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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
##### Keywords:
quasiregular; Laplace-Beltrami
##### Citations:
Zbl 0319.35013; Zbl 0162.381; Zbl 0162.383; Zbl 0348.35014
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