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Estimation of the best constant involving the \(L^2\) norm of the higher-order Wente problem. (English) Zbl 1095.53007

The Wente problem arose in the study of surfaces with constant mean curvature. In higher dimensions this is the problem of regularity of a solution \(\varphi\) of the equation \((-\Delta)^p\varphi= \det\nabla u\) in \(\mathbb{R}^{2p}\) with \(\lim_{|x|\to+\infty}\varphi(x)=0\) and \(u\in W^{1,2p}(\mathbb{R}^{2p},\mathbb{R}^{2p})\). The first author proved in 2001 that the solution \(\varphi\) is in \(L^\infty (\mathbb{R}^{2p})\) and \(\overline\Delta^{k/2}\varphi\) is in \(L^{2p/k}(\mathbb{R}^{2p})\) for \(1\leq k\leq p\), with the following estimates: \(\|\varphi\|_\infty+\|\overline \Delta^{k/2}\varphi\|_{2p/k}\leq C\|\Delta u\|^{2p}_{2p}\), where \(\|\overline \Delta^{k/2}\varphi\|_{2p/k}\) is \(\|\Delta^{k/2}\varphi \|_{2p/k}\) if \(k\) is even, and is \(\|\nabla(\Delta^{(k-1)/2})\varphi \|_{2p/k}\) if \(k\) is odd. Now it is proved that the best constant \(C_p\) is achieved by a one parameter family of functions \(\overline\varphi\) and \(\overline u\) given by \[ \overline\varphi(x) =\frac{2}{(2p)!(1+cr^2)}, \quad\overline u=\frac{2\sqrt cx}{1+cr^2}, \] where \(c>0\) is some arbitrary positive constant. In the case \(p=2\) a more explicit expression of the best constant is given.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C40 Global submanifolds
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