) such that \(w,w^{-1}\in L^ 1_{loc}(\Omega_ 0)\) and \[ (1)\quad 0\leq w(x)\leq J(x,f_ h)^{1-2/n}\leq \Lambda w(x)\text{ a.e. in }\Omega_ 0, \] where \(\Lambda\geq 1\) and \(h\in {\mathbb{N}}\). In particular f satisfies (1), too. Let \(\Omega \subset \subset \Omega_ 0\), consider the quadratic forms \[ (2)\quad F_ h(u,\Omega)=\int_{\Omega}<a(f_ h,x)\nabla u,\nabla u>dx \] and \[ (3)\quad F(u,\Omega)=\int_{\Omega}<a(f,x)\nabla u,\nabla u>dx. \] To solve \(\Delta_{f_ h}u=g\) we have to minimize \(F_ h(v,\Omega)+<g,v>\) on a suitable Sobolev space X, \(g\in X^*\). Define \(X=W^ 1_ 0(\Omega,w)\) as the completion of \(C^{\infty}_ 0(\Omega)\) under the inner product \[ <\phi,\psi >=\int_{\Omega}<\nabla \phi,\nabla \psi >w dx+\int_{\Omega}\phi \psi w^{n/(n-2)} dx. \] The dual of \(W^ 1_ 0(\Omega,w)\) as a space of distributions \(X^*\) is denoted by \(W^{-1}(\Omega,w)\). The main theorem states that the solutions to the problem \[ (4)\quad \min \{F_ h(v,\Omega)+<g,v>: v\in W^ 1_ 0(\Omega,w)\} \] converge in \(L^ 1(\Omega)\) to the solution of (4) with \(F_ h\) replaced by F. To this reviewer the proof given in the paper is complete only when all the \(f_ h\) are quasiconformal mappings. The technique used is \(\Gamma\)-convergence of functionals. The authors claim that \[ (5)\quad F=\Gamma^-(L^ 1(\Omega))\lim_{n\to \infty}F_ h. \] To prove it in the quasiregular case they use a lemma (Lemma 3.3) and suggest its proof which is very similar to Proposition 3.2 in [P. Marcellini and C. Sbordone, J. Math. Pures Appl., IX. Ser. 56, 157-182 (1977; Zbl 0348.35014)], which uses higher integrability for w and 1/w than assumed in (1). Once (5) is established, the main theorem follows from the general theory of \(\Gamma\)-convergence. Another point that needs further clarification is whether \(W^ 1_ 0(\Omega,w)\) is compactly embedded in \(L^ 1(\Omega)\) in the quasiregular case, asserted by the authors but not proved.
Reviewer: J.J.Manfredi

MSC:
30C62 Quasiconformal mappings in the complex plane
35J70 Degenerate elliptic equations
49J99 Existence theories in calculus of variations and optimal control
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