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On pseudospheres. (English) Zbl 0734.31006

Let \(H^{n-1}(E)\) denote the (n-1)-dimensional Hausdorff measure of a subset E of \({\mathbb{R}}^ n\). A bounded domain D in \({\mathbb{R}}^ n\) (n\(\geq 2)\) with \(0\in D\) and \(H^{n-1}(\partial D)<+\infty\) is called a pseudosphere if \(\partial D\) is homeomorphic to the unit sphere S in \({\mathbb{R}}^ n\) and there is a constant a such that \(g(0)=a\int_{\partial D}g dH^{n-1},\) whenever g is harmonic on D and continuous on \(\bar D.\) It has long been known that in \({\mathbb{R}}^ 2\) there exist pseudospheres which are not circles, but the constructions of such plane pseudospheres do not generalize readily to higher dimensions. Answering a question of H. S. Shapiro, the authors prove that there exist pseudospheres in \({\mathbb{R}}^ n\) (n\(\geq 3)\) which are not spheres.
Indeed, the main theorem asserts more: there exists a pseudosphere D in \({\mathbb{R}}^ n\) (n\(\geq 3)\) such that D is not a sphere and such that there is a homeomorphism f: \({\mathbb{R}}^ n\to {\mathbb{R}}^ n\) with \(f(S)=\partial D\) and \[ c^{-1}\| x-y\|^{1/\beta}\leq \| f(x)-f(y)\| \leq c\| x-y\|^{\beta} \] whenever \(0<\beta <1\) and \(\| x-y\| \leq 1/2\); here c is a positive constant depending only on n and \(\beta\).

MSC:

31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
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